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A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces

Abstract

In this paper we present a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in \(\mathbb{R }^d\). For two-dimensional surfaces embedded in \(\mathbb{R }^3\), these types of problems have received growing interest in biology, chemistry, and computer graphics to model such things as diffusion of chemicals on biological cells or membranes, pattern formations in biology, nonlinear chemical oscillators in excitable media, and texture mappings. Our kernel method is based on radial basis functions and uses a semi-discrete approach (or the method-of-lines) in which the surface derivative operators that appear in the PDEs are approximated using collocation. The method only requires nodes at “scattered” locations on the surface and the corresponding normal vectors to the surface. Additionally, it does not rely on any surface-based metrics and avoids any intrinsic coordinate systems, and thus does not suffer from any coordinate distortions or singularities. We provide error estimates for the kernel-based approximate surface derivative operators and numerically study the accuracy and stability of the method. Applications to different non-linear systems of PDEs that arise in biology and chemistry are also presented.

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Notes

  1. 1.

    The condition \(\tau > 3/2\) ensures that functions within the kernel’s native space are continuous on \(\mathbb{R }^3\).

  2. 2.

    See the discussion preceeding Proposition 2 in “Appendix A” for details.

  3. 3.

    See Theorem 2 for details.

  4. 4.

    With this integral operator comes the pseudodifferential operators \(T^{-r}, \,r>0\). A function \(f\) is in the native space if and only if \(T^{-1/2}f \in L_2(\mathbb{M })\) [35, Proposition 4.9], so we expect functions such that \(T^{-1}f\in L_2(\mathbb{M })\) to be twice as smooth.

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Acknowledgments

Research support for G. B. Wright was provided, in part, by grants DMS-0934581, DMS-0540779, and DMS-1160379 from the National Science Foundation.

Author information

Correspondence to Grady B. Wright.

Appendices

Appendix A: Convergence Results

In this section we present the convergence results for our discrete differential operators. The results presented here are for the special case when \(\mathbb{M }\subset \mathbb{R }^3\) is a two dimensional surface, although similar results hold in higher dimensions. The arguments will depend heavily on error estimates for kernel interpolants on smooth, embedded submanifolds of \(\mathbb{R }^d\) given recently in [35]. We will frequently reference results from that paper throughout this section. Sobolev error estimates for the surface gradient and divergence operators will immediately follow the results in [35]. However, the estimates for the discrete surface Laplacian will require more work.

Proposition 1

Let \(1\le q \le \infty , \,\phi \) be a positive definite kernel satisfying (25) with \(\tau > 3/2 + 1\), and define \(s = \tau - 1/2\). Then there is a constant \(h_{\mathbb{M }}\) depending only on \(\mathbb{M }\) such that if a finite node set \(X\subset \mathbb{M }\) satisfies \(h\le h_\mathbb{M }\), for all \(f\in \mathcal N _\phi (\mathbb{M })\) and \(\mathbf{f }\in (\mathcal N _\phi (\mathbb{M }))^3\) we have

$$\begin{aligned} \Vert G_\mathbb{M }f-\nabla _{\mathbb{M }} f\Vert _{\mathbf{L }_q(\mathbb{M })}&\le C h^{s-1-2(1/2-1/q)_+}\Vert f\Vert _\mathcal{N _\phi (\mathbb{M })},\\ \Vert D_{\mathbb{M }}\mathbf{f }-\nabla _{\mathbb{M }}\cdot \mathbf{f }\Vert _{L_q(\mathbb{M })}&\le C h^{s-1-2(1/2-1/q)_+}\Vert \mathbf{f }\Vert _\mathcal{N _\phi (\mathbb{M })}, \end{aligned}$$

where \((x)_+ = x\) if \(x\ge 0\) and is zero otherwise.

Proof

We will focus on the discrete gradient operator. We have

$$\begin{aligned} \Vert G_\mathbb{M }f-\nabla _{\mathbb{M }} f\Vert _{\mathbf{L }_q(\mathbb{M })} =\Vert \nabla _{\mathbb{M }}I_{\phi }f-\nabla _{\mathbb{M }} f\Vert _{\mathbf{L }_q(\mathbb{M })} \le C\Vert I_{\phi }f - f\Vert _{W^{1}_q(\mathbb{M })}, \end{aligned}$$

where \(W^1_q(\mathbb{M })\) is the \(L_q\) Sobolev space of order 1 (see Sect. 2, [35]). The assumptions on \(\tau \) allow us to apply Theorem 4.6 in [35], and the results follow. The error bounds in Theorem 4.6 in [35] carry over to the vector-valued case, thus bounds for the divergence operator error are obtained in a similar manner. \(\square \)

When \(\phi \) satisfies (26), i.e. \(\phi \) has finite smoothness, the estimates come in two types: the first concerns targets that may be too rough to be within the native space, and the second applies to functions with additional smoothness. This additional smoothness is measured with the inverse of the integral operatorFootnote 4

$$\begin{aligned} Tf(x) := \int _{\mathbb{M }}\phi (y,x)f(y)d\mu (y). \end{aligned}$$

We denote a vector version of this operator by \(\mathbf{T }\), which simply applies \(T\) to each component of a \(3\)-dimensional vector field. Similar to the proof above, the result below follows from Theorem 4.12 and Corollary 4.10 in [35].

Proposition 2

Let \(1\le q \le \infty \), and let \(\phi \) be a positive definite kernel satisfying (26) with \(\tau > 3/2 + 1\), and define \(s = \tau - 1/2\). Then there is a constant \(h_{\mathbb{M }}\) depending only on \(\mathbb{M }\) such that if a finite node set \(X\subset \mathbb{M }\) satisfies \(h\le h_\mathbb{M }\), we have the following:

  1. 1.

    Rough target functions. Let \(\beta \) be such that \(s \ge \beta > 2\). Then for all \(f \in H^\beta (\mathbb{M })\) and \(\mathbf{f }\in \mathbf{H }^\beta (\mathbb{M })\) we have

    $$\begin{aligned} \Vert G_\mathbb{M }f-\nabla _{\mathbb{M }} f\Vert _{\mathbf{L }_q(\mathbb{M })}&\le C h^{\beta -1-2(1/2-1/q)_+}\rho ^{s -\beta }\Vert f\Vert _{H^{\beta }(\mathbb{M })},\\ \Vert D_{\mathbb{M }}\mathbf{f }-\nabla _{\mathbb{M }}\cdot \mathbf{f }\Vert _{L_q(\mathbb{M })}&\le C h^{\beta -1-2(1/2-1/q)_+}\rho ^{s - \beta }\Vert \mathbf{f }\Vert _{\mathbf{H }^{\beta }(\mathbb{M })}. \end{aligned}$$
  2. 2.

    Smooth target functions. For all \(f\in L_2(\mathbb{M })\) such that \(T^{-1}f\in L_2(\mathbb{M })\) and \(\mathbf{f }\in \mathbf{L }_2(\mathbb{M })\) such that \(T^{-1}\mathbf{f }\in \mathbf{L }_2(\mathbb{M })\), we have

    $$\begin{aligned} \Vert G_\mathbb{M }f-\nabla _{\mathbb{M }} f\Vert _{\mathbf{L }_q(\mathbb{M })}&\le C h^{2s-1-2(1/2 - 1/q)_+}\Vert T^{-1}f\Vert _{L_2(\mathbb{M })},\\ \Vert D_{\mathbb{M }}\mathbf{f }-\nabla _{\mathbb{M }}\cdot \mathbf{f }\Vert _{\mathbf{L }_q(\mathbb{M })}&\le C h^{2s-1-2(1/2 - 1/q)_+}\Vert \mathbf{T }^{-1}\mathbf{f }\Vert _{\mathbf{L }_2(\mathbb{M })}. \end{aligned}$$

Obtaining convergence rates for the discrete Laplace operator is not as straightforward, and will require extra tools. The first is the “zeros lemma” from Narcowich, Ward and Wendland [54]. We will use the manifold version stated in [35, Lemma 4.5], with parameters adapted to our situation.

Proposition 3

Let \(1\le q\le \infty , \,t\in \mathbb{R }\) with \(t > 1\). Let \(\mu \in \mathbb{N }\) satisfy \(0\le \mu \le \lceil t - 2(1/2 - 1/q)_{+}\rceil -1\). Also, let \(X\subset \mathbb{M }\) be a discrete set with mesh norm \(h_{X,\mathbb{M }}\). Then there is a constant depending only on \(\mathbb{M }\) such that if \(h_{X,\mathbb{M }}\le C_{\mathbb{M }}\) and if \(u\in H^{t}(\mathbb{M })\) satisfies \(u|_{X}=0\), then

$$\begin{aligned} |u|_{W_{q}^{\mu }(\mathbb{M })}\le Ch^{t-\mu -2(1/2-1/q)_{+}}|u|_{H^{t}(\mathbb{M })}. \end{aligned}$$

Proposition 3 also holds with full Sobolev norms, and a similar result holds for vector fields.

Since the operator \(L_{\mathbb{M }}\) is a mixture of differential and interpolation operators, it will be beneficial to bound the norm of the interpolation operator and its associated error in several different Sobolev spaces. In particular, we will use the following.

Lemma 1

Let \(\phi \) satisfy (26) with \(\tau > 3/2\), and define \(s = \tau - 1/2\). Let \(\beta \) be such that \(s \ge \beta > 1\). Then there is a constant \(h_{\mathbb{M }}\) depending only on \(\mathbb{M }\) such that if a finite node set \(X\subset \mathbb{M }\) satisfies \(h\le h_\mathbb{M }\) then for all \(f\in H^{\beta }(\mathbb{M })\) we have

$$\begin{aligned} \Vert I_{\phi }f - f\Vert _{H^{\beta }(\mathbb{M })} \le C \rho ^{s - \beta }\Vert f\Vert _{H^{\beta }(\mathbb{M })}\quad \text{ and} \quad \Vert I_{\phi }f\Vert _{H^{\beta }(\mathbb{M })} \le C \rho ^{s - \beta }\Vert f\Vert _{H^{\beta }(\mathbb{M })}. \end{aligned}$$

Proof

The last estimate in the proof of Theorem 4.12 in [35] is

$$\begin{aligned} \Vert f - I_{\phi }f\Vert _{H^{\beta }(\mathbb{M })} \le C \rho ^{s - \beta }\Vert f\Vert _{H^{\beta }(\mathbb{M })}. \end{aligned}$$

Applying a triangle inequality and observing that \(1\le \rho \), we get

$$\begin{aligned} \Vert I_{\phi }f\Vert _{H^{\beta }(\mathbb{M })} \le \Vert f \!-\! I_{\phi }f\Vert _{H^{\beta }(\mathbb{M })} \!+\! \Vert f\Vert _{H^{\beta }(\mathbb{M })} \le C (\rho ^{s \!-\! \beta }\!+\!1)\Vert f\Vert _{H^{\beta }(\mathbb{M })} \le C\rho ^{s - \beta }\Vert f\Vert _{H^{\beta }(\mathbb{M })}. \end{aligned}$$

\(\square \)

Now we are poised to present the following theorem.

Theorem 1

Let \(1\le q \le \infty , \,\phi \) satisfy (26) with \(\tau > 3/2 + 2\), and define \(s = \tau - 1/2\). Let \(\beta \) be such that \(s \ge \beta > 3\). Then there is a constant \(h_{\mathbb{M }}\) depending only on \(\mathbb{M }\) such that if a finite node set \(X\subset \mathbb{M }\) satisfies \(h\le h_\mathbb{M }\), then for all \(f\in H^\beta (\mathbb{M })\) we have the estimate

$$\begin{aligned} \Vert L_{\mathbb{M }}f-\Delta _\mathbb{M }f\Vert _{L_q(\mathbb{M })}\le C h^{\beta -2-2(1/2-1/q)_+}\rho ^{2(s - \beta ) + 1}\Vert f\Vert _{H^{\beta }(\mathbb{M })}. \end{aligned}$$
(32)

Proof

First, we have

$$\begin{aligned} \Vert L_{\mathbb{M }}f-\Delta _\mathbb{M }f\Vert _{L_q(\mathbb{M })}\le \Vert L_{\mathbb{M }}f-\Delta _\mathbb{M }I_{\phi }f\Vert _{L_q(\mathbb{M })} + \Vert \Delta _\mathbb{M }f - \Delta _\mathbb{M }I_{\phi }f\Vert _{L_q(\mathbb{M })}. \end{aligned}$$
(33)

For the rightmost term, the assumption \(\beta > 3\) allows us to use [35, Theorem 4.12] to get:

$$\begin{aligned} \Vert \Delta _\mathbb{M }f - \Delta _\mathbb{M }I_{\phi }f\Vert _{L_q(\mathbb{M })}&\le C \Vert f - I_{\phi }f\Vert _{W^{2}_q(\mathbb{M })} \le C h^{\beta -2 -2(1/2-1/q)_+}\rho ^{s-\beta }\Vert f\Vert _{H^{\beta }(\mathbb{M })}. \end{aligned}$$

For the other term in (33), we have

$$\begin{aligned} \Vert L_{\mathbb{M }}f-\Delta _\mathbb{M }I_{\phi }f\Vert _{L_q(\mathbb{M })}&= \Vert \nabla _\mathbb{M }\cdot (I_{\Phi }(\nabla _\mathbb{M }I_{\phi }f)) -\Delta _\mathbb{M }I_{\phi }f\Vert _{L_q(\mathbb{M })} \nonumber \\&= \Vert \nabla _\mathbb{M }\cdot (I_{\Phi }(\nabla _\mathbb{M }I_{\phi }f)) -\nabla _\mathbb{M }\cdot (\nabla _\mathbb{M }I_{\phi }f)\Vert _{L_q(\mathbb{M })} \nonumber \\&\le C\Vert I_{\Phi }(\nabla _\mathbb{M }I_{\phi }f) -\nabla _\mathbb{M }I_{\phi }f\Vert _{\mathbf{W }_q^{1}(\mathbb{M })}. \end{aligned}$$
(34)

Note that the last estimate is the interpolation error for the target function \(\mathbf{g }:= \nabla _\mathbb{M }I_{\phi }f\).

We claim that \(\mathbf{g }\) is in \(\mathbf{H }^{t}(\mathbb{M })\) for all \(t < 2s - 2\). Since \(\phi \) satisfies (26), \(I_{\phi }f \in H^{\nu }(\mathbb{R }^3)\) for all \(\nu < 2\tau - 3/2\). By the trace theorem restricting to \(\mathbb{M }\) puts \(I_{\phi }f\) in \(H^{\nu - 1/2}(\mathbb{M })\). Thus \(\mathbf{g }\in \mathbf{H }^{\nu - 3/2}(\mathbb{M })\) for all \(\nu < 2\tau - 3/2\), which is equivalent to \(\mathbf{g }\in \mathbf{H }^{t}(\mathbb{M })\) for all \(t < 2s - 2\). In particular, \(\mathbf{g }\in \mathbf{H }^{\beta - 1}(\mathbb{M })\). Also, since \(\beta > 2\) then we have

$$\begin{aligned} 1 \le \lceil (\beta - 1) - 2(1/2 - 1/q)_+ \rceil - 1, \end{aligned}$$

and that \(\beta -1 > 1\). This allows us to estimate (34) with Proposition 3 (with parameter \(\mu = 1\) and target smoothness \(t = \beta -1\)) to get

$$\begin{aligned} \Vert I_{\Phi }(\nabla _\mathbb{M }I_{\phi }f) -\nabla _\mathbb{M }I_{\phi }f\Vert _{\mathbf{W }_q^{1}(\mathbb{M })}&\le C h^{(\beta - 1) - 1-2(1/2-1/q)_+}\Vert I_{\Phi }\mathbf{g }- \mathbf{g }\Vert _{\mathbf{H }^{\beta -1}(\mathbb{M })}.\\&= C h^{\beta - 2 -2(1/2-1/q)_+}\Vert I_{\Phi }\mathbf{g }- \mathbf{g }\Vert _{\mathbf{H }^{\beta -1}(\mathbb{M })}. \end{aligned}$$

Further, since \(\beta - 1>1\), we can also apply Lemma 1, and with two applications of it we get

$$\begin{aligned} \Vert I_{\Phi }\mathbf{g }- \mathbf{g }\Vert _{\mathbf{H }^{\beta -1}(\mathbb{M })}&\le C\rho ^{s-\beta +1}\Vert \mathbf{g }\Vert _{\mathbf{H }^{\beta -1}(\mathbb{M })} = C\rho ^{s-\beta +1}\Vert \nabla _{\mathbb{M }}I_{\phi }f\Vert _{\mathbf{H }^{\beta -1}(\mathbb{M })}\\&\le C\rho ^{s-\beta +1}\Vert I_{\phi }f\Vert _{H^{\beta }(\mathbb{M })}\le C\rho ^{2(s-\beta )+1}\Vert f\Vert _{H^{\beta }(\mathbb{M })}\nonumber . \end{aligned}$$
(35)

This completes the proof. \(\square \)

Continuing with our error analysis, we now shift our attention to target functions that are very smooth. First we will need a lemma.

Lemma 2

Let \(\phi \) satisfy (26) with \(\tau > 3/2\), and let \(s = \tau - 1/2\). Then for all \(f\in L_2(\mathbb{M })\) such that \(T^{-1}f \in L_2(\mathbb{M })\) we have the following estimate

$$\begin{aligned} \Vert f - I_{\phi }f\Vert _{H^s(\mathbb{M })}\le C h^s\Vert T^{-1}f\Vert _{L_2(\mathbb{M })}. \end{aligned}$$

Also, for all \(\mathbf{f }\in \mathbf{L }_2(\mathbb{M })\) such that \(\mathbf{T }^{-1}\mathbf{f }\in \mathbf{L }_2(\mathbb{M })\) we have

$$\begin{aligned} \Vert \mathbf{f }- I_{\Phi }\mathbf{f }\Vert _{\mathbf{H }^s(\mathbb{M })}\le C h^s\Vert \mathbf{T }^{-1}\mathbf{f }\Vert _{\mathbf{L }_2(\mathbb{M })}. \end{aligned}$$

Proof

The first estimate is established in the proof of Corollary 4.10 in [35]. By working component-wise the second estimate follows from the first. \(\square \)

Theorem 2

Let \(1\le q \le \infty , \,\phi \) satisfy (26) with \(\tau > 3/2 + 2\), and define \(s = \tau - 1/2\). Let \(f\in L_2(\mathbb{M })\) be such that \(T^{-1}f \in L_2(\mathbb{M })\) and \(\mathbf{T }^{-1}\nabla _{\mathbb{M }}f \in \mathbf{L }_2(\mathbb{M })\). Then there is a constant \(h_{\mathbb{M }}\) depending only on \(\mathbb{M }\) such that if a finite node set \(X\subset \mathbb{M }\) satisfies \(h\le h_\mathbb{M }\), we have the estimate

$$\begin{aligned} \Vert L_{\mathbb{M }}f-\Delta _\mathbb{M }f\Vert _{L_q(\mathbb{M })}\le C h^{2s-2-2(1/2-1/q)_+}\rho (\Vert T^{-1}f\Vert _{L_2(\mathbb{M })} + \Vert \mathbf{T }^{-1}\nabla _{\mathbb{M }}f\Vert _{\mathbf{L }_2(\mathbb{M })}). \end{aligned}$$

Proof

We begin as in the proof of the previous theorem. We have

$$\begin{aligned} \Vert L_{\mathbb{M }}f-\Delta _\mathbb{M }f\Vert _{L_q(\mathbb{M })}\le \Vert L_{\mathbb{M }}f-\Delta _\mathbb{M }I_{\phi }f\Vert _{L_q(\mathbb{M })} + \Vert \Delta _\mathbb{M }f - \Delta _\mathbb{M }I_{\phi }f\Vert _{L_q(\mathbb{M })}. \end{aligned}$$
(36)

To estimate the rightmost term we can use the error estimates in [35, Corollary 4.10] to get:

$$\begin{aligned} \Vert \Delta _\mathbb{M }f - \Delta _\mathbb{M }I_{\phi }f\Vert _{L_q(\mathbb{M })}&\le C \Vert f - I_{\phi }f\Vert _{W^{2}_q(\mathbb{M })} \le C h^{2s-2-2(1/2-1/q)_+}\Vert T^{-1}f\Vert _{L_2(\mathbb{M })}. \end{aligned}$$

For the other term in (36), we proceed as in the proof of Theorem 1 to get

$$\begin{aligned} \Vert L_\mathbb{M }f-\Delta _\mathbb{M } I_{\phi }f\Vert _{{L_{q}(\mathbb{M })}} \le C\Vert I_{\Phi }(\mathbf{g }) -\mathbf{g }\Vert _{\mathbf{W }_{q}^{1}(\mathbb{M })}, \end{aligned}$$
(37)

where \(\mathbf{g }= \nabla _\mathbb{M }I_{\phi }f\), and as before we know that this function is in \(\mathbf{H }^{t}(\mathbb{M })\) for all \(t < 2s - 2\). In particular, \(\mathbf{g }\in \mathbf{H }^{s-1}(\mathbb{M })\) and we can estimate (37) with Proposition 3 (with \(t=s-1\) and \(\mu =1\)) to get

$$\begin{aligned} \Vert I_{\Phi }(\mathbf{g }) -\mathbf{g }\Vert _{\mathbf{W }_q^{1}(\mathbb{M })}&\le C h^{s - 2 -2(1/2-1/q)_+}\Vert I_{\Phi }\mathbf{g }- \mathbf{g }\Vert _{\mathbf{H }^{s-1}(\mathbb{M })}. \end{aligned}$$

This is where the proof detours from that of the previous theorem. To estimate \(\Vert I_{\Phi }\mathbf{g }- \mathbf{g }\Vert _{\mathbf{H }^{s-1}(\mathbb{M })}\), note that it is bounded it by following quantity:

$$\begin{aligned} \underbrace{\Vert I_{\Phi }(\nabla _{\mathbb{M }}I_{\phi }f) - I_{\Phi }(\nabla _{\mathbb{M }}f)\Vert _{\mathbf{H }^{s-1}(\mathbb{M })}}_{I} + \underbrace{\Vert I_{\Phi }(\nabla _{\mathbb{M }}f) - \nabla _{\mathbb{M }}f\Vert _{\mathbf{H }^{s-1}(\mathbb{M })}}_{II} + \underbrace{\Vert \nabla _{\mathbb{M }}f - \nabla _{\mathbb{M }}I_{\phi }f\Vert _{\mathbf{H }^{s-1}(\mathbb{M })}}_{III}. \end{aligned}$$

We will bound each term individually. First we concentrate on \(I\). An application of Lemma 1, which we may apply since \(s - 1 > 1\), gives us

$$\begin{aligned} \Vert I_{\Phi }(\nabla _{\mathbb{M }}I_{\phi }f) - I_{\Phi }(\nabla _{\mathbb{M }}f)\Vert _{\mathbf{H }^{s-1}(\mathbb{M })}&= \Vert I_{\Phi }(\nabla _{\mathbb{M }}I_{\phi }f - \nabla _{\mathbb{M }}f)\Vert _{\mathbf{H }^{s-1}(\mathbb{M })} \\&\le C\rho \Vert \nabla _{\mathbb{M }}I_{\phi }f - \nabla _{\mathbb{M }}f\Vert _{\mathbf{H }^{s-1}(\mathbb{M })} = C\rho (III). \end{aligned}$$

Thus a bound for \(III\) will result in a bound for \(I\). To bound \(III\), we can apply Lemma 2 to get

$$\begin{aligned} \Vert \nabla _{\mathbb{M }}I_{\phi }f - \nabla _{\mathbb{M }}f\Vert _{\mathbf{H }^{s-1}(\mathbb{M })}&\le C \Vert I_{\phi }f - f\Vert _{\mathbf{H }^{s}(\mathbb{M })} \le h^{s}\Vert T^{-1}f\Vert _{L_2(\mathbb{M })}. \end{aligned}$$

To bound \(II\), we again employ Lemma 2:

$$\begin{aligned} \Vert I_{\Phi }(\nabla _{\mathbb{M }}f) - \nabla _{\mathbb{M }}f\Vert _{\mathbf{H }^{s-1}(\mathbb{M })}&\le \Vert I_{\Phi }(\nabla _{\mathbb{M }}f) - \nabla _{\mathbb{M }}f\Vert _{\mathbf{H }^{s}(\mathbb{M })} \le C h^s\Vert \mathbf{T }^{-1}\nabla _{\mathbb{M }}f\Vert _{\mathbf{L }_2(\mathbb{M })}. \end{aligned}$$

This completes the proof. \(\square \)

Appendix B: Surfaces and Node Sets from the Numerical Experiments

B.1 Unit Sphere

This manifold is, of course, described implicitly by

$$\begin{aligned} \mathbb{M } = \left\lbrace \mathbf{x }= (x,y,z) \in \mathbb{R }^3 \;\left| x^2 + y^2 + z^2 = 1 \right. \right\rbrace . \end{aligned}$$
(38)

The node sets we use for discretizing the unit sphere are the minimum energy (ME) node sets of Womersley and Sloan [69]. These node sets are approximately uniformly distributed over the surface of the sphere and have the nice property that the mesh norm \(h\) and the separation radius \(q\) decrease uniformly like the inverse of the square root of the number of nodes \(N\), i.e. \(h,q \sim \frac{1}{\sqrt{N}}\). Additionally, the nodes in these sets are not oriented along any vertices or lines, which emphasizes the ability of our method to handle arbitrary node layouts. They have been used quite successfully in many other RBF applications, e.g. [25, 26, 28, 33, 34, 55, 70].

B.2 Red Blood Cell

This manifold is a mathematical model for human red blood cells in static equilibrium conditions. The model was first proposed in [19] and has been used in many subsequent studies (e.g. [48]). The model can be described parametrically as follows:

$$\begin{aligned} \mathbb{M } \!=\! \left\lbrace (x,y,z)\in \mathbb{R }^3\;\left| x \!=\! r_0\cos \lambda \cos \theta ,\; y \!=\! r_0\sin \lambda \cos \theta ,\; z \!=\! \frac{1}{2}\sin \theta \left(c_0 \!+\! c_2\cos ^2\theta \!+\! c_4\cos ^4\theta \right)\right.\right\rbrace \!, \nonumber \\ \end{aligned}$$
(39)

where \(-\pi /2 \le \theta \le \pi /2, \,-\pi \le \lambda < \pi , \,r_0 = 3.91/3.39, \,c_0 = 0.81/3.39, \,c_2 = 7.83/3.39\), and \(c_4=-4.39/3.39\). The node sets we used for discretizing this manifold were obtained using a radial projection of the ME points for the surface of the sphere described above.

B.3 Bumpy Sphere

This manifold is constructed from the “bumpy sphere” surface [3] using the following procedure. First, \(N=5256\) points on the bumpy sphere surface were obtained from http://shapes.aimatshape.net. This surface is homeomorphic to the unit sphere, so spherical coordinates for each of the \(N=5256\) points on the bumpy sphere were obtained. Upon normalizing the original nodes by the maximum distance of any of the nodes from the origin, a parametric model for the surface was constructed using the RBF geometric modeling technique from [61]. The original (normalized) \(N=5256\) points on the bumpy sphere are used as the computational nodes in all the numerical experiments and the parametric model is used for computing the normal vectors to the surface.

B.4 Torus

This manifold is given by the implicit equation:

$$\begin{aligned} \mathbb{M } = \left\lbrace \mathbf{x }= (x,y,z) \in \mathbb{R }^3 \;\left| \left(1 - \sqrt{x^2 + y^2}\right)^2 + z^2 - \frac{1}{9} = 0 \right. \right\rbrace . \end{aligned}$$
(40)

The node sets used for discretizing this manifold were obtained by arranging the nodes so that their Reisz energy (with a power of 2) is near minimal as described in Hardin and Saff’s seminal article [41]. As with the ME sphere nodes, these ME torus nodes sets provide a near uniform discretization of the manifold with an approximately uniform decrease in the mesh norm \(h\) and the separation radius \(q\). The node sets used in the experiments were kindly given to us by Drs. Douglas Hardin and Edward Saff and Ms. Ayla Gafni from Vanderbilt University.

B.5 Dupin’s Cyclide

This manifold is given by the implicit equation:

$$\begin{aligned} \mathbb{M } = \left\lbrace \mathbf{x }= (x,y,z) \in \mathbb{R }^3 \;\left| \left(x^2 + y^2 + z^2 - d^2 + b^2 \right)^2 - 4\left(a x + c d \right)^2 - 4 b^2 y^2 = 0 \right. \right\rbrace , \nonumber \\ \end{aligned}$$
(41)

where \(a=2, \,b=1.9, \,d=1\), and \(c^2 = a^2 - b^2\). The node set for discretizing this manifold was obtained from the algorithm of Palais, Palais, and Karcher (PPK) [56], which will be included in a future version of the amazing mathematical visualization software package 3-D-XplorMath [1]. This algorithm generates “uniformly random” point clouds for surfaces and works for both implicit and parametrically defined surfaces. To obtain the node set displayed in Fig. 1e we used the PPK algorithm to generate a point cloud consisting of 9562 points. We then thinned this point set by removing points that were too close together in order to increase the separation radius. The final node set ended up at \(N=4948\). Both choices for the starting and ending number of nodes were chosen somewhat arbitrarily.

B.6 Bretzel2

This manifold is given by the implicit equation:

$$\begin{aligned} \mathbb{M } = \left\lbrace \mathbf{x }= (x,y,z) \in \mathbb{R }^3 \;\left| \left(x^2(1-x^2)-y^2 \right)^2 + \frac{1}{2} z^2 = \frac{1}{40} \right. \right\rbrace . \end{aligned}$$
(42)

Unlike the the other surfaces, this one does not have an explicit parameterization. The node set for this domain was also obtained from the PPK algorithm. In this case we started with a point cloud of 10278 points and thinned it to end up with a node set of \(N=5041\) nodes.

Appendix C: Example Matlab Code

The code below simulates the Turing system (30) on the red blood cell surface using the parameters for spots in Table 1. The results should be similar to those shown in the top right plot of Fig. 6. The code below uses SBDF2 as the numerical time integration method to simplify the presentation. For the code to work, the \(N=4096\) ME node set for the sphere must be downloaded from [69], or from the second authors’ webpage http://math.boisestate.edu/~wright/research/.

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Fuselier, E.J., Wright, G.B. A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces. J Sci Comput 56, 535–565 (2013). https://doi.org/10.1007/s10915-013-9688-x

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Keywords

  • Radial basis functions
  • Mesh-free
  • Manifold
  • Collocation
  • Method-of-lines
  • Pattern formation
  • Turing patterns
  • Spiral waves

Mathematics Subject Classification(2000)

  • 58J45
  • 35K57
  • 41A05
  • 41A25
  • 41A30
  • 41A63
  • 65D25
  • 65M20
  • 65M70
  • 46E22
  • 35B36