Robust mixed finite element methods for a quad-curl singular perturbation problem

TL;DR

This paper develops robust mixed finite element methods for a quad-curl singular perturbation problem, providing uniform error estimates and optimal convergence rates, verified through numerical experiments.

Contribution

It introduces lower order H(grad curl)-nonconforming finite elements extended to nonconforming finite element Stokes complexes for the first time.

Findings

Achieved sharp, uniform error estimates with respect to the perturbation parameter.

Demonstrated optimal convergence rates using Nitsche's technique in boundary layer cases.

Numerical results confirm the theoretical convergence rates.

Abstract

Robust mixed finite element methods are developed for a quad-curl singular perturbation problem. Lower order H(grad curl)-nonconforming but H(curl)-conforming finite elements are constructed, which are extended to nonconforming finite element Stokes complexes and the associated commutative diagrams. Then H(grad curl)-nonconforming finite elements are employed to discretize the quad-curl singular perturbation problem, which possess the sharp and uniform error estimates with respect to the perturbation parameter. The Nitsche's technique is exploited to achieve the optimal convergence rate in the case of the boundary layers. Numerical results are provided to verify the theoretical convergence rates. In addition, the regularity of the quad-curl singular perturbation problem is established.