The Closest Point Method for Surface PDEs with General Boundary Conditions

TL;DR

This paper extends the closest point method to handle surface PDEs with diverse boundary conditions, improving its applicability and robustness for complex problems.

Contribution

It introduces a unified extrapolation framework for inhomogeneous boundary conditions within the CPM, enhancing its versatility.

Findings

Demonstrates high accuracy through numerical convergence studies

Shows robustness in solving elliptic, Steklov eigenvalue, and nonlinear problems

Validates the method's effectiveness across various PDE types

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.