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- eigenvalue problem (3)
- adaptive finite element method (2)
- adaptive finite element method (AFEM) (2)
- AFEM (1)
- AMLS (1)
- Krylov subspace method (1)
- a posteriori error estimates (1)
- balanced AFEM algorithm (1)
- convection-diffusion-reaction operator (1)
- eigenfunction (1)
- eigenvalue (1)
- eigenvalue problems (1)
- elliptic eigenvalue problem (1)
- finite element method (1)
- finite element method (FEM) (1)
- functional backward error (1)
- functional condition number (1)
- homotopy (1)
- hp-adaptive mesh refinement (1)
- non-self-adjoint operator (1)
- partial differential equation (1)
- saturation (1)
- self-adjoint eigenvalue problem (1)
This paper presents three different adaptive algorithms for eigenvalue problems
associated with non-selfadjoint partial differential
operators. The basis for the developed algorithms is a homotopy method.
The homotopy method starts from a well-understood selfadjoint problem,
for which well-established adaptive methods are available.
Apart from the adaptive grid refinement, the progress of the homotopy as
well as the solution of the iterative method are adapted to balance the contributions
of the different error sources.
The first algorithm balances the homotopy, discretization and approximation errors with respect
to a fixed step-size $\tau$ in the homotopy.
The second algorithm combines the adaptive step-size control for the homotopy
with an adaptation in space that ensures an error below a fixed tolerance $\varepsilon$.
The third algorithm allows the complete adaptivity in space,
homotopy step-size as well as the iterative algebraic eigenvalue solver.
All three algorithms are compared in numerical examples.
We introduce functional perturbation results for PDE eigenvalue problems including the functional backward error
and the functional condition number. These results are used to establish a combined a posteriori error estimator embodying the discretization and the approximation error for the simple eigenpair.
Based on known perturbation results in $H^{1}(\Omega)$ and $H^{-1}(\Omega)$ norms and a standard residual a posteriori error estimator, a balancing AFEM algorithm is proposed. The stopping criterion for the eigensolver is based on the equilibrating strategy, i.e., iterations proceed as long as the discrete part of the error estimator dominates the continuous part. All our statements are illustrated with several numerical examples.
This paper discusses adaptive finite element methods (AFEMs) for the solution of elliptic eigenvalue problems associated with partial differential operators. An adaptive method based on nodal-patch refinement leads to an asymptotic error reduction property for the computed sequence of simple eigenvalues and eigenfunctions. This justifies the use of the proven saturation property for a class of reliable and efficient hierarchical a posteriori error estimators. Numerical experiments confirm that the saturation property is present even for very coarse meshes for many examples; in other cases the smallness assumption on the initial mesh may be severe.
We consider a new adaptive finite element (AFEM) algorithm for elliptic PDE-eigenvalue problems.
In contrast to other approaches we incorporate the iterative solution of the resulting finite dimensional
algebraic eigenvalue problems into the adaptation process.
In this way we can balance the costs of the adaption process
for 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 for the adaptation process it is possible to restrict to
very few iterations
of a Krylov subspace solver for the eigenvalue problem on coarse meshes.
We present several examples and 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.
We present new residual estimates based on Kato's square root theorem for spectral approximations of diagonalizable non-self-adjoint differential operators of convection-diffusion-reaction type. These estimates are incorporated as part of an hp-adaptive finite element algorithm for practical spectral computations, where it is shown that the
resulting a posteriori error estimates are reliable. Provided experiments demonstrate the efficiency and reliability of our approach.
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.