Convergence Rate and Quasi-Optimal Complexity of Adaptive Finite Element Computations for Multiple Eigenvalues

TL;DR

This paper establishes the convergence rate and quasi-optimal complexity of an adaptive finite element method tailored for multiple eigenvalue problems in second order elliptic equations, extending previous results to more complex eigenvalue scenarios.

Contribution

It introduces eigenspace approximation techniques and extends existing convergence and complexity results to handle multiple eigenvalues in elliptic PDEs.

Findings

Proves convergence rate for adaptive finite element eigenvalue approximation.

Establishes quasi-optimal complexity bounds.

Extends prior work to multiple eigenvalue problems.

Abstract

In this paper, we study an adaptive finite element method for multiple eigenvalue problems of a class of second order elliptic equations. By using some eigenspace approximation technology and its crucial property which is also presented in this paper, we extend the results in \cite{dai-xu-zhou08} to multiple eigenvalue problems, we obtain both convergence rate and quasi-optimal complexity of the adaptive finite element eigenvalue approximation.