The Closest Point Method for Surface PDEs with General Boundary Conditions

TL;DR

The paper advances the Closest Point Method by enabling surface PDEs on open surfaces with general boundary conditions through a novel extrapolation mechanism for $\partial_n u|_{\bm{y}} = j(\bm{y},u(\bm{y}))$, integrated into an embedding framework with a penalty to enforce the extension. The resulting scheme achieves $O(\Delta x^3)$ boundary error and second-order overall accuracy, as validated on Poisson and Steklov problems and demonstrated on a Gray–Scott system on curved geometries. The approach is tested on a unit hemisphere and a Möbius strip, including a generalized Steklov eigenproblem and Robin-type boundary effects, showing robust convergence and capacity to capture boundary-driven dynamics. Overall, this work broadens CPM’s applicability to open surface PDEs with complex BCs, offering a practical, accurate tool for simulations on embedded manifolds in applications such as fluid interfaces and membrane modeling.

Abstract

We generalize the closest point method (CPM) to solve surface partial differential equations with general boundary conditions. The proposed extrapolation method provides a unified framework for treating a broad class of inhomogeneous Neumann and Robin boundary conditions within the framework of CPM. The accuracy and robustness of the method are demonstrated through numerical convergence studies of an elliptic problem, Steklov eigenvalue problems, and a nonlinear reaction-diffusion system.

Figures (3)

• Figure 1:
• Figure 2: Steklov eigenfunctions for different geometries. Top: Unit hemisphere, a selection of the Steklov eigenfunctions, numerically computed with mesh size $\Delta x = 0.025$ ($89,989$ grid points). The rightmost plot shows the second-order accuracy of these selected eigenvalues. Bottom: Möbius strip, a selection of the Steklov eigenfunctions, numerically computed with mesh size $\Delta x = 0.025$ ($72,062$ grid points).
• Figure 3: Gray-Scott model with Robin boundary condition. Numerical solution of the substrate $u$ on the surface of the Möbius strip at different time points. Top row: $\kappa=0$ (homogeneous Neumann condition), with total simulation time $T=4000$. Bottom row: $\kappa=10$, $T=10000$.

Theorems & Definitions (8)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof
• Remark 1
• Remark 2