Adaptive FEM with optimal convergence rate for non-self-adjoint eigenvalue problems
This provides incremental improvements for computational mathematicians and engineers solving complex eigenvalue problems in fields like quantum mechanics or structural analysis.
The paper tackles non-self-adjoint eigenvalue problems by developing adaptive finite element methods with new theoretical and computable error estimators for multiple and clustered eigenvalues, proving their equivalence and demonstrating optimal convergence rates through numerical experiments.
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.