Spurious solutions for high order curl problems

TL;DR

This paper examines high order curl problems, revealing that classical finite elements often produce spurious solutions, while mixed formulations aligned with cohomological structures provide correct solutions.

Contribution

It demonstrates the limitations of classical conforming finite elements and advocates for mixed formulations based on complexes to accurately solve high order curl problems.

Findings

Classical conforming finite elements lead to spurious solutions

Mixed formulations in complexes solve high order curl problems correctly

Cohomological structures explain numerical behaviors

Abstract

We investigate numerical solutions of high order curl problems with various formulations and finite elements. We show that several classical conforming finite elements lead to spurious solutions, while mixed formulations with finite elements in complexes solve the problems correctly. To explain the numerical results, we clarify the cohomological structures in high order curl problems by relating the partial differential equations to the Hodge-Laplacian boundary problems of the gradcurl-complexes.