Finite Element Approximation of the Laplace-Beltrami Operator on a Surface with Boundary

TL;DR

This paper presents a finite element method for solving the Laplace-Beltrami operator on surfaces with boundaries, incorporating boundary conditions via Nitsche's method, and provides optimal error estimates.

Contribution

It introduces a novel finite element approach for surface Laplace-Beltrami problems with boundary conditions, including rigorous error analysis.

Findings

Optimal order error estimates in energy and L2 norms

Method effectively handles nonhomogeneous Dirichlet boundary conditions

Applicable to triangulated surfaces with boundary

Abstract

We develop a finite element method for the Laplace-Beltrami operator on a surface with boundary and nonhomogeneous Dirichlet boundary conditions. The method is based on a triangulation of the surface and the boundary conditions are enforced weakly using Nitsche's method. We prove optimal order a priori error estimates for piecewise continuous polynomials of order $k \geq 1$ in the energy and $L^2$ norms that take the approximation of the surface and the boundary into account.