$\bf H(\mathrm{curl}^2)$ conforming element for Maxwell's transmission eigenvalue problem using fixed-point approach

TL;DR

This paper introduces ${\bf H}(\mathrm{curl}^2)$ conforming elements to solve Maxwell's transmission eigenvalue problem, providing optimal error estimates for eigenvalues and eigenfunctions in relevant norms.

Contribution

It develops a new ${\bf H}(\mathrm{curl}^2)$ conforming element and establishes optimal error estimates for numerical solutions of Maxwell's transmission eigenvalue problem.

Findings

Successfully handles real and complex eigenvalues.

Provides optimal error estimates in ${\bf H}(\mathrm{curl}^2)$-norm and ${\bf H}(\mathrm{curl})$-semi-norm.

Demonstrates effectiveness of the fixed-point approach.

Abstract

Using newly developed ${\bf H}(\mathrm{curl}^2)$ conforming elements, we solve the Maxwell's transmission eigenvalue problem. Both real and complex eigenvalues are considered. Based on the fixed-point weak formulation with reasonable assumptions, the optimal error estimates for numerical eigenvalues and eigenfunctions (in the ${\bf H}(\mathrm{curl}^2)$-norm and ${\bf H}(\mathrm{curl})$-semi-norm) are established.