# Adaptive numerical solution of PDE eigenvalue problems

**Volker Mehrmann
**

Matheon

Institut für Mathematik

Technische Universität Berlin

Berlin, Germany

### Abstract:

We consider a new adaptive finite element (AFEM) algorithm for self-adjoint elliptic PDE eigenvalue problems. In contrast to other approaches we incorporate the inexact solutions of the resulting finite dimensional algebraic eigenvalue problems into the adaptation process. In this way we can balance the costs of the adaptive refinement of the mesh with the costs for the iterative eigenvalue method. We present error estimates that incorporate the discretization errors, approximation errors in the eigenvalue solver and roundoff errors, and use these for the adaptation process. We show that it is also possible to restrict to very few iterations of a Krylov subspace solver for the eigenvalue problem on coarse meshes. Several examples are presented to show that this new approach achieves much better complexity than previous AFEM approaches which assume that the algebraic eigenvalue problem is solved to full accuracy. Finally we will also discuss non-self-adjoint convection diffusion problems. We will introduce homotopy methods that adapt in three ways, the PDE mesh, the iterative solver and the homotopy stepping.

This is joint work with Agnieska Miedlar (and partly with Joscha Gedicke and Carsten Carstensen)