The effect of quadrature rules on finite element solutions of Maxwell variational problems. Consistency estimates on meshes with straight and curved elements

TL;DR

This paper analyzes how different numerical quadrature rules affect the error convergence in finite element solutions of Maxwell variational problems, providing comprehensive error estimates for both polygonal and curved domains.

Contribution

It offers a complete a priori error analysis including conditions on mesh refinement and quadrature precision to ensure optimal convergence rates, especially on curved domains.

Findings

Quadrature rules significantly influence error convergence rates.

Sufficient conditions for mesh and quadrature precision are established.

Error contributions from numerical quadrature are isolated on curved domains.

Abstract

We study the effects of numerical quadrature rules on error convergence rates when solving Maxwell-type variational problems via the curl-conforming or edge finite element method. A complete {\em a priori} error analysis for the case of bounded polygonal and curved domains with non-homogeneous coefficients is provided. We detail sufficient conditions with respect to mesh refinement and precision for the quadrature rules so as to guarantee convergence rates following that of exact numerical integration. On curved domains, we isolate the error contribution to numerical quadrature rules.