65N25 Eigenvalue problems
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- eigenvalue (4)
- adaptive finite element method (3)
- eigenvalue problem (3)
- AFEM (2)
- adaptive finite element method (AFEM) (2)
- eigenvalue problems (2)
- finite element method (2)
- AMLS (1)
- DWR (1)
- Krylov subspace method (1)
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.
In this paper we present an efficient algorithm for the calculation of photonic crystal band structures and band structures of photonic crystal waveguides. Our method relies on the fact that the dispersion curves of the band structure are smooth functions of the quasi-momentum in the one-dimensional Brillouin zone. We show the derivation and computation of the group velocity, the group velocity dispersion, and any higher derivative of the dispersion curves. These derivatives are then employed in a Taylor expansion of the dispersion curves. We control the error of the Taylor expansion with the help of a residual estimate and introduce an adaptive scheme for the selection of nodes in the one-dimensional Brillouin zone at which we solve the underlying eigenvalue problem and compute the derivatives of the dispersion curves. The proposed algorithm is not only advantageous as it decreases the computational effort to compute the band structure but also because it allows
for the identification of crossings and anti-crossings of dispersion curves, respectively. This identification is not possible with the standard approach of solving the underlying eigenvalue problem at a discrete set of values of the quasi-momentum without taking the mode parity into account.
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.
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 discuss the possibility of computing eigenpairs of some prototypical linear second-order self-adjoint elliptic partial differential operator (or its high-resolution finite element discretization) by numerical upscaling techniques. We compute a low-dimensional generalized finite element space that preserves small eigenvalues in a superconvergent way. The approximate eigenpairs are then obtained by solving the corresponding low-dimensional algebraic eigenvalue problem. The rigorous error bounds are based on two-scale decompositions of H1 by means of a certain Clement-type quasi-interpolation operator.
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 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.
This paper presents a combined adaptive finite element method with an iterative algebraic eigenvalue solver for the Laplace eigenvalue problem of quasi-optimal computational complexity. The analysis is based on a direct approach for eigenvalue problems and allows the use of higher order conforming finite element spaces with fixed polynomial degree k>0. The optimal adaptive finite element eigenvalue solver (AFEMES) involves a proper termination criterion for the algebraic eigenvalue solver and does not need any coarsening. Numerical evidence illustrates the optimal computational complexity.
A posteriori error estimators for non-symmetric eigenvalue model problems are discussed in [Heuveline and Rannacher, A posteriori error control for finite element approximations of elliptic eigenvalue problems, 2001] in the context of the dual-weighted residual method (DWR). This paper directly analyses the variational formulation rather than the non-linear ansatz of Becker and Rannacher for some convection-diffusion model problem and presents error estimators for the eigenvalue error based on averaging techniques. In the case of linear P1 finite elements and globally constant coefficients, the error estimates of the residual and averaging error estimators are refined. Moreover, several postprocessing techniques attached to the DWR paradigm plus two new dual-weighted error estimators are compared in numerical experiments. The first new estimator utilises an auxiliary Raviart-Thomas mixed finite element method and the second exploits an averaging technique in combination with ideas of DWR.
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.
A refined a posteriori error analysis for symmetric eigenvalue problems and the convergence of the first-order adaptive finite element method (AFEM) is presented. The $H^1$ stability of the $L^2$ projection provides reliability and efficiency of the edge-contribution of standard residual-based error estimators for $P_1$ finite element methods. In fact, the volume contributions and even oscillations can be omitted for Courant finite element methods. This allows for a refined averaging scheme and so improves [Dong Mao, Lihua Shen and Aihui Zhou, Adaptive finite element algorithms for eigenvalue problems based on local averaging type a posteriori error estimates, Advanced in Computational Mathematics, 2006, 25: 135-160]. The proposed AFEM
monitors the edge-contributions in a bulk criterion and so enables a contraction property up to higher-order terms and global convergence. Numerical experiments exploit the remaining $L^2$ error contributions and confirm our theoretical findings. The averaging schemes show a high accuracy and the AFEM leads to optimal empirical convergence rates.