Recent advances in the nonconforming FEM approximation of elliptic PDE eigenvalue problems include the guaranteed lower eigenvalue bounds (GLB) and its adaptive finite element computation. The first part of the talk explains the derivation of GLB for the simplest second-order (and fourth-order) eigenvalue problems with relevant applications, e.g., for the localization of the critical load in the buckling analysis of the Kirchhoff plates. The second part mentions an optimal adaptive mesh-refining algorithm for the effective eigenvalue computation for the Laplace (and bi-Laplace) operator with optimal convergence rates in terms of the number of degrees of freedom relative to the concept of nonlinear approximation classes. Numerical experiments in the third part of the presentation shows benchmarks in which the naive adaptive mesh-refining and the post processed GLB do not lead to efficient GLB. The fourth part outlines a new extra-stabilised scheme based on extended Crouzeix-Raviart (resp. Morley) finite elements that directly computes approximations as GLB and that allows optimal convergence rates at the same time.

The presentation is on joint work with Dr. Sophie Puttkammer.