Adaptivity for Guaranteed (Bi-) Harmonic Eigenvalue Bounds
A talk in the BI.discrete Workshop series by
Carsten Carstensen from HU Berlin
| Abstract: | 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 in the finite-dimensional case. The first part introduces the derivation of GLB for the simplest second-order and fourth-order eigenvalue problems with relevant applications for the localization of in the critical load in the buckling analysis of the Kirchhoff plates. The second part discusses an optimal adaptive mesh-refining algorithm for the effective eigenvalue computation for the Laplace operator with optimal convergence rates in terms of the number of degrees of freedom relative to the concept of nonlinear approximation classes. The two approaches do not automatically provide efficient GLB in model examples. The maximal mesh-size parameter in the post-processed GLB may destroy the sharp estimation for some highly refined triangulations. Therefore a new methodology is required for sharp GLB. The third part introduces a modified scheme with fine-tuned extra stabilization that allows for adaptive simulations with optimal convergence rates. The topics reflect earlier joint work with Joscha Gedicke (Bonn) and Dietmar Gallistl (Jena) and recent joint work with Sophie Puttkammer (Berlin). [1] C.~Carstensen and D.~Gallistl, Guaranteed lower eigenvalue bounds for the biharmonic equation, Numer. Math. 126 (2014) [2] C. Carstensen, D. Gallistl, M. Schedensack, Adaptive nonconforming Crouzeix-Raviart FEM for eigenvalue problems, Math. Comp. 84 (2015) [3] C. Carstensen, J. Gedicke, An adaptive finite element eigenvalue solver of asymptotic quasi-optimal computational complexity, SINUM 50 (2012) [4] ------, Guaranteed lower bounds for eigenvalues, Math. Comp. 83 (2014) [5] C.~Carstensen and S. Puttkammer, Direct guaranteed lower eigenvalue bounds with optimal a priori convergence rates for the bi-Laplacian, arXiv:2105.01505 (2021) [6] ------, Adaptive guaranteed lower eigenvalue bounds with optimal convergence rates, in preparation (2021) [7] C. Carstensen, Q. Zhai, and R. Zhang, A skeletal FEM can compute lower eigenvalue bounds, SINUM 58 (2020) [8] D. Gallistl, An optimal adaptive FEM for eigenvalue clusters, Numer. Math. 130 (2015) [9] ------, Adaptive nonconforming finite element approximation of eigenvalue clusters, Comput. Methods Appl. Math.14 (2014) [10] ------, Morley finite element method for the eigenvalues of the biharmonic operator, IMA J. Numer. Anal. 35 (2015) Attendance is only possible after registration with the organizers and with 3G-certificate. Within the CRC this talk is associated to the project(s): A7, B7 |