Year of Publication: 2016
Authors: C. Conrads; V. Mehrmann; A. Miedlar

We discuss adaptive numerical methods for the solution of eigenvalue
problems arising either from the finite element discretization of a partial
differential equation (PDE) or from discrete finite element modeling. When a
model is described by a partial differential equation, the adaptive finite element
method starts from a coarse finite element mesh which, based on a posteriori
error estimators, is adaptively refined to obtain eigenvalue/eigenfunction approximations
of prescribed accuracy. This method is well established for classes
of elliptic PDEs, but is still in its infancy for more complicated PDE models.
For complex technical systems, the typical approach is to directly derive finite
element models that are discrete in space and are combined with macroscopic
models to describe certain phenomena like damping or friction. In this case
one typically starts with a fine uniform mesh and computes eigenvalues and
eigenfunctions using projection methods from numerical linear algebra that
are often combined with the algebraic multilevel substructuring to achieve an
adequate performance. These methods work well in practice but their convergence
and error analysis is rather difficult. We analyze the relationship
between these two extreme approaches. Both approaches have their pros and
cons which are discussed in detail. Our observations are demonstrated with
several numerical examples.

Corporate Authors: C. Conrads; V. Mehrmann; A. Miedlar
Volume: 658
Publication Language: eng
Start Page: 197
Type of Publication: Journal Article