Volumetric variational principles for a class of partial differential equations defined on surfaces and curves

Abstract

In this paper, we propose simple numerical algorithms for partial differential equations (PDEs) defined on closed, smooth surfaces (or curves). In particular, we consider PDEs that originate from variational principles defined on the surfaces; these include Laplace–Beltrami equations and surface wave equations. The approach is to systematically formulate extensions of the variational integrals and derive the Euler–Lagrange equations of the extended problem, including the boundary conditions that can be easily discretized on uniform Cartesian grids or adaptive meshes. In our approach, the surfaces are defined implicitly by the distance functions or by the closest point mapping. As such extensions are not unique, we investigate how a class of simple extensions can influence the resulting PDEs. In particular, we reduce the surface PDEs to model problems defined on a periodic strip and the corresponding boundary conditions and use classical Fourier and Laplace transform methods to study the well-posedness of the resulting problems. For elliptic and parabolic problems, our boundary closure mostly yields stable algorithms to solve nonlinear surface PDEs. For hyperbolic problems, the proposed boundary closure is unstable in general, but the instability can be easily controlled by either adding a higher-order regularization term or by periodically but infrequently “reinitializing” the computed solutions. Some numerical examples for each representative surface PDEs are presented.

Appendix

1.1 Extension of surface gradient and surface divergence in $\mathcal {\mathbb {R}}^{3}$

Let $\Omega \subset \mathbb {R}^{3}$ be a bounded open set with $C^{2}$ boundary $\Gamma =\partial \Omega$. For simplicity, we assume there exists a signed-distance function $d_{\Gamma }:\mathbb {R}^{3}\rightarrow \mathbb {R}$ such that $\Gamma$ is the zero-level set of $d_{\Gamma }$. Without loss of generality, we also assume that $d_{\Gamma }<0$ on the interior of $\Gamma$ and $d_{\Gamma }>0$ on the exterior. The normal vector $\mathbf {n}(x)=\nabla d_{\Gamma }(x)$ is the unit outer normal vector field of $\Gamma$ and the projection $\Pi =I-\mathbf {n}\otimes \mathbf {n}$ which maps vectors in $\mathbb {R}^{3}$ onto the tangent space of $\Gamma$ at x. For any smooth function u defined on $\Gamma$, the tangent gradient of u is defined by

$$\begin{aligned} \nabla _{\Gamma }u(x)=\Pi \nabla \tilde{{u}}(x),\quad \forall x\in \Gamma , \end{aligned}$$

where $\tilde{{u}}$ is any $C^{1}$ extension of u(x) in a neighborhood of x. For any smooth vector field F defined on $\Gamma$, the surface divergence $\nabla _{\Gamma }\cdot F$ is defined analogously. Recall that $\eta$-level set of $d_{\Gamma }$ is $\Gamma _{\eta }=\{x\in \mathbb {R}^{3}\,|\,d(x)=\eta \}$ and $T_{\epsilon }=\cup _{|\eta |<\epsilon }\Gamma _{\eta }$ is the $\epsilon -$narrowband of $\Gamma$. The following theorem relates constant-along-normal extension of the surface gradient and divergence to the Eulerian gradient and divergence of normal-extended function.

Theorem A.1

Suppose u is a smooth function defined on $\Gamma$. Let $P_{\Gamma }$ denote the closest point mapping and $\overline{u}$ denote the constant-along-normal extension of u in $T_{\epsilon }$. Then for any $z\in T_{\epsilon }$, we have

$$\begin{aligned} \overline{\nabla _{\Gamma }u(z)}\equiv \nabla _{\Gamma }u\circ P_{\Gamma }(z)=A(z;\mu )\nabla \overline{u}(z)\equiv A(z;\mu )\nabla u(P_{\Gamma }z), \end{aligned}$$

(A.1)

where $A(z;\mu )=A_{0}(z)+\mu A_{1}(z),$

$$\begin{aligned} A_{0}(z):=&\sigma _{1}^{-1}\mathbf {t}_{1}\otimes \mathbf {t}_{1}+\sigma _{2}^{-1}\mathbf {t}_{2}\otimes \mathbf {t}_{2},\\ A_{1}(z):=&\mathbf {n}\otimes \mathbf {n}. \end{aligned}$$

Here $\mathbf {t}_{1}$, $\mathbf {t}_{2}$ are the two orthonormal tangent vectors corresponding to the directions that yield the principle curvatures of $\Gamma$, $\mathbf {n}$ is the unit normal vector of $\Gamma$, $\sigma _{1}$ and $\sigma _{2}$ are two largest singular values of $DP_{\Gamma }$ and $\mu$ is any real number.

Suppose F is a smooth vector field defined on $\Gamma$. Let $\overline{F}$ denote its constant-along-normal extension in $T_{\epsilon }$. Then for any $z\in T_{\epsilon }$ , we have

$$\begin{aligned} \overline{\nabla _{\Gamma }\cdot F(z)}\equiv (\nabla _{\Gamma }\cdot F)\circ P_{\Gamma }(z)=\sigma _{1}^{-1}\sigma _{2}^{-1}\nabla \cdot (B(z;\mu )\overline{F}(z))=J^{-1}\nabla \cdot (B(z;\mu )\overline{F}(z)), \end{aligned}$$

(A.2)

where $B(z;\nu )=B_{0}(z)+\nu B_{1}(z),$

$$\begin{aligned} B_{0}(z):=&\sigma _{2}\mathbf {t}_{1}\otimes \mathbf {t}_{1}+\sigma _{1}\mathbf {t}_{2}\otimes \mathbf {t}_{2}=\sigma _{1}\sigma _{2}A_{0}(z)=JA_{0}(z),\\ B_{1}(z):=&\mathbf {n}\otimes \mathbf {n}, \end{aligned}$$

where $\nu$ is any real number. In particular, if we choose $\nu =J\mu$, then $B(z;\nu )=JA(z;\mu )$.

Proof

Let $(s_{1},s_{2})$ be a local coordinate system for $\Gamma$ such that $\mathbf {t}_{1}=\frac{{\partial \Gamma (s_{1,}s_{2})}}{\partial s_{1}}$ and $\mathbf {t}_{2}=\frac{{\partial \Gamma (s_{1,}s_{2})}}{\partial s_{2}}$ are orthogonal unit eigenvectors of $D^{2}d_{\Gamma }$ corresponding to two principle curvatures $\kappa _{1}$ and $\kappa _{2}$ respectively. Notice that $(s_{1},s_{2},\eta )$ forms curvilinear coordinates on $T_{\epsilon }$ with the coordinate transformation

$$\begin{aligned} z(s_{1},s_{2},\eta )=\Gamma (s_{1},s_{2})+\eta \mathbf {n}(s_{1},s_{2}), \end{aligned}$$

(A.3)

where $\mathbf {n}(s_{1,}s_{2})=\mathbf {n}(\Gamma (s_{1,}s_{2}))$. By using $\frac{\partial \mathbf {n}}{\partial s_{i}}=\kappa _{i}\mathbf {t}_{i}$, we obtain $\frac{\partial z}{\partial s_{1}}=(1+\eta \kappa _{1}(s_{1,}s_{2}))\mathbf {t}_{1}$, $\frac{\partial z}{\partial s_{2}}=(1+\eta \kappa _{2}(s_{1,}s_{2}))\mathbf {t_{2}}$ and $\frac{\partial z}{\partial \eta }=\mathbf {n}$. By formula of gradient in orthogonal curvilinear coordinate systems, we have

$$\begin{aligned} \nabla \overline{u}&=(1+\eta \kappa _{1})^{-1}\,{\frac{\partial \overline{u}}{\partial s_{1}}\mathbf {t_{1}}}+{(1+\eta \kappa _{2})^{-1}\, {\frac{\partial \overline{u}}{\partial s_{1}}\mathbf {t_{2}}}}+{\frac{\partial \overline{u}}{\partial \eta }}\mathbf {n}=(1+\eta \kappa _{1})^{-1}\,{\frac{\partial \overline{u}}{\partial s_{1}}\mathbf {t_{1}}}\\ {}&\quad +{(1+\eta \kappa _{2})^{-1}\, {\frac{\partial \overline{u}}{\partial s_{1}}\mathbf {t_{2}}}} \end{aligned}$$

It follows the tangential gradient of u is

$$\begin{aligned} \nabla _{\Gamma }u(P_{\Gamma }z)&=\frac{\partial \overline{u}}{\partial s_{1}}\mathbf {t_{1}}+\frac{\partial \overline{u}}{\partial s_{1}}\mathbf {t_{2}}\\&=((1+d_{\Gamma }(z)\kappa _{1}(P_{\Gamma }z))\mathbf {t}_{1}\otimes \mathbf {t}_{1}+(1+d_{\Gamma }(z)\kappa _{2}(P_{\Gamma }z))\mathbf {t}_{2}\otimes \mathbf {t}_{2})\nabla \overline{u}(z) \end{aligned}$$

By using the fact $(1+d(z)\kappa _{i}(P_{\Gamma }z))=(1-d(z)\kappa _{i}(z))^{-1}=\sigma _{i}$ and $\mathbf {n}\cdot \nabla v=0$, we prove (A.1).

First notice that $\mathbf {n}\cdot \overline{F}=0$ since $\overline{F}$ is the normal extension of F. Let $F^{1}$ and $F^{2}$ denote the components of F in $\mathbf {t_{1},}\mathbf {t_{2}}$ respectively. That is, $F(x)=F^{1}(x)\mathbf {t_{1}}+F^{2}(x)\mathbf {t_{2}}$ for $x\in \Gamma$. By formula of divergence in orthogonal curvilinear coordinate systems, we have

$$\begin{aligned} \nabla \cdot \overline{F}=(1+\eta \kappa _{1})^{-1}(1+\eta \kappa _{2})^{-1}\left( \frac{\partial ((1+\eta \kappa _{2})\overline{F^{1}})}{\partial s_{1}}+\frac{\partial ((1+\eta \kappa _{1})\overline{F^{2}})}{\partial s_{2}}\right) \end{aligned}$$

Therefore it follows

$$\begin{aligned} \overline{\nabla _{\Gamma }\cdot F}=\frac{\partial \overline{F^{1}}}{\partial s_{1}}+\frac{\partial \overline{F^{2}}}{\partial s_{2}}=(1+\eta \kappa _{1})(1+\eta \kappa _{2})\nabla \cdot (B_{0}\overline{F}). \end{aligned}$$

By $\mathbf {n}\cdot \overline{F}=0$ , we have $B_{1}(z;\nu )\overline{F}=0$. This shows (A.2) holds.

Remark

In fact, by using $\mathbf {n}\cdot \overline{F}=0$ and $\frac{\partial \overline{F^{1}}}{\partial \eta }\text {=}\frac{\partial \overline{F^{2}}}{\partial \eta }=0$, we can choose more general $B_{1}$ as following

$$\begin{aligned} B_{1}(z;\nu ):=\mathbf {\nu _{1}n}\otimes \mathbf {n}+\nu _{2}\mathbf {t_{1}}\otimes \mathbf {n}+\mathbf {\nu _{3}t_{2}}\otimes \mathbf {n}+\nu _{4}\mathbf {n}\otimes \mathbf {t_{1}}+\nu _{5}\mathbf {n}\otimes \mathbf {t_{2}}, \end{aligned}$$

where $\nu _{i}$ are arbitrary real numbers.

Remark

If $\Gamma$ is a curve in $\mathbb {R}^{2},$ then it can be shown analogously that

$$\begin{aligned} \nabla _{\Gamma }u(P_{\Gamma }z)=(\sigma ^{-1}\mathbf {t}\otimes \mathbf {t}+\mu \mathbf {n}\otimes \mathbf {n})\nabla v(z), \end{aligned}$$

(A.4)

and

$$\begin{aligned} \nabla _{\Gamma }\cdot F=\sigma ^{-1}\nabla \cdot (\mathbf {\mathbf {t}\otimes \mathbf {t}+\nu n}\otimes \mathbf {n)}\,F). \end{aligned}$$

(A.5)

1.2 Second-order finite difference approximation for normal extension of surface Laplacian

In this paper, all numerical experiments are done by second-order finite difference schemes. We present the detail about finite difference scheme to approximate the normal extension of surface Laplacian $\Delta _{\Gamma }u$. For simplicity, we demonstrate two- dimensional case and assume that $\Delta x=\Delta y=h$, but it can be easily generalized to higher-dimensional cases with nonuniform Cartesian grids. Recall that the normal extended equation for surface Laplacian is given by

$$\begin{aligned} \sigma ^{-1}\nabla \cdot A\nabla v=\sigma ^{-1}[(A^{11}v_{x})_{x}+(A^{12}v_{y})_{x}+(A^{21}v_{x})_{y}+(A^{22}v_{y})_{y}], \end{aligned}$$

where $A=\sigma ^{-1}\mathbf{t }\otimes \mathbf{t }+\mu {\mathbf{n }\otimes }{\mathbf{n }}=\left[ \begin{array}{cc} A^{11} &{}\quad A^{12}\\ A^{21} &{}\quad A^{22} \end{array}\right]$. We use central difference to approximate all terms as following

$$\begin{aligned}{}[(A^{11}v_{x})_{x}]_{ij}&=\frac{1}{h}\left( [A^{11}v_{x}]_{i+\frac{1}{2},j}-[A^{11}v_{x}]_{i-\frac{1}{2},j}\right) +O(h^{2})\\&=\frac{1}{h^{2}}\left( A_{i+\frac{1}{2},j}^{11}(v_{i+1,j}-v_{i,j})-A_{i-\frac{1}{2},j}^{11}(v_{i,j}-v_{i-1,j})\right) +O(h^{2}),\\ [(A^{12}v_{y})_{x}]_{ij}&=\frac{1}{2h}([A^{12}v_{y}]_{i+1,j}-[A^{12}v_{y}]_{i-1,j})+O(h^{2})\\&=\frac{1}{\text {4}h^{2}}(A_{i+1,j}^{12}(v_{i+1,j+1}-v_{i+1,j-1})-A_{i-1,j}^{12}(v_{i-1,j+1}-v_{i-1,j-1}))+O(h^{2}),\\ [(A^{21}v_{x})_{y}]_{ij}&=\frac{1}{2h}([A^{21}v_{x}]_{i,j+1}-[A^{21}v_{x}]_{i,j-1})+O(h^{2})\\&=\frac{1}{\text {4}h^{2}}(A_{i,j+1}^{21}(v_{i+1,j+1}-v_{i-1,j+1})-A_{i,j-1}^{12}(v_{i+1,j-1}-v_{i-1,j-1}))+O(h^{2}),\\ [(A^{22}v_{y})_{y}]_{ij}&={{\frac{1}{h}}}\left( [A^{22}v_{y}]_{i,j+\frac{1}{2}}-[A^{22}v_{y}]_{i,j-\frac{1}{2}}\right) +O(h^{2})\\&=\frac{1}{h^{2}}\left( A_{i,j+\frac{1}{2}}^{22}(v_{i,j+1}-v_{i,j})-A_{i,j-\frac{1}{2}}^{22}(v_{i,j}-v_{i,j-1})\right) +O(h^{2}). \end{aligned}$$

If $v_{l,k}$ is taking value at a ghost node, replace $v_{l,k}$ by linear combination of other $v_{i,j}$ at inner nodes as discussed in Sect. 2.4.