On the numerical analysis of eigenvalue problems

This thesis “on the numerical analysis of eigenvalue problems” consists of five major aspects of the numerical analysis of adaptive finite element methods for eigenvalue problems. The first three consider the symmetric Laplace eigenvalue problem while the last two concern a non-symmetric convection-diffusion eigenvalue problem. The first part presents a combined adaptive finite element method with an iterative algebraic eigenvalue solver for a symmetric eigenvalue problem of asymptotic 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. The asymptotic quasi-optimal adaptive finite element eigenvalue solver involves a proper termination criterion for the algebraic eigenvalue solver and does not need any coarsening. Numerical evidence illustrates the asymptotic quasi-optimal computational complexity in 2 and 3 dimensions. The second part introduces fully computable two-sided bounds on the eigenvalues of the Laplace operator on arbitrarily coarse meshes based on some approximation of the corresponding eigenfunction in the nonconforming Crouzeix-Raviart finite element space plus some postprocessing. The efficiency of the guaranteed error bounds involves the global mesh-size and is proven for the large class of graded meshes. Numerical examples demonstrate the reliability of the guaranteed error control even for coarse meshes and inexact solve of the algebraic eigenvalue problem. This motivates an adaptive algorithm which monitors the discretisation error, the maximal mesh-size, and the algebraic eigenvalue error. The accuracy of the guaranteed eigenvalue bounds is surprisingly high with efficiency indices as small as 1.4. The third part presents an adaptive finite element method (AFEM) based on nodal-patch refinement that leads to an asymptotic error reduction property for the adaptive sequence of simple eigenvalues and eigenfunctions of the Laplace operator. The proven saturation property yields reliability and efficiency for a class of hierarchical a posteriori error estimators. Numerical experiments confirm that the saturation property is present even for very coarse meshes for many examples, but in other cases the smallness assumption on the initial mesh is severe. The fourth part considers a posteriori error estimators for convection-diffusion eigenvalue problems as discussed by Heuveline and Rannacher (2001) in the context of the dual-weighted residual method (DWR). This presentation directly addresses the variational formulation rather than the non-linear ansatz of Becker and Rannacher. Two different postprocessing techniques attached to the DWR paradigm plus two new dual-weighted a posteriori error estimators are presented. 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. The six a posteriori error estimators compete in three numerical examples and illustrate reliability and efficiency and the dependence of generic constants on the size of the eigenvalue or the convection coefficient.