Paper Info
Title
An iterative adaptive finite element method for elliptic eigenvalue problems
Abstract
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.
Year
DOI
Venue
2012
10.1016/j.cam.2012.05.002
J. Computational Applied Mathematics
Keywords
Field
DocType
adaptive finite element method,generalized eigensolver,novel adaptive algorithm,elliptic eigenvalue problem,higher-order finite element,iterative adaptive finite element,n eigenpairs,conventional adaptive fem algorithm,elliptic partial differential equation,mesh refinement step,iterative method,generalized eigenproblem,hp,partial differential equation
Discretization,Mathematical optimization,Iterative method,Mathematical analysis,Finite element method,Adaptive algorithm,Elliptic partial differential equation,Partial differential equation,Eigenvalues and eigenvectors,Mathematics,hp-FEM
Journal
Volume
Issue
ISSN
236
18
0377-0427
Citations 
PageRank 
References 
1
0.40
6
Authors
2
Name
Order
Citations
PageRank
Pavel Solin1346.19
Stefano Giani2369.55