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. Like guaranteed upper eigenvalue bounds with conforming finite element methods, GLB arise naturally from the min-max principle, also named after Courant, Fischer, Weyl. The first part introduces the derivation of GLB for the simplest second-order and fourth-order eigenvalue problems with relevant applications, e.g., for the localization of in the critical load in the buckling analysis of the Kirchhoff plates. The second part studies 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. The third part presents a modified hybrid high-order (HHO) eigensolver in the spirit of Carstensen, Ern, and Puttkammer [Numer. Math. 149, 2021] that directly computes guaranteed lower eigenvalue bounds under the idealized hypothesis of exact solve of the generalized algebraic eigenvalue problem and a mild explicit condition on the maximal mesh-size in a simplicial mesh. The error analysis allows for a priori quasi-best approximation and L2 error estimates as well as a stabilization-free reliable and efficient a posteriori error control. The associated adaptive mesh-refining algorithm performs well in computer benchmarks with striking numerical evidence for optimal higher convergence rates.The topics reflect joint work with Sophie Puttkammer (Berlin), Ngoc Tien Tran (Augsburg), and Benedikt Gräßle (Berlin).

Carsten Carstensen studied civil engineering and mathematics in Hannover and received two diploma and two doctor degrees from what is today called Leibniz Universität Hannover. After two years as a postdoc with H.M. Ball in Edinburgh, he accepted professorships in Darmstadt, Kiel, Vienna, and Berlin. He was a foreign faculty member in Seoul, a longterm visiting professor at the IITB in Mumbai. He received the Richard-von-Mises Price in 1995 and is a correspondent member of the Akademie der Wissenschaften und der Literatur Mainz and a Honorary Member of The Academy of Romanian Scientists. He served in the editorial board of Math. Comp, SINUM, Journal of Numerical Mathematics, and is the Editor-in-Chief of Computational Methods in Applied Mathematics. He wrote almost 300 papers (over 50 in Numerische Mathematik and more than 40 in SINUM) and supervised over 20 PhD students. His current interest is on adaptive algorithms for mesh refinement for non-standard finite element methods in computational PDE.