Convergence of adaptive finite element methods for eigenvalue problems

TL;DR

This paper proves that adaptive finite element methods reliably converge when solving second order elliptic eigenvalue problems, regardless of initial conditions or marking strategies, for various eigenvalue multiplicities.

Contribution

It establishes convergence of adaptive finite element methods for elliptic eigenvalue problems with arbitrary degree Lagrange elements, covering simple and multiple eigenvalues, under minimal refinement strategies.

Findings

Proves convergence for simple and multiple eigenvalues.

Works with any degree of Lagrange finite elements.

Applies to all reasonable marking strategies and initial triangulations.

Abstract

In this article we prove convergence of adaptive finite element methods for second order elliptic eigenvalue problems. We consider Lagrange finite elements of any degree and prove convergence for simple as well as multiple eigenvalues under a minimal refinement of marked elements, for all reasonable marking strategies, and starting from any initial triangulation.