# Multigrid methods based on shifted inverse iteration for the Maxwell eigenvalue problem

## TL;DR

This paper develops multigrid methods based on shifted inverse iteration for solving the Maxwell eigenvalue problem with discontinuous material properties, providing convergence proofs and optimal error estimates validated by numerical experiments.

## Contribution

It introduces two multigrid algorithms tailored for Maxwell eigenvalue problems with discontinuous coefficients, with proven convergence and optimal error bounds.

## Key findings

- Proved uniform convergence of the discrete solution operator.
- Established asymptotically optimal error estimates.
- Numerical experiments confirm theoretical results.

## Abstract

In this paper two types of multgrid methods, i.e., the Rayleigh quotient iteration and the inverse iteration with fixed shift, are developed for solving the Maxwell eigenvalue problem with discontinuous relative magnetic permeability and electric permittivity. With the aid of the mixed form of source problem associated with the eigenvalue problem, we prove the uniform convergence of the discrete solution operator to the solution operator in $L^2(\Omega)$ using discrete compactness of edge element space. Then we prove the asymptotically optimal error estimates for both multigrid methods. Numerical experiments confirm our theoretical analysis.