ABSTRACT

Recent advances in the nonconforming FEM approximation of symmetric PDE eigenvalue problems include guaranteed lower eigenvalue bounds (GLB) and their adaptive finite element computation. While the first generation of post-processed GLB from nonconforming FEM is unconditional and hence allows a verified eigenvalue localization even on coarse meshes, it is incompatible with adaptive mesh refinement if the eigenfunction localizes. The second generation of GLB circumvent this difficulty with new extrastabilized schemes that directly compute approximations as GLB. An 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 is highlighted. The talk concludes with an introduction to skeletal schemes for higher-order GLB, in particular for fourth-order problems. This enable a high-precision localization of eigenvalues with rates comparable to adaptive Argyris FEM approximations. The presentation concerns model examples for the Laplace, bi-Laplace and Schrodinger eigenvalue problem and is based on recent joint work with Tim Stiebert and former students.