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

TL;DR

This paper develops an adaptive finite element method with proven optimal convergence rates for non-self-adjoint eigenvalue problems, introducing new error estimators and algorithms validated by numerical experiments.

Contribution

It introduces new theoretical and computable error estimators for eigenvalues and clusters, and proposes an adaptive algorithm with optimal convergence for non-self-adjoint problems.

Findings

Proved equivalence of theoretical and computable error estimators.

Demonstrated optimal convergence rate of the adaptive algorithm.

Validated theoretical results with numerical experiments.

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.