An iterative adaptive finite element method for elliptic eigenvalue problems

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:236

Article

Solin, Pavel^2,3  Giani, Stefano¹ 

[1] Univ Nottingham, Sch Math Sci, Nottingham NG7 2RD, England

[2] Univ Nevada, Dept Math & Stat, Reno, NV 89557 USA

[3] Inst Thermomech, Prague, Czech Republic

关键词: Partial differential equation;    Eigenvalue problem;    Iterative method;    Adaptive higher-order finite element method;    hp-FEM;    Reproducible research;   

DOI  :  10.1016/j.cam.2012.05.002

来源: Elsevier

【 摘 要 】

We consider the task of resolving accurately the nth eigenpair of a generalized eigenproblem rooted in some elliptic partial differential equation (PDE), using an adaptive finite element method (FEM). Conventional adaptive FEM algorithms call a generalized eigensolver after each mesh refinement step. This is not practical in our situation since the generalized eigensolver needs to calculate n eigenpairs after each mesh refinement step, it can switch the order of eigenpairs, and for repeated eigenvalues it can return an arbitrary linear combination of eigenfunctions from the corresponding eigenspace. In order to circumvent these problems, we propose a novel adaptive algorithm that only calls a generalized eigensolver once at the beginning of the computation, and then employs an iterative method to pursue a selected eigenvalue-eigenfunction pair on a sequence of locally refined meshes. Both Picard's and Newton's variants of the iterative method are presented. The underlying partial differential equation (PDE) is discretized with higher-order finite elements (hp-FEM) but the algorithm also works for standard low-order FEM. The method is described and accompanied with theoretical analysis and numerical examples. Instructions on how to reproduce the results are provided. (C) 2012 Elsevier B.V. All rights reserved.

【 授权许可】

Free