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Adaptive FEM with optimal convergence rate for non-self-adjoint eigenvalue problems

Shixi Wang, Hai Bi, Yidu Yang

Abstract

In this paper, we first discuss the optimal convergence of the adaptive finite element methods for non-self-adjoint eigenvalue problems. We present new theoretical error estimators and computable error estimators for multiple and clustered eigenvalues with the help of the error estimators of finite element solutions for the corresponding source problems, and prove the equivalence between these two estimators. We propose an adaptive algorithm for the eigenvalue cluster and demonstrate that it achieves the optimal convergence rate.We also provide numerical experiments to support our theoretical findings.

Adaptive FEM with optimal convergence rate for non-self-adjoint eigenvalue problems

Abstract

In this paper, we first discuss the optimal convergence of the adaptive finite element methods for non-self-adjoint eigenvalue problems. We present new theoretical error estimators and computable error estimators for multiple and clustered eigenvalues with the help of the error estimators of finite element solutions for the corresponding source problems, and prove the equivalence between these two estimators. We propose an adaptive algorithm for the eigenvalue cluster and demonstrate that it achieves the optimal convergence rate.We also provide numerical experiments to support our theoretical findings.
Paper Structure (6 sections, 10 theorems, 100 equations, 5 figures)

This paper contains 6 sections, 10 theorems, 100 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Computable and theoretical error estimators
  4. Optimal convergence of the adaptive scheme
  5. Numerical experiments
  6. Conclusion comments

Key Result

Lemma 2.1

Assume that the condition (C1) or (C2) is valid. Then when the initial mesh size $h_0$ is small enough, there hold When the eigenvalue cluster $\{\lambda_{j}\}_{j\in J}$ is composed of non-defective eigenvalues, i.e., eigenvalues for which the algebraic multiplicity equals the geometric multiplicity, then for $k\in J$,

Figures (5)

  • Figure 1: The initial mesh (left) and adaptively refined mesh by using the $P_{3}$ element (right) with the bulk parameter $\theta=0.5$.
  • Figure 2: The global error estimators $\sum_{j\in J}(\eta_{\ell}(u_{j,\ell}, \mathcal{T}_\ell)^2 + \eta_{\ell}^*(u^*_{j,\ell}, \mathcal{T}_\ell)^2)$ for the clustered eigenvalues $\{\lambda_{j,\ell}\}_{j\in J}$ ($n=0$ and $N=12$) by using the $P_1$ element (top left), the $P_2$ element (top right), and the $P_3$ element (bottom), respectively.
  • Figure 3: The error curves of the clustered eigenvalues $\{\lambda_{j,\ell}\}_{j\in J}$ ($n=0$ and $N=12$) on adaptively refined meshes obtained by the $P_1$ element when the bulk parameters $\theta=0.25$ (top left), $\theta=0.5$ (top right), $\theta=0.75$ (bottom left), and $\theta=1$ (bottom right), respectively.
  • Figure 4: The error curves of the clustered eigenvalues $\{\lambda_{j,\ell}\}_{j\in J}$ ($n=0$ and $N=12$) on adaptively refined meshes obtained by the $P_2$ element when the bulk parameters $\theta=0.25$ (top left), $\theta=0.5$ (top right), $\theta=0.75$ (bottom left), and $\theta=1$ (bottom right), respectively.
  • Figure 5: The error curves of the clustered eigenvalues $\{\lambda_{j,\ell}\}_{j\in J}$ ($n=0$ and $N=12$) on adaptively refined meshes obtained by the $P_3$ element when the bulk parameters $\theta=0.25$ (top left), $\theta=0.5$ (top right), $\theta=0.75$ (bottom left), and $\theta=1$ (bottom right), respectively.

Theorems & Definitions (18)

  • Remark 2.1
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.2
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Lemma 4.1
  • Lemma 4.2
  • ...and 8 more