Adaptive Multilevel Methods for the Maxwell Eigenvalue Problem

TL;DR

This paper introduces an adaptive multilevel preconditioned method for efficiently solving Maxwell eigenvalue problems with singularities, demonstrating mesh-independent convergence and confirmed by numerical experiments.

Contribution

It presents a novel adaptive multilevel preconditioning approach for Maxwell eigenvalue problems, achieving quasi-optimal convergence independent of mesh sizes.

Findings

Convergence factor is independent of mesh sizes and levels.

Numerical experiments confirm theoretical convergence and efficiency.

Method effectively handles complex domain singularities.

Abstract

In this paper, we propose an adaptive multilevel preconditioned Helmholtz-Jacobi-Davidson (PHJD) method for the Maxwell eigenvalue problem with singularities. The key idea in this work is to employ the local multilevel method for preconditioning the Jacobi-Davidson correction equation. It is shown that our convergence factor is quasi-optimal, which means the convergence factor is independent of mesh sizes and mesh levels provided the coarse mesh is sufficiently fine. Numerical experiments on complex domains are carried out to confirm the theoretical results and demonstrate the efficiency of the proposed method.