# Spectral theory and Galerkin approximations

**Christian Engström
**

UMIT Research Lab

Department of Mathematics and Mathematical Statistics

Umeå University

### Abstract:

Block operator matrices are used to study coupled systems of partial differential equations and problems with a nonlinear dependence of the spectral parameter. In this talk, I focus on Galerkin approximations of rational operator functions and block operator matrices. Simple examples illustrate the principle difficulties and show that a Galerkin approximation can produce results that are completely wrong.