# Analysis of the finite element method for the Laplace--Beltrami equation on surfaces with regions of high curvature using graded meshes

## TL;DR

This paper derives error estimates for finite element approximations of the Laplace--Beltrami equation on curved surfaces, demonstrating that graded meshes can effectively handle high curvature regions without increased error.

## Contribution

It introduces a novel analysis showing error estimates are independent of high curvature regions when using graded meshes for finite element methods.

## Key findings

- Error estimates are independent of high curvature regions.
- Graded meshes improve approximation accuracy on curved surfaces.
- Numerical experiments confirm theoretical results.

## Abstract

We derive error estimates for the piecewise linear finite element approximation of the Laplace--Beltrami operator on a bounded, orientable, $C^3$, surface without boundary on general shape regular meshes. As an application, we consider a problem where the domain is split into two regions: one which has relatively high curvature and one that has low curvature. Using a graded mesh we prove error estimates that do not depend on the curvature on the high curvature region. Numerical experiments are provided.