$L^{\infty}$-error estimate for the finite element method on two dimensional surfaces

TL;DR

This paper establishes an $L^{ abla}$-error estimate of order $O(h^2 | ext{log} h|)$ for finite element solutions of a PDE on a 2D surface, extending Euclidean error analysis to curved geometries.

Contribution

It provides the first $L^{ abla}$-error estimate for finite element methods on surfaces with flat triangles, matching Euclidean error bounds.

Findings

$L^{ abla}$-error is $O(h^2 | ext{log} h|)$ on surfaces.

Error bounds are comparable to Euclidean cases.

Analysis applies to triangulations with flat triangles.

Abstract

We approximate the solution of the equation $$ -\Delta_S u+u = f $$ on a two-dimensional, embedded, orientable, closed surface $S$ where $-\Delta_S$ denotes the Laplace Beltrami operator on $S$ by using continuous, piecewise linear finite elements on a triangulation of $S$ with flat triangles. We show that the $L^{\infty}$-error is of order $O(h^2|\log h|)$ as in the corresponding situation in an Euclidean setting.