Flux reconstructions in the Lehmann-Goerisch method for lower bounds on eigenvalues

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:340

Article

Vejchodsky, Tomas¹ 

[1] Czech Acad Sci, Inst Math, Zitna 25, CZ-11567 Prague 1, Czech Republic

关键词: Eigenproblem;    Guaranteed;    Symmetric;    Elliptic operators;    Finite element method;    Conforming;   

DOI  :  10.1016/j.cam.2018.02.034

来源: Elsevier

【 摘 要 】

The standard application of the Lehmann-Goerisch method for lower bounds on eigenvalues of symmetric elliptic second-order partial differential operators relies on determination of fluxes (sigma) over tilde (i) that approximate co-gradients of exact eigenfunctions scaled by corresponding eigenvalues. Fluxes (sigma) over tilde (i) are usually computed by solving a global saddle point problem with mixed finite element methods. In this paper we propose a simpler global problem that yields fluxes a; of the same quality. The simplified problem is smaller, it is positive definite, and any H(div, Omega) conforming finite elements, such as Raviart-Thomas elements, can be used for its solution. In addition, these global problems can be split into a number of independent local problems on patches, which allows for trivial parallelization. The computational performance of these approaches is illustrated by numerical examples for Laplace and Steklov type eigenvalue problems. These examples also show that local flux reconstructions enable computation of lower bounds on eigenvalues on considerably finer meshes than the traditional global reconstructions. (C) 2018 Elsevier B.V. All rights reserved.

【 授权许可】

Free