CSE Special Seminar
Speaker: Carsten Carstensen (Humboldt-Universität zu Berlin, Germany)
Date: 2023.09.07 / 1:30pm
Place: ASTC 516 (첨단관 516호)
Talk title: Lower eigenvalue bounds of the Laplacian
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. The first part 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. There exists examples where the
naive adaptive mesh-refining and the post processed GLB do not lead to
efficient GLB. The third part gives a remedy with an extra-stabilised new
scheme that directly computes approximations as GLB.
References:
C. Carstensen and D. Gallistl, Guaranteed lower eigenvalue bounds for the
biharmonic equation, Numer. Math. 126 (2014), 33–51.
C. Carstensen and J. Gedicke, Guaranteed lower bounds for eigenvalues, Math.
Comp. 83 (2014), 2605–2629.
C. Carstensen and S. Puttkammer, Direct guaranteed lower eigenvalue bounds
with optimal a priori convergence rates for the bi-Laplacian,
SIAM J. Numer. Anal., volume 61, pp. 812–836, 2023
C. Carstensen and S.Puttkammer, Adaptive guaranteed lower eigenvalue
bounds with optimal convergence rates, (2022). preprint (arXiv:2203.01028).
C. Carstensen, Q. Zhai, and R. Zhang, A skeletal finite element method can
compute lower eigenvalue bounds, SIAM J. Numer. Anal. 58 (2020), 109–124.