Mesh Sensitivity Analysis for Finite Element Solution of Linear Elliptic Partial Differential Equations

TL;DR

This paper provides a comprehensive analysis of how finite element solutions for linear elliptic PDEs are affected by mesh deformations, establishing bounds that demonstrate solution stability across various mesh types and dimensions.

Contribution

It introduces a general bound on the sensitivity of finite element solutions to mesh deformations applicable to any dimension and unstructured meshes.

Findings

Finite element solutions change continuously with mesh deformation.

The derived bounds are valid for arbitrary unstructured simplicial meshes.

Results apply to general linear elliptic PDEs and finite element methods.

Abstract

Mesh sensitivity of finite element solution for linear elliptic partial differential equations is analyzed. A bound for the change in the finite element solution is obtained in terms of the mesh deformation and its gradient. The bound shows how the finite element solution changes continuously with the mesh. The result holds in any dimension and for arbitrary unstructured simplicial meshes, general linear elliptic partial differential equations, and general finite element approximations.