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Adaptive Multilevel Methods for the Maxwell Eigenvalue Problem

Qigang Liang, Xuejun Xu, Qingquan Zhang

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.

Adaptive Multilevel Methods for the Maxwell Eigenvalue Problem

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.

Paper Structure

This paper contains 9 sections, 4 theorems, 63 equations, 15 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model problems and preliminaries
  3. The adaptive multilevel PHJD method
  4. Local multilevel preconditioner
  5. Algorithm
  6. Convergence analysis
  7. A rigorous proof of the convergence result.
  8. Numerical results
  9. The proof of Lemma \ref{['lem:4_10']}

Key Result

Theorem 2.1

Let $(\lambda_{1,l},\bm{u}_{1,l})$$(0\le l\le L)$ be the first eigenpair of eq:2_3 with $\|\bm{u}_{1,l}\|_{\bm{\mathrm{curl}}} = 1\ (\|\bm{u}_{1,l}\|_b = 1)$. Then there exists an eigenpair $(\lambda_{1},\bm{u}_{1})$$(\bm{u}_{1} \in W_1)$ with $\|\bm{u}_1\|_{\bm{\mathrm{curl}}} = 1\ (\|\bm{u}_{1}\| where $\delta_l(\lambda_1)$ denotes the gap between $W_1$ and $W_1^l$, namely

Figures (15)

  • Figure 1: The local refined mesh on adaptive levels 1 and 15
  • Figure 2: Reduction of a posteriori error
  • Figure 3: Increasing of CPU time
  • Figure 4: The local refined mesh on adaptive levels 1 and 15
  • Figure 5: Reduction of a posteriori error
  • ...and 10 more figures

Theorems & Definitions (18)

  • Remark 2.1
  • Theorem 2.1
  • Remark 2.2
  • Remark 3.1
  • Definition 3.1: Local multilevel preconditioner $B_L^j$
  • Remark 3.2
  • Remark 3.3
  • Theorem 4.1
  • Remark 4.1
  • Lemma 4.1
  • ...and 8 more