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Adaptive Numerical Solution of Eigenvalue Problems arising from Finite Element Models. AMLS vs. AFEM
(2015)
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.