Numerical Turing Patterns Formation on 3D Surfaces with a Linear Finite Element Method

1 Introduction

Reaction-diffusion systems are mathematical models that allow us to describe how the concentration of one or more substances distributed in a medium changes under the influence of two physical processes: a chemical reaction, which transform the substances into each other, and diffusion, which causes the substances to spread in the medium. Applications for these systems have been found in a wide range of scientific and technological areas, from biology to physics, as well as chemistry, and even in the social sciences.

A fascinating aspect of a great variety of reaction-diffusion systems is their ability to generate spatio-temporal patterns from the solution of their governing equations. These patterns, such as those studied by Turing in 1952 (see [1]), not only reveal ordered structures in biological systems, such as the models of morphogenesis that explain how complex patterns can form on an animal skin, like spots on a leopard or stripes on a zebra, but can also be found in more abstract contexts, such as in the distribution of resources in complex networks as well as in the organization of neural patterns in the brain.

The adaptability and versatility of reaction-diffusion systems have allowed these models to be applied to more complex problems that require an analysis similar to that used in the study of Turing patterns. For example, they have been used to model tumour growth, cell organization in tissues, and ecosystem dynamics. The ability of these systems to capture both the complexity of local dynamics and the global interaction within a system makes them valuable tools in scientific research.

In this work we explore the numerical generation of Turing patterns on three dimensional surfaces using as a guideline the well known conditions that arise form the analysis of reaction-diffusion systems on planar regions. Several numerical schemes have been applied to solve numerically the reaction–diffusion system on surfaces. For instance, the finite element method is employed in [2], while a lifted local Galerkin method on implicit surfaces is proposed in [3], a finite difference method has been introduced in [4], and mesh-free kernel method like the ration basis function (RBF) is proposed in [5]. In this work we aim to introduce with some detail a simple but effective scheme based on a linear finite element method for space dicretization combined with a semi-implicit (some time named IMPEX) Euler time dicretization. It is applied directly on the discrete surface in Cartesian coordinates and yield accurate results in an efficient way, as shown with some numerical examples.

2 Mechanism for Turing pattern formation on planar regions

The model proposed by Alan Turing to describe patterns in nature was a model based on reaction-diffusion systems. Turing's main idea lies in the following sentence:

"A system in a stable state can be guided to an unstable state through diffusion, and in that state, the patterns will appear.".

The general form of reaction-diffusion systems that allow generating spatial patterns through their temporal solutions is the following:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\partial{u}}{\partial{t}} = D_{u}\nabla ^{2}u + \gamma \,f(u,v), (1) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\partial{v} }{\partial{t}} = D_{v}\nabla ^{2}v + \gamma \,g(u,v), (2) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): (\hat{\mathbf{n}}\cdot \nabla )u = (\hat{\mathbf{n}}\cdot \nabla )v = 0, \quad \forall \,\mathbf{x}\in \partial{\Omega }, (3) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u(\mathbf{x},0) = u_{0}(\mathbf{x}),\,\, u(\mathbf{x},0) = v_{0}(\mathbf{x}), \quad \forall \,\mathbf{x}\in \Omega , (4)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \nabla ^2}

is the Laplace operator and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma{>0}}
is known as scale factor that determines the size and the complexity of the patterns generated by the Turing conditions, as it will be shown later. Functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u(\mathbf{x},t)}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle v(\mathbf{x},t)}

represent the concentrations of two substances mixing in a medium Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega \subset \mathbb{R}^2}
at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t}
whose initial concentrations are given by (4), respectively. The diffusion coefficients Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle D_{u},D_{v}>0}
are assumed constant, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{\mathbf{n}}}
denotes the outer unit normal vector to the boundary Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \partial \Omega }
of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega }

. Zero flow conditions (3) are introduced for open surfaces (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \partial \Omega \neq \emptyset } ) to ensure pattern formation due to self-organization only, avoiding the influence of external forces.

2.1 Turing conditions for pattern formation

Turing conditions are those that ensure the generation of spatial patterns from the temporal solutions of the reaction-diffusion systems (1-4), and they can be derived from Turing's idea already expressed before. First, under the absence of diffusion, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle D_u = D_v = 0} , we assume the existence of an equilibrium point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{w}^\ast = (u^\ast , v^\ast )^T}

and linearize the system (1-2) around Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{w} = (u,v)^T}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dot{\mathbf{w}} \approx \gamma \,\mathbb{J}\, (\mathbf{w}-\mathbf{w}^{\ast })

(5)

with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbb{J}= \begin{pmatrix} f_{u}(\mathbf{w}^\ast ) & f_{v}(\mathbf{w}^\ast ) \\ g_{u}(\mathbf{w}^\ast ) & g_{v}(\mathbf{w}^\ast ) \end{pmatrix}.

By setting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{z} = \mathbf{w}-\mathbf{w}^\ast }

and assuming equality in (5) we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \dot{\mathbf{z}} = \gamma \,\mathbb{J}\, \mathbf{z}}

, and then proposing a non trivial solution of (1-4), say Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{z} = \mathbf{K}e^{\lambda t}} , with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{K}}

eigenvector of the Jacobian matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{J}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \lambda }
it's associated eigenvalue, we get 

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \lambda = \displaystyle \frac{\gamma }{2}\left[\hbox{Tr}(\mathbb{J})\,\pm \,\sqrt{\left[\hbox{Tr}(\mathbb{J})\right]^{2}-4\,\hbox{det}(\mathbb{J})}\,\right].

(6)

Since we look for a stable point, then from the Hartman-Grobman theorem ([6], p. 120), ([7], p. 155) we must assume that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{w}^\ast }

is a hyperbolic equilibrium point, that is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Re(\lambda )\not=0}

, and to ensure such stability it is necessary that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \hbox{Tr}(\mathbb{J}) < 0, (7) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \hbox{ det}(\mathbb{J}) > 0. (8)

Thus, obtaining the first conditions. Next, as a second step, diffusion is reintroduced in (1-2), that is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle D_u, D_v >0} , and assuming that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{w}^\ast = (u^\ast , v^\ast )^T}

is still an equilibrium point, then it must satisfy Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f(\mathbf{w}^\ast ) = g(\mathbf{w}^\ast ) = 0}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \nabla ^2u^\ast = \nabla ^2v^\ast = 0}

. But now, we want that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{w}^\ast }

be an unstable equilibrium point (diffusion destabilize the system). To derive the mathematical condition for this new situation, the following linear system with diffusion is analysed. 

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dot{\mathbf{z}} = \mathbb{D}\nabla ^2\mathbf{z} + \gamma \,\mathbb{J}\, \mathbf{z},

(9)

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \nabla ^2\mathbf{z} = \nabla ^2(\mathbf{w}-\mathbf{w}^\ast )}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{D}= \begin{pmatrix} D_u & 0 \\ 0 & D_v \end{pmatrix}}

. This time we consider a separated solution

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): z(\mathbf{x},t) = \mathbf{Z}(\mathbf{x}) \, T(t),

(10)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{Z}(\mathbf{x}) = \left(Z_1(\mathbf{x}), Z_2(\mathbf{x})\right)^T}

is a vectorial function in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{R}^2}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T(t)}
is a scalar function. From (9) we obtain 

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): T(t) = Ce^{\lambda{t}}, \quad C\in \mathbb{R},

(11)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\nabla }^{2}\mathbf{Z} + {\mathbb{D}}^{-1}(\gamma \,\mathbb{J}-\lambda \,{\mathbb{I}}_2)\mathbf{Z} = \mathbf{0}.

(12)

It is worth mentioning that the constant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \lambda }

in (11) is not the same as the one in (6), but it plays an equivalent role, as will be see below. We denote by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k^{2}}
the eigenvalues1 of matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{A} = \mathbb{D}^{-1}(\gamma \,\mathbb{J}-\lambda \,\mathbb{I}_2)}

. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Z_k(\mathbf{x})}

is the associated eigenfunction, then from (12) it must solve the following Helmholtz equation with zero-flux at the boundary

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\nabla }^{2}\mathbf{Z}_k + k^{2}\mathbf{Z}_k = \mathbf{0}, (13) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): (\hat{\mathbf{n}}\cdot \nabla )\mathbf{Z}_k = \mathbf{0}. (14)

Solutions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{Z}_k}

are commonly oscillatory and depend exclusively on system's geometry. For bounded domains, values Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}
are discrete and can be enumerated, they represent the oscillatory mode of the solution and are called the wave numbers. Therefore, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{z}_k (\mathbf{x}, t) = e^{\lambda _k t} \mathbf{Z}_k(\mathbf{x})}
is a solution of (9) with zero flux on the boundary, and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \lambda _{k} = \displaystyle \frac{\left[\gamma \hbox{Tr}(\mathbb{J}) - k^{2}\hbox{Tr}(\mathbb{D})\right]\,\pm \,\sqrt{\left[k^{2}\hbox{Tr}(\mathbb{D}) - \gamma \hbox{Tr}(\mathbb{J})\right]^{2}-4h(k^{2})}}{2},

(15)

with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): h(k^2)=\hbox{det}(\mathbb{D})k^{4} - \gamma (D_{v}f_{u}+D_{u}g_{v})k^{2} + \gamma ^{2}\hbox{det}(\mathbb{J}).

(16)

The roots of the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h(k^2)}

determine an interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (k_1^2, k_2^2)}

, and any wave number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k^2}

inside this interval will allow the generation of spatial patterns in the medium Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega }

. Those roots are positive and are given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): k_{1}^{2},k_{2}^{2} = \displaystyle \frac{\gamma }{2\,D_v}\left[(d\,f_{u} + g_{v}) \,\mp \,\sqrt{(d\,f_{u}+g_{v})^{2}-4\,d\, \det (\mathbb{J})}\,\right].

(17)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle d = D_v / D_u >0}

is the diffusivity ratio. This means that only the modes with wave number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}
between these two values are able to produce spatial patterns, all other modes, out this range, have an exponential decay in time and eventually vanish. Thus, the equilibrium point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{w}^\ast }
is unstable if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Re[\lambda _k]>0}
for some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k^2 \in (k_1^2, k_2^2)}

. This occurs when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h(k^2)<0} , and from (16) and (17) the following inequalities must be satisfied

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): d\,f_{u} + g_{v} > 0, (18) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): (d\, f_{u} + g_{v})^{2} - 4d\,\det (\mathbb{J}) > 0, (19) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): k_{1}^{2} < k^{2} <k_{2}^{2}. (20)

Combining these conditions with (7-8), we obtain the so called Turing conditions for pattern generation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): f_{u}+g_{v} < 0, (21) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): f_{u} \, g_{v}-f_{v} \, g_{u} >0, (22) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): d\,f_{u} + g_{v} > 0, (23) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): (d\, f_{u} + g_{v})^{2} - 4d\,\det (\mathbb{J}) > 0, (24) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): k_{1}^{2} < k^{2} <k_{2}^{2}. (25)

Furthermore, from these conditions and some simple algebraic calculations, it can be shown that the following must hold: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle d > 1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f_u > 0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle g_v <0}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f_vg_u < 0}

.

(¹) The squared exponent is for algebraic convenience when performing calculations.

3 Finite element approximation of the Laplace-Beltrami operator

In this part, we will focus on the numerical approximation of the following elliptic equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \alpha \, \phi - D_{\Sigma } \, \nabla _{\Sigma }^{2} \, \phi = f,

(26)

with homogeneous Neumann boundary conditions when the bounded surface Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Sigma }

is open. Of course if the surface is closed (like an sphere), there are no boundary conditions. In this equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha }
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle D_{\Sigma }}
are positive constants, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \nabla _{\Sigma }^{2}}
denotes the so called Laplace-Beltrami operator, defined on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Sigma }

, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f}

is a source term. This numerical approximation will be performed by a linear finite element method. This methodology will be the basis, with some minimal adjustments, to numerically solve the reaction-diffusion systems for the generation of Turing patterns on those surfaces, as shown in next sections.

3.1 The Laplace-Beltrami operator (LBO)

The Laplace-Beltrami operator is a generalization of the usual Laplace operator. This operator acts on functions defined on surfaces in the Euclidean space and, more generally, on Riemannian surfaces and manifolds. We will use it for functions defined on three-dimensional surfaces in the Euclidean space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{R}^3} .

Definición 1: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Sigma \subset \mathbb{R}^3}

be a smooth oriented surface, and let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{\mathbf{n}} = \hat{\mathbf{n}}(\mathbf{x})}
be a unit normal vector at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{x}}

, exterior to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Sigma } , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi \,:\,\Sigma \subset R^3\longmapsto \,\mathbb{R}}

a function defined on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Sigma }
of class Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{C}^1}

. The tangential gradient of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi }

at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{x} \in \Sigma }
is defined by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \nabla _{\Sigma }\phi (\mathbf{x}) = \nabla{\phi (\mathbf{x})} - \langle{\nabla{\phi (\mathbf{x})},\hat{\mathbf{n}}}\rangle \,\hat{\mathbf{n}}

(27)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \langle{\cdot ,\cdot }\rangle }

denotes the inner product in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{R}^3}

.

Geometrically, the tangential gradient is the orthogonal projection of the gradient vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \nabla{\phi (\mathbf{x})}}

onto the tangent plane at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{x}}
with normal vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{\mathbf{n}}(\mathbf{x})}
as shown in figure 1. The projection matrix to the normal vector is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{P} = \hat{\mathbf{n}}\,\hat{\mathbf{n}}^{T}}

, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \nabla _{\Sigma }\phi (\mathbf{x})}

can also be expressed as:   

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \nabla _{\Sigma }\phi (\mathbf{x}) = (\mathbb{I}- \mathbb{P})\nabla{\phi (\mathbf{x})},

(28)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{I}}

represents the identity matrix of size Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 3\times{3}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{I}- \mathbb{P}}
is the complementary orthogonal projection to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbb{P}

.

Figure 1: Tangential gradient Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \nabla _{\Sigma }\phi (\mathbf{x}) .

Definición 2: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi \,:\,\Sigma \subset{\mathbb{R}}^{3}\longmapsto \,\mathbb{R}}

be of class Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{C}^{2}}

. We define the Laplace-Beltrami operator of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi }

denoted by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \nabla _{\Sigma }^{2}\phi }

, by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \nabla _{\Sigma }^{2}\phi = \hbox{div}(\nabla _{\Sigma }\phi ),

(29)

or equivalently, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \nabla _{\Sigma }^{2}\phi = \nabla _{\Sigma }\cdot \nabla _{\Sigma }\phi } . In many texts, this operator is denoted by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta _{\Sigma }\phi } .

3.2 Weak form of equation (26)

In order to numerically solve the elliptic equation (26) using a finite element method, we must find its weak or variational formulation. It is obtained by multiplying that equation by a test function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle v}

and integrating over the surface Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Sigma }

. After applying integration by parts we obtain the following problem:

Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi \in \mathcal{L}^{2}(\Sigma )}

such that, for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle v\in \mathcal{H}^{1}(\Sigma )}

, it satisfies

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \alpha \displaystyle \int _{\Sigma }\phi \,v\,d\Sigma + D_{\Sigma }\displaystyle \int _{\Sigma }{\nabla }_{\Sigma }\phi \cdot{\nabla }_{\Sigma }v\,d\Sigma = \displaystyle \int _{\Sigma }f\,v\,d\Sigma ,

(30)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathcal{H}^{1}(\Sigma ) = \left\{v\in \mathcal{L}^{2}(\Sigma )\,:\,\displaystyle \int _{\Sigma }|{\nabla }_{\Sigma }v|^{2}\,d\Sigma{<\infty}\right\}

and where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle d\Sigma }

represents the surface differential.

Note. The integral equation (30) does not include boundary integral terms, because either the surface is closed (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \partial \Sigma = \emptyset } ) or it is open with zero-flux boundary conditions. So, the above variational formulation is the same for both cases.

3.2.1 Approximation of the variational problem

To obtain a finite element approximation to the variational problem (30) we use a Ritz-Galerkin approach, where the Hilbert space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{H}^{1}(\Sigma )}

is approximated by a finite-dimensional subspace. One possible way is approximating each function in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{H}^{1}(\Sigma )}
by a continuous piecewise polynomial. One of the most simple approximations is obtained with continuous piecewise polynomials of degree one. So, if we denote by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{P}_1}
the set of polynomials in two variables of degree 1, and we approximate the surface Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Sigma }
by a triangular mesh Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{T}_{h}}

, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{H}^{1}(\Sigma )}

is approximated by the following finite-dimensional space of piecewise linear functions:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): H_{h} = \left\{v\in \mathcal{C}^{0}(\widehat{\Sigma })\,:\,v\,|_{_T}\in \mathbb{P}_{1},\,\,\forall \,T\in \mathcal{T}_{h}\right\},

(31)

We denote by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{V}_{h}}

the set of vertices of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{T}_{h}}
and for each vertex (node point) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P\in \mathcal{V}_{h}}
there is a `hat function', denoted by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \varphi _{P} \in H_{h}}

, that satisfies the following basic property for each vertex Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q\in \mathcal{V}_{h}}

(see Figure 2):

Figure 2: Pyramidal or 'hat' function.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \varphi _{P}(Q) = \begin{cases} 1 \quad \hbox{if} \quad Q = P, \\ 0 \quad \hbox{if} \quad Q \neq P. \end{cases}

The set of hat (pyramidal) functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta _h = \left\{\varphi _{P}\, |\, P\in \mathcal{V}_{h} \right\}}

form a basis of the finite element space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H_{h}}

. Furthermore, the dimension Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_I}

of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H_h}
is number of vertex nodes in the triangulation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{T}_{h}}

.

We can now discretize the variational problem (30) as follows: Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi _{h}\in H_{h}}

such that    

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \alpha \displaystyle \int _{\widehat \Sigma }\phi _{h}\,v_{h}\,d\Sigma + D_{\Sigma }\displaystyle \int _{\widehat \Sigma }{\nabla }_{\Sigma }\phi _{h}\cdot{\nabla }_{\Sigma }v_{h}\,d\Sigma = \displaystyle \int _{\widehat \Sigma }f\,v_{h}\,d\Sigma , \quad \quad \forall \,v_{h} \in H_{h}.

(32)

It is enough that (32) be satisfied for each basis function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \varphi _{P}} . Furthermore, by expressing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi _{h}\in {H_{h}}}

as linear combination of the basis functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi _{h}(\mathbf{x}) = \displaystyle \sum _{P\in \mathcal{V}_{h}}\phi _{h}(P) \, \varphi _{P}(\mathbf{x})}

, we get the following reformulation of (32):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \sum _{P\in \mathcal{V}_{h}} \left(\alpha \displaystyle \int _{\widehat \Sigma }\varphi _{P}\, \varphi _{Q}\,d\Sigma + D_{\Sigma } \int _{\widehat \Sigma } \nabla _{\Sigma }\varphi _{P} \cdot \nabla _{\Sigma }\varphi _{Q} \,d\Sigma \right)\, \phi _{h}(P) = \int _{\widehat \Sigma } f\,\varphi _{Q}\,d\Sigma , \quad \forall \, Q\in \mathcal{V}_{h},

(33)

which can be expressed in the following form (after enumerating the vertices in the mesh):

Find the values Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi _{i}=\phi _h(P_i)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i=1,2,\ldots ,N_{I}}

such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{align}\displaystyle \sum _{i=1}^{N_{I}}\left(\alpha \displaystyle \int _{\widehat \Sigma }\varphi _{i}\, \varphi _{j}\,d\Sigma + D_{\Sigma }\displaystyle \int _{\widehat \Sigma }{\nabla }_{\Sigma }\varphi _{i}\cdot{\nabla }_{\Sigma }\varphi _{j} \,d\Sigma \right)\, \phi _{i} = \displaystyle \int _{\widehat \Sigma }f\,\varphi _{j} \, d\Sigma , \quad \forall \, j=1,2,\ldots ,N_{I}. \end{align}

(34)

Written in matrix form, we obtain

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): (\alpha \,\mathbb{M}+ D_{\Sigma }\mathbb{K}) \, \boldsymbol{\phi } = \mathbf{F},

(35)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\phi }}

is the vector of unknowns Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \{ \phi _{i}\} }

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathbb{M}}=(M_{ij})} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{K}=(K_{ij}) }

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{F}=(F_{1},\ldots ,F_{N_{I}})^{T}}

, with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): M_{ij} = \displaystyle \int _{\widehat \Sigma }\varphi _{i}\, \varphi _{j}\,d\Sigma , (36) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): K_{ij} = \displaystyle \int _{\widehat \Sigma }{\nabla }_{\Sigma }\varphi _{i}\cdot{\nabla }_{\Sigma }\varphi _{j}\,d\Sigma , (37) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): F_{j} = \displaystyle \int _{\widehat \Sigma }f\,\varphi _{j}\,d\Sigma{.} (38)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{M}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{K}}
are called the mass matrix and stiffness matrix, respectively, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{F}}
is known as the load vector. Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \displaystyle \widehat{\Sigma } = \bigcup _{T\in \mathcal{T}_h} T}

, we obtain

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): M_{ij} = \displaystyle \sum _{T\in \mathcal{T}_{h}^{ij}}\int _{T}\varphi _{i}\, \varphi _{j}\,dT, (39) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): K_{ij} = \displaystyle \sum _{T\in \mathcal{T}_{h}^{ij}} \int _{T}{\nabla }_{_T}\varphi _{i}\cdot{\nabla }_{_T}\varphi _{j}\,dT (40)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{T}_h^{ij}}

represents the triangular elements containing the adjacent vertices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_{i}}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_{j}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\nabla }_{T}}
is the tangential gradient over the triangle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T}

.

3.2.2 Building local basis functions on each element

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ne}

be the number of triangular elements in the mesh. For each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle e=1,2,\ldots ,ne}

, the triangular element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T_{e}\in \mathcal{T}_{h}}

has vertices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_{1}^{e}}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_{2}^{e}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_{3}^{e}} , with coordinates Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (x_{1}^{e}, y_{1}^{e},z_{1}^{e})} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (x_{2}^{e},y_{2}^{e},z_{2}^{e})} ,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (x_{3}^{e},y_{3}^{e},z_{3}^{e})} , respectively. The connectivity matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \{ g(e,\lambda )\} } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 1\leq{e}\leq{ne}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 1\leq{\lambda }\leq{3}} , relates local node numbering to the global one. More precisely, the local vertices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \lambda=1,2,3}

of an element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T_{e}}
have global numbering Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i=g(e,1)}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle j=g (e,2)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k=g(e,3)} , respectively. So, a basis function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \varphi _{i}}

restricted to the element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T_{e}}
can be denoted by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\varphi }_{\lambda }^{e}(\mathbf{x}) = {\varphi }_{i}(\mathbf{x})|_{T_{e}}, \quad \hbox{with} \quad i=g(e,\lambda{).}

These local basis function thus have the following properties

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \varphi _{\lambda }^{e}(x,y,z)} is a polynomial of degree Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \le 1} on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T_{e}} .
2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\varphi }_{\lambda }^{e}(x_{\mu }^{e},y_{\mu }^{e},z_{\mu }^{e})=\delta _{\lambda{\mu }}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 1\leq{\lambda ,\mu }\leq{3}} .
3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \varphi _1^e (\mathbf{x}) \!+\! {\varphi }_{2}^{e}(\mathbf{x}) \!+\! {\varphi }_{3}^{e}(\mathbf{x}) \!=\! 1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \forall \mathbf{x}\!=\! (x,y,z) \in T_e} .

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{x}}

can be expressed as a weighted average of the vertices of the triangle with weights determined by the local basis functions. That is,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{x} = \varphi _{1}^{e}(\mathbf{x})\, P_{1}^{\,e} + \varphi _{2}^{e}(\mathbf{x})\, P_{2}^{\,e} + \varphi _{3}^{e}(\mathbf{x})\, P_{3}^{\,e}

(41)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_{1}^{\,e}=(x_{1}^{e},y_{1}^{e},z_{1}^{e})} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_{2}^{\,e}=(x_{2}^{e},y_{2}^{e},z_{2}^{e})}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_{3}^{\,e}=(x_{3}^{e},y_{3}^{e},z_{3}^{e})}
are the coordinates of the vertices of triangle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T_e}

.

3.2.3 Reference element and affine transformation

We consider a reference unit triangle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \widehat{T}}

in the coordinates Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \xi }

-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta }

and a triangular element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T_{e}\in \mathcal{T}_{h}}
in the Cartesian system Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x}

-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y} -Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z}

as in Figure 3.

Figure 3: Affine transformation from the reference element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \widehat{T} to the physical element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): T_{e} .

Assuming a counter-clockwise node numbering, the affine transformation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{r}_{e}\,:\,\widehat{T}\longmapsto \,T_{e}}

 is given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{r}_{e}(\xi ,\eta ) = \hat{\varphi }_{1}(\xi ,\eta )(x_{1}^{e},y_{1}^{e},z_{1}^{e})^{T}+\hat{\varphi }_{2}(\xi ,\eta )(x_{2}^{e},y_{2}^{e},z_{2}^{e})^{T}+\hat{\varphi }_{3}(\xi ,\eta )(x_{3}^{e},y_{3}^{e},z_{3}^{e})^{T},

(42)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \hat{\varphi }_{1}(\xi ,\eta ) = 1-\xi{-\eta}, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \hat{\varphi }_{2}(\xi ,\eta ) = \xi , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \hat{\varphi }_{3}(\xi ,\eta ) = \eta{.}

The Jacobian matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{J}_e}

of the affine transformation (42) is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbb{J}_e = \left[\begin{array}{cc}x_{2}^{e}-x_{1}^{e} & x_{3}^{e}-x_{1}^{e} \\ y_{2}^{e}-y_{1}^{e} & y_{3}^{e}-y_{1}^{e} \\ z_{2}^{e}-z_{1}^{e} & z_{3}^{e}-z_{1}^{e} \end{array} \right].

(43)

The column vectors of this matrix are the fundamental vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{r}_{\xi }^{e}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{r}_{\eta }^{e}} . This vectors, along the normal vector to the triangular element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T_e}

are:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{r}_{\xi }^{e} = \begin{pmatrix}x_{2}^{e}-x_{1}^{e} \\ y_{2}^{e}-y_{1}^{e} \\ z_{2}^{e}-z_{1}^{e} \end{pmatrix}, \qquad \mathbf{r}_{\eta }^{e} = \begin{pmatrix}x_{3}^{e}-x_{1}^{e} \\ y_{3}^{e}-y_{1}^{e} \\ z_{3}^{e}-z_{1}^{e} \end{pmatrix}, \qquad \hat{\mathbf{n}}^e = \mathbf{r}_{\xi }^{e}\times \mathbf{r}_{\eta }^{e}

(44)

3.2.4 Assembly of mass and stiffness matrices using the reference element

Assembly of the mass matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbb{M} . From equation (39), and the connectivity matrix relating local to global nodes, we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): M_{ij} = \sum _{T_e\in \mathcal{T}_{h}^{ij}} \int _{T_e}\varphi _\lambda ^e \, \varphi _{\mu }^e \,dT_e,

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i=g(e,\lambda )}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle j=g(e,\mu )}
adjacent nodes. These integrals are computed in the reference element employing the affine transformation (42), so

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): M_{ij} = \displaystyle \sum _{T_{e}\in \mathcal{T}_{h}^{ij}}\iint _{\widehat{T}}\hat{\varphi }_{\lambda } \, \hat{\varphi }_{\mu }\, ||\mathbf{r}_{\xi }^{e}\times \mathbf{r}_{\eta }^{e}||\,d\xi \,d\eta = \displaystyle \sum _{T_{e}\in \mathcal{T}_{h}^{ij}} ||\mathbf{r}_{\xi }^{e}\times \mathbf{r}_{\eta }^{e}|| \, \widehat{M}_{\lambda \mu },

(45)

where, for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 1 \le \lambda , \mu \le 3}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \widehat{M}_{\lambda \mu }=\displaystyle \iint _{\widehat{T}}\hat{\varphi }_{\lambda }(\xi ,\eta )\hat{\varphi }_{\mu }(\xi ,\eta )\,d\xi \,d\eta ,

(46)

are the coefficients of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 3\times 3}

local mass matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \widehat{M}}

. Double integral notation has been used to emphasize that they are surface integrals. Coefficients Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \widehat{M}_{\lambda \mu }}

may be computed exactly or approximated by Simpson's rule or a Gaussian quadrature only once and stored in memory.

Assembly of the stiffness matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbb{K} . From (40), we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): K_{ij} = \sum _{T_e\in \mathcal{T}_{h}^{ij}} \int _{T_{e}}\nabla _{T_{e}}\varphi _{\lambda }^{e} \cdot \nabla _{T_{e}}\varphi _{\mu }^{e} \,dT_e,

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i=g(e,\lambda )}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle j=g(e,\mu )}
as before.  Here Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \nabla _{T_{e}}\varphi _{\lambda }^{e}=(\mathbb{I}-\mathbb{P}_{e})\nabla \varphi _{\lambda }^{e}}

, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{P}_{e} = \widehat{\mathbf{n}}^e (\widehat{\mathbf{n}}^e)^T}

is the projection matrix over the normal vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \widehat{\mathbf{n}}^e}
to the triangular element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T_{e}}

. Then, we can write

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): K_{ij} = \!\!\! \sum _{T_e\in \mathcal{T}_{h}^{ij}} \! \int _{T_{e}} \!\! (\mathbb{I}-\mathbb{P}_{e}) \nabla \varphi _{\lambda }^{e} \cdot (\mathbb{I}-\mathbb{P}_{e})\nabla \varphi _{\mu }^{e} \,dT_e.

(47)

Again, these integrals are calculated in the reference element using the affine transformation. Observing that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \nabla \hat{\varphi }_{\lambda } = \mathbb{J}_e^T \, \nabla \varphi _{\lambda }^ {e}} , and that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \nabla \varphi _\lambda ^e}

is a linear combination of the local fundamental vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{r}_\xi ^e}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{r}_\eta ^e} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{\mathbf{n}}^e} , we obtain

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): (\mathbb{I}-\mathbb{P}_{e})\nabla \varphi _{\lambda }^{e} = \mathbb{J}_e \, (\mathbb{J}_e^T \mathbb{J}_e)^{-1} \nabla \hat{\varphi }_{\lambda }, \lambda = 1,2,3.

(48)

Therefore,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): K_{ij} = \sum _{T_e\in \mathcal{T}_{h}^{ij}} \iint _{\hat{T}} (\nabla \hat{\varphi }_\lambda )^T (\mathbb{J}_e^T \mathbb{J}_e)^{-1} \nabla \hat{\varphi }_\mu \, ||\mathbf{r}_\xi ^e \times \mathbf{r}_\mu ^e || \, d\xi d\eta = \sum _{T_e\in \mathcal{T}_{h}^{ij}} ||\mathbf{r}_\xi ^e \times \mathbf{r}_\mu ^e || \, \widehat{K}_{\lambda \mu }^e,

(49)

where, for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 1 \le \lambda , \mu \le 3} ,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \widehat{K}_{\lambda \mu }^e = \displaystyle \frac{1}{2}(\nabla \hat{\varphi }_\lambda )^T (\mathbb{J}_e^T \mathbb{J}_e)^{-1} \nabla \hat{\varphi }_\mu ,

(50)

and the gradients are the constant column vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \nabla \hat{\varphi }_1 (\xi ,\eta ) = (-1,-1)^T} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \nabla \hat{\varphi }_2 (\xi ,\eta ) = (1,0)^T}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \nabla \hat{\varphi }_3 (\xi ,\eta ) = (0,1)^T}

.

In summary, the coefficients of the matrix

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbb{A} = \alpha \, \mathbb{M} + D_{\Sigma } \, \mathbb{K},

associated to the finite element approximation of the elliptic equation (26) are computed by means of the formula:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): a_{ij} = \sum _{T_e \in \mathcal{T}_h^{ij}} || \mathbf{r}_\xi ^e \times \mathbf{r}_\eta ^e || \, \left(\alpha \, \widehat{M}_{\lambda \mu } + D_\Sigma \, \widehat{K}_{\lambda \mu }^e \right),

(51)

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i = g(e,\lambda )}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle j = g(e,\mu )}

.

3.2.5 Calculation of the coefficients F_j of the load vector F

Using again the relationship between the global index and the local index we can approximate the load vector (38). First, the source term Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f(x,y,z)}

is approximated on each triangular element as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): f(x,y,z) \approx \displaystyle \sum _{\lambda=1}^{3}f\left(x_{\lambda }^{e},y_{\lambda }^{e},z_{\lambda }^{e}\right)\varphi _{\lambda }^{e}(x,y,z),

then, we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): F_j = \displaystyle \sum _{T_{e}\in \mathcal{T}_{h}^{ij}}||\mathbf{r}_{\xi }^{e}\times \mathbf{r}_{\eta }^{e}||\displaystyle \sum _{\lambda=1}^{3}f\left(x_{\lambda }^{e},y_{\lambda }^{e},z_{\lambda }^{e}\right)\widehat{M}_{\lambda \mu },

(52)

assuming that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle j=g(e,\mu )}

in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T_e}

, so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i=g(e,\lambda )}

run over the adjacent nodes.

4 Full discretization of the reaction diffusion model for 3–D surfaces

Coming back to the generation of Turing patterns on three-dimensional surfaces, we must solve the following reaction-diffusion system on a given surface Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Sigma } .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \displaystyle \frac{\partial{u}}{\partial{t}} = D_{u}\nabla _{\Sigma }^{2} u + \gamma \,f(u,v) \hbox{ in}\quad \Sigma , (53) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \displaystyle \frac{\partial{v}}{\partial{t}} = D_{v}\nabla _{\Sigma }^{2} v + \gamma \,g(u,v) \hbox{ in}\quad \Sigma , (54) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): (\hat{\mathbf{n}}\cdot \nabla )u = (\hat{\mathbf{n}}\cdot \nabla ) v = 0, \hbox{ on}\quad \partial{\Sigma }, (55) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u(\mathbf{x},0) = u_0(\mathbf{x}),\,\, v(\mathbf{x},0) = v_0(\mathbf{x}), \quad \forall \,\mathbf{x}\in \Sigma{.} (56)

A similar analysis to the one done in Section (3.1) may be performed for this system. However, the analysis is not as simple as before, because surface geometry and topology (curvature, for instance) play an important role in pattern formation and in pattern dynamics. Nowadays there is a limited understanding about this phenomena [8]. In fact, a static pattern on a flat plane can become a propagating pattern on a curved surface [9]. Also, it is well-known that geometric aspects influence the transport properties of particles that diffuse on curved surfaces, as shown in [10] and some reference therein. For these reasons, we will take advantage of what has been done for planar regions and use it as a guideline to generate Turing patterns on curved surfaces. Therefore, we will consider the Turing conditions (21-25) for planar regions and, with some slight variation of the parameter values, depending of the particular surface, we will generate some Turing patterns by solving numerically the above reaction-diffusion equations and show some results with the numerical methodology developed in the previous sections.

4.1 Semi-implicit scheme for discretizing the parabolic form of the Laplace-Beltrami equation

A full discretization of the reaction-diffusion system (53)-(56) is may be obtained by a combination of finite difference dicretization with respect to time and finite element discretization with respect to space variables. One of the simplest, but still very effective, schemes is obtained by means of Euler time discretization. The time-interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [0,T]}

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T>0}
is divided into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N}
subintervals of small length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta t = T/N}

, say Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [(n-1)\Delta t, n\Delta t]} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n = 1,\ldots ,N} . From the initial conditions we obtain the starting values Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (u^{0}(\mathbf{x}), v^{0}(\mathbf{x})) = (u_{0}(\mathbf{x}), v_{0}(\mathbf{x}))} , and assuming that approximations Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (u^{1}(\mathbf{x}),v^{1}(\mathbf{x}) ) \approx (u(\mathbf{x},\Delta{t}), v(\mathbf{x},\Delta{t}))} ,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \ldots } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (u^{n}(\mathbf{x}), v^{n}(\mathbf{x}))\approx (u(\mathbf{x},n\,\Delta{t}), v(\mathbf{x},n\,\Delta{t}))} , are already known, we can compute the next approximation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (u^{n+1}(\mathbf{x}), v^{n+1}(\mathbf{x}) )}

by solving the system

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \displaystyle \frac{u^{n+1}-u^{n}}{\Delta{t}} - D_u \, \nabla _{\Sigma }^{2} \, u^{n+1} = \gamma \, f(u^{n},v^n), \quad (57) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \displaystyle \frac{v^{n+1}-v^{n}}{\Delta{t}} - D_v \, \nabla _{\Sigma }^{2} \, v^{n+1} = \gamma \, g(u^{n+1},v^n), (58)

where the boundary conditions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (\hat{\mathbf{n}}\cdot \nabla ) u^{n+1} = 0}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (\hat{\mathbf{n}}\cdot \nabla ) v^{n+1} = 0}
must be satisfied if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \partial \Sigma \neq \emptyset }

. It is easy to see that these equations are of elliptic type (26). For instance for the first equation we can set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u=u^{n+1}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha=1/\Delta{t}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f=\gamma \, f(u^n,v^n) + \alpha \, u^{n}}

, obtaining a system of the form (26). So, the numerical solution of both equations at each time step is obtained by the finite element method developed in Section 4. The full discretization scheme is called semi-implicit (see [11]) because the diffusion terms are evaluated implicitly while the reaction terms are evaluated explicitly, at each time step.

4.2 Validation of the proposed methodology

According to the full discretization scheme, we must solve the elliptic equation (26) at every time step. So, it it very important to have an effective and accurate numerical solution to this equation. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Sigma }

be the unit sphere, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha = 0}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle D_{\Sigma } = 1}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f(x,y,z) = 12x y z}

, then the exact solution of (26) is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi (x,y,z) = x y z} . We can compare the numerical solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi _h}

with this exact solution and find the accuracy of the numerical method. We generate triangular meshes on the unit sphere by the subdivision of an inscribed icosahedron, and we denote by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_s}
the number of the subdivisions. We show the generated meshes for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_s = 0,3,4,5}

in Figure 4.

Figure 4: Triangular meshes

The rate of convergence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r}

of the linear finite element method is obtained from the error estimate

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Vert \phi - \phi _h \Vert _{L^2(\Sigma )} \le C \Vert \phi \Vert _{L^2(\Sigma )} h^r,

(59)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi _h}

is the numerical solution on a triangular mesh with mesh-size Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h}
(the average diameter of the triangular elements), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C}
is a constant independent of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h}

. It we show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r}

is close to 2, then our numerical method achieves the optimal convergence rate. This parameter is estimated from the relation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\Vert \phi - \phi _h \Vert _{L^2(\Sigma )}}{\Vert \phi - \phi _{h/2}\Vert _{L^2(\Sigma )} } \approx 2^r,

(60)

for two meshes, one with mesh-size Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h}

and a refinement with mesh-size Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h/2}

, for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h}

small enough. We obtained Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r = 2 .0155}
with mesh-sizes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h/2}

, corresponding Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_s = 4}

and 5, respectively.

On the other hand, for parabolic PDE the methodology has been tested in the context of optimal control on a toroidal surface and on a sphere, as shown in references [12,13].

5 Numerical results

Example 1. We consider the Schnakenberg model, which is a classical and simple model to generate spot patterns, among other structures. Its reaction terms give rise to one of the simplest oscillating reactions between two chemical species, and was introduced by Schnakenberg in 1979 [14]. These reaction terms, in dimensionless form, are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): f(u,v) = \gamma \, (a-u+u^{2}v), (61) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): g(u,v) =\gamma \, (b-u^{2}v), (62)

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a,b,\gamma }

positive constants. We assume homogeneous Neumann-type boundary conditions for the case of open and bounded surfaces, and initial conditions as in (55–56). The stable equilibrium of the chemical reaction is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u^{\ast } = a+b, \quad v^{\ast } = \displaystyle \frac{b}{(a+b)^{2}}.

(63)

The Jacobian matrix at the equilibrium point (63) is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbb{J}= \begin{pmatrix}\displaystyle \frac{b-a}{a+b} & (a+b)^{2} \\ -\displaystyle \frac{2b}{a+b} & -(a+b)^{2} \end{pmatrix}

(64)

Considering Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a<b}

the conditions (21-24) translate into

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0<b-a<(a+b)^{3}} ,
2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle D_\Sigma (b-a)>(a+b)^{3}} ,
3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left[D_\Sigma (b-a)-(a+b)^{3}\right]^{2}>4D_\Sigma (a+b)^{4}} .

For more details we recommend references [15,16,17].

We present numerical results for four different 3-D surfaces for which we show coarse meshes in Figure 5: The first two are closed surfaces and the other two are open. Actual meshes to compute numerical solutions are refinement of these meshes (the number of nodes and triangular elements are included along with each numerical result below. These meshes were generated using their parametric equations:

• `Blood cell'¹: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle -\pi /2 \leq \theta \leq \pi /2}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0 \leq \phi \leq 2\pi } ,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x = r_0\,\hbox{ cos}\,\phi \,\hbox{ cos}\,\theta Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): y = r_0\,\hbox{ sin}\,\phi \,\hbox{ cos}\,\theta Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): z = \displaystyle \frac{1}{2}\hbox{ sin}\,\theta \, (c_0 + c_2\hbox{ cos}^{2}\theta + c_4\hbox{ cos}^{4}\theta )

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_0 = 3.91/3.39} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c_0 = 0.81/3.39} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c_2 = 7.83/3.39}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c_4 = -4.39/3.39}

.

• `Strawberry': Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{x^2 + y^2 + (z - \sqrt{x^2 + y^2})^2 = 1}}

.

• `Coiled toroid': Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0\leq \theta \leq 2\pi }

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0\leq \phi \leq 4\pi } ,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x = (3 + \hbox{ cos}\,\theta ) \, \hbox{ cos}\,\phi Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): y = (3 + \hbox{ cos}\,\theta ) \, \hbox{ sin}\,\phi Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): z = \hbox{ sin}\,\theta - 0.325\,\phi

• `Helical shell' ²: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0\leq \theta \leq 2\pi }

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t\in [-40,-1]}

,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x = e^{0.05\,t}(1.5 + \hbox{ cos}\,\theta ) \, \hbox{ cos}\,t Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): y = -e^{0.05\,t}(1.5 + \hbox{ cos}\,\theta ) \, \hbox{ sin}\,t Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): z = e^{0.05\,t}(-8.5 + 1.2\,\hbox{ sin}\,\theta )

Figure 5: Surfaces used for the Schnakenberg model to generate spot patterns.

For all these cases we fix the parameter values Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle D_\Sigma = 20} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a = 0.126779}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle b = 0.792366}

, for which it is known that generate spot like patterns in a planar region for different values Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma } . The initial conditions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (u_0, v_0)}

were generated by a 10% perturbation of the equilibrium Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (u^\ast , v^\ast )}

.

• `Blood cell'. Discretization parameters: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta t = 10^{-4}}

, mesh with 40962 nodes and 81920 triangular elements. Scale factor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma = 1000}

.

Figure 6: Concentrations at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t = 0 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t=4 .

• `Strawberry surface' Discretization parameters: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta t = 10^{-4}}

, mesh with 40962 nodes and 81920 triangular elements. Scale factor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma = 500}

.

Figure 7: Concentrations at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t = 0 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t = 1. .

• 'Coiled toroid'. Discretization parameters: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta t = 10^{-4}}

, mesh with 48690 nodes and 97200 elements. Scale factor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma = 100}

.

Figure 8: Concentrations at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t = 0 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t = 4. .

• `Helical shell'. Discretization parameters: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta t = 10^{-4}}

, mesh with 90150 nodes and 180000 elements. Scale factor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma = 500}

.

Figure 9: Concentrations at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t = 0 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t = 2 .

Example 2. Now, we consider the BVAM model³ taken from [20], which expressed in its Laplace-Beltrami form for surface domains is given by:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\partial{u}}{\partial{t}} = D_\Sigma \delta \nabla _{\Sigma }^{2}u + \gamma \left[\alpha u(1 - r_1v^2) + v(1 - r_2u)\right], Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\partial{v}}{\partial{t}} = \delta{\nabla }_{\Sigma }^{2}v + \gamma \left[\beta v\left(1 + \displaystyle \frac{\alpha r_1}{\beta }uv\right)+ u(\epsilon + r_2v)\right],

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha ,\beta ,\delta ,\epsilon ,D}

are constants. The scale factor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma >0}
has been added to allow for more convenient manipulation of pattern sizes. The stable equilibria of the chemical reaction, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{w}^\ast = (u^\ast ,v^\ast )^T}

, must satisfy

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): v^\ast = -\displaystyle \frac{(\alpha + \epsilon )}{1 + \beta }u^\ast , \quad \beta \not = 1.

(65)

For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \epsilon = -\alpha } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle v^\ast = 0} . For simplicity, we consider Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{w}^\ast = (0,0)^T}

as the equilibrium point. In this case the Jacobian matrix at the equilibrium point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{w}^\ast = (0,0)^T}
is given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbb{J}= \begin{pmatrix}\alpha & 1 \\ \epsilon & \beta \end{pmatrix}.

(66)

Thus, the conditions (21-24) for this case are:

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha + \beta <0} ,
2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha \beta - \epsilon >0} ,
3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle D\alpha + \beta >0} ,
4. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left(D\alpha + \beta \right)^{2}>4D(\alpha \beta - \epsilon )} ,

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha , \delta >0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta , \epsilon <0}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0< D < 1}

.

Again, we present numerical results for four different 3-D surfaces shown in Figure 10, three of them are closed surfaces and the other one is open. These meshes were generated using their parametric equations:

• Unit sphere: standard parametrization with radius Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R=1}

• Toroid: usual parametrization with major radius Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R = 2.5}

and minor radius Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r = 1}

.

• Dupin's cyclide⁴: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0\leq \phi ,\theta \leq 2\pi }

,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x = \displaystyle \frac{d(c - a\,\hbox{cos}\,\phi \,\hbox{cos}\,\theta ) + b^2\hbox{cos}\,\phi }{a - c\,\hbox{cos}\,\phi \,\hbox{cos}\,\theta } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): y = \displaystyle \frac{b\,\hbox{sin}\,\phi \,(a - d\,\hbox{cos}\,\theta )}{a - c\,\hbox{cos}\,\phi \,\hbox{cos}\,\theta } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): z = \displaystyle \frac{b\,\hbox{sin}\,\theta \,(c\,\hbox{cos}\,\phi - d)}{a - c\,\hbox{cos}\,\phi \,\hbox{cos}\,\theta }

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a = 2} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle b = 1.95} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c^2 = a^2 - b^2} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle d = c + 0.2} .

• Wavy cylinder without covers: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0\leq \theta \leq 2\pi }

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0\leq z \leq 2}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x = R(z)\,\hbox{ cos}\theta Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): y = R(z)\,\hbox{ sin}\theta Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): z = z

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R(z) = 0.5\,\hbox{cos}^{2}(3z) + 1} .

Figure 10: Surfaces used for the BVAM model to generate Turing patterns.

The following numerical results were obtained with parameter values Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \epsilon } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_2} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle D_\Sigma }

within the range proposed in [18]. In particular, the following parameters are common for the four cases: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha = 0.899}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta = -0.91} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta = 0.516} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \epsilon = -0.899} .

• Sphere. Discretization parameters: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta t = 10^{-2}}

, mesh with 40962 nodes and 81920 triangular elements. Scale factor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma = 10} . Parameters: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_1 = 3.5} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_2 = 0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle D_\Sigma=2.1\times{10^{-3}}}

.

Figure 11: Concentrations at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t = 0 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t=600 .

• Toroid. Discretization parameters: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta t = 10^{-2}}

, mesh with 20000 nodes and 40000 triangular elements. Scale factor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma = 0.5} . Parameters: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_1 = 0.02} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_2 = 0.2} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle D_\Sigma=4.5\times{10^{-3}}}

.

Figure 12: Concentrations at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t = 0 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t = 800 .

• Dupin's cyclide. Discretization parameters: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta{t} = 10^{-2}}

, mesh with 22500 nodes and 45000 elements. Scale factor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma = 10} . Parameters: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_1 = 3.5} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_2 = 0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle D_\Sigma=2.1\times{10^{-3}}}

.

Figure 13: Concentrations at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t = 0 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t =1350 .

• Wavy cylinder without covers. Discretization parameters: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta t = 10^{-2}}

, mesh with 29040 nodes and 57600 elements. Scale factor value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma = 2} . Parameters: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_1 = 0.02} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_2 = 0.2} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle D_\Sigma=4.5\times{10^{-3}}}

.

Figure 14: Concentrations at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t = 0 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t = 500 .

(¹) Equations obtained from [18]

(²) Equations obtained from [19]

(³) Named after the initials of its authors: Barrio, Varea, Aragón and Maini.

(⁴) Equations obtained from [21].

6 Conclusions

• When transferring the mathematical conditions for pattern generation from planar regions to surfaces, the Laplace-Beltrami operator was used and computational simulations were performed. In these simulations, spatial patterns similar to those already known in planar regions were observed. In some cases, it was necessary to adjust the value of the scale factor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma }

, which is closely related to the wave modes in the analytical solution; this adjustment was crucial to obtain patterns on surfaces.

• The numerical methodology for the solution of the Laplace-Beltrami equation, using the finite element method, allowed the development of a simple way for calculating the mass matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{M}}

, the stiffness matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{K}}

and the load vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{F}}

. This resulted in an optimized code to perform the computational simulations. A great advantage of this implementation is that the calculations to determine the mass matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{M}} , stiffness Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{K}}

and the load vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{F}}
depend only on the coordinates of the nodes of the triangular mesh associated with the surface and on the gradients of the functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{\varphi }_1}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{\varphi }_2}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{\varphi }_3}

, which, as previously seen, are constant.

• Although the strategy for solving the system Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (\alpha \mathbb{M}+ D_\Sigma \mathbb{K})\boldsymbol{\phi } = \mathbf{F}}

obtained in (35) was not presented, the Cholesky-type factorization was used in the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{A}= \alpha \mathbb{M}+ D_\Sigma \mathbb{K}}
because it is a positive definite symmetric matrix. Additionally, since matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{A}}
is sparse, different methods are available to further speed up the calculations (see [22], [23]). In our case, the Minimum Degree Permutation Method was used, which significantly optimized the computation time.       

• It is important to mention that, for the most part, the triangular meshes are generated from known parametrizations of the surfaces. It is crucial that these meshes are of good quality to obtain optimal numerical results, as poor-quality meshes can lead to significant numerical errors. Reference [24] can be consulted for an introduction to the study of triangular mesh quality. It is worth mention that parametrizations are not used for the numerical solution of the reaction diffusion-models, instead they are solved directly in Euclidean coordinates, as indicated in Section 3.

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