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Local Multilevel Preconditioned Jacobi-Davidson Method for Elliptic Eigenvalue Problems on Adaptive Meshes

Jianing Guo, Qigang Liang, Xuejun Xu

TL;DR

The paper tackles computing the principal eigenpair of elliptic operators with singularities on adaptively refined meshes. It develops an adaptive multilevel preconditioned Jacobi-Davidson (PJD) method with a local smoothing strategy to solve the JD correction equations efficiently. Theoretical results establish a uniform convergence rate independent of mesh level and degrees of freedom, with robustness to highly discontinuous coefficients, and numerical experiments confirm $O(N)$ runtime and stable iteration counts across challenging test cases. This yields a scalable, accurate solver for AFEM-based eigenvalue problems in domains with singular behavior.

Abstract

In this work, we propose an efficient adaptive multilevel preconditioned Jacobi-Davidson (PJD) method for eigenvalue problems with singularity. Our multilevel method utilizes a local smoothing strategy to solve the preconditioned Jacobi-Davidson algebraic systems arising from adaptive finite element methods (AFEM). As a result, the algorithm holds optimal computational complexity $O(N)$. The theoretical analysis reveals that our method has a uniform convergence rate with respect to mesh levels and degrees of freedom. Further, the convergence rate is not affected by highly discontinuous coefficients within the domain. Numerical results verify our theoretical findings.

Local Multilevel Preconditioned Jacobi-Davidson Method for Elliptic Eigenvalue Problems on Adaptive Meshes

TL;DR

The paper tackles computing the principal eigenpair of elliptic operators with singularities on adaptively refined meshes. It develops an adaptive multilevel preconditioned Jacobi-Davidson (PJD) method with a local smoothing strategy to solve the JD correction equations efficiently. Theoretical results establish a uniform convergence rate independent of mesh level and degrees of freedom, with robustness to highly discontinuous coefficients, and numerical experiments confirm runtime and stable iteration counts across challenging test cases. This yields a scalable, accurate solver for AFEM-based eigenvalue problems in domains with singular behavior.

Abstract

In this work, we propose an efficient adaptive multilevel preconditioned Jacobi-Davidson (PJD) method for eigenvalue problems with singularity. Our multilevel method utilizes a local smoothing strategy to solve the preconditioned Jacobi-Davidson algebraic systems arising from adaptive finite element methods (AFEM). As a result, the algorithm holds optimal computational complexity . The theoretical analysis reveals that our method has a uniform convergence rate with respect to mesh levels and degrees of freedom. Further, the convergence rate is not affected by highly discontinuous coefficients within the domain. Numerical results verify our theoretical findings.

Paper Structure

This paper contains 12 sections, 11 theorems, 99 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model problem
  3. The adaptive multilevel PJD method
  4. Preliminaries on multilevel subspaces
  5. The main convergent result
  6. The estimate of $R_1^{j}$
  7. The estimate of $R_2^{j}$
  8. Numerical experiments
  9. L-shaped domain
  10. Crack domain
  11. Highly discontinuous coefficients eigenvalue problem
  12. Conclusion

Key Result

Lemma 1

(see babuvska1989finiteMR2652780). For ($\lambda_{1},u_{1}$), there exists a discrete eigenpair $(\lambda_{1,0},u_{1,0})$ on $V_{0}$ such that where the constant $C$ depends on the gap of $\lambda_1$.

Figures (11)

  • Figure 1: $\widetilde{\mathcal{M}}_l\text{ consists of the big dots in the right figure}.$
  • Figure 2: The local refined mesh on adaptive level 12
  • Figure 3: A posteriori estimator
  • Figure 4: CPU time
  • Figure 5: The local refined mesh on adaptive level 12
  • ...and 6 more figures

Theorems & Definitions (22)

  • Lemma 1
  • Lemma 2
  • Lemma 3: The stability of decomposition
  • proof
  • Lemma 4: Strengthened Cauchy-Schwarz inequality
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 12 more