Finite-Element Domain Approximation for Maxwell Variational Problems on Curved Domains

TL;DR

This paper investigates how to accurately approximate curved domains in finite element methods for Maxwell equations, establishing conditions that guarantee convergence of solutions despite domain approximation errors.

Contribution

It introduces conditions on domain approximation quality that ensure convergence rates for finite element solutions on curved domains.

Findings

Derived error convergence rates for domain approximations.

Established conditions ensuring solution accuracy despite mesh inexactness.

Provided theoretical foundations for finite element domain approximation in Maxwell problems.

Abstract

We consider the problem of domain approximation in finite element methods for Maxwell equations on curved domains, i.e., when affine or polynomial meshes fail to exactly cover the domain of interest. In such cases, one is forced to approximate the domain by a sequence of polyhedral domains arising from inexact meshes. We deduce conditions on the quality of these approximations that ensure rates of error convergence between discrete solutions -- in the approximate domains -- to the continuous one in the original domain.