Numerical Approximation of Eigenvalue Problems by Adaptive Finite Element Methods

Abstract

In this article we present an algorithm for the approximation through adaptive finite element methods of solutions to second order elliptic eigenvalue problems, considering Lagrange finite elements of any degree. We show the convergence of the algorithm 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. Finally, we discuss briefly the quasi-optimality of the adaptive method and conclude with some numerical experiments that illustrate the advantages of adaptivity and the relationship between order of convergence and regularity.