Virtual Element Method for the Laplace-Beltrami equation on surfaces

TL;DR

This paper introduces a high-order Virtual Element Method (VEM) for solving the Laplace-Beltrami equation on surfaces, effectively handling arbitrary polygonal meshes and nonconforming mesh configurations with proven error estimates.

Contribution

It combines VEM with Surface Finite Element Methods to extend surface PDE solutions to complex, nonconforming meshes with rigorous error analysis and practical numerical validation.

Findings

Optimal H^1 error estimate for linear VEM

Effective handling of nonconforming mesh pasting

Numerical experiments confirm convergence and applicability

Abstract

We present and analyze a Virtual Element Method (VEM) of arbitrary polynomial order $k\in\mathbb{N}$ for the Laplace-Beltrami equation on a surface in $\mathbb{R}^3$. The method combines the Surface Finite Element Method (SFEM) [Dziuk, Elliott, \emph{Finite element methods for surface PDEs}, 2013] and the recent VEM [Beirao da Veiga et al, \emph{Basic principles of Virtual Element Methods}, 2013] in order to handle arbitrary polygonal and/or nonconforming meshes. We account for the error arising from the geometry approximation and extend to surfaces the error estimates for the interpolation and projection in the virtual element function space. In the case $k=1$ of linear Virtual Elements, we prove an optimal $H^1$ error estimate for the numerical method. The presented method has the capability of handling the typically nonconforming meshes that arise when two ore more meshes are pasted along a straight line. Numerical experiments are provided to confirm the convergence result and to show an application of mesh pasting.