Inexact Adaptive Finite Element Methods for Elliptic PDE Eigenvalue Problems

Since decades modern technological applications lead to challenging PDE eigenvalue problems, e.g., vibrations of structures, modeling of photonic gap materials, analysis of the hydrodynamic stability, or calculations of energy levels in quantum mechanics [6, 58, 78, 95]. Recently, a lot of research is devoted to the so-called Adaptive Finite Element Methods (AFEM) [10]. In most AFEM approaches it is assumed that the resulting finite dimensional algebraic problem (linear system or eigenvalue problem) is solved exactly and computational costs for this part of the method as well as the fact that they are solved in finite precision arithmetic are typically ignored. The goal of this work is to analyze the influence of the accuracy of the algebraic approximation on the adaptivity process. Efficient and reliable adaptive algorithms should take into consideration not only discretization errors, but also iteration errors and especially for non-symmetric problems the conditioning of the eigenvalues. Our new AFEMLA algorithm extends the standard AFEM approaches to incorporate approximation errors into the adaptation process. Furthermore, we show that the adaptive mesh refinement may be steered by the discrete residual vector, e.g., when the problem is stated in a discrete formulation where only the underlying matrices and meshes are available. Moreover, we discuss how to reduce the computational effort of the iterative solver by adapting the size of the Krylov subspace. With classical perturbation results we prove upper bounds for the eigenvalue and the eigenfunction error. Under certain assumptions similar results are obtained for convection-diffusion problems. Following [9], 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. Based on perturbation results in the H1( )and H 1( )-norm derived in [65] and a standard residual a posteriori error estimator a balanced AFEM algorithm is proposed. The eigensolver stopping criterion is based on the equilibrating strategy, i.e., iterations proceed as long as the discrete part of the error estimator dominates the continuous part. A completely new approach combining the adaptive finite element method with the homotopy method is introduced to determine the particular eigenvalue of the convection-diffusion problem. The adaptive homotopy approach derived here emphasizes the need of the multi-way adaptation based on three different errors, the homotopy, the discretization and the iteration error. All our statements are illustrated with several numerical examples.