A mixed hybrid finite element method for the Helmholtz equation

In this paper, a high-order hybrid mixed finite element method for solving the two-dimensional Helmholtz equation with high wave numbers is presented. The novelty of this method is in using a discrete eigenfunction basis for solving the hybrid problem. Such a basis allows an inexpensive elimination...

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Bibliographic Details
Published in:Journal of modern optics 2011-03, Vol.58 (5-6), p.424-437
Main Authors: Hannukainen, A., Huber, M., Schöberl, J.
Format: Article
Language:English
Subjects:
Degrees of freedom
Eigenfunctions
Eigenvalues
Finite element analysis
Finite element method
finite elements
Helmholtz equation
Helmholtz equations
hybridization techniques
Linear systems
Mathematical analysis
Mathematical models
Partial differential equations
Polynomials
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Summary:In this paper, a high-order hybrid mixed finite element method for solving the two-dimensional Helmholtz equation with high wave numbers is presented. The novelty of this method is in using a discrete eigenfunction basis for solving the hybrid problem. Such a basis allows an inexpensive elimination of the inner degrees of freedom, which considerably reduces the size of the resulting linear system. On a rectangular grid with hanging nodes, the eigenfunction basis is constructed by solving a one-dimensional eigenvalue problem for each pair of edge length and polynomial order in the mesh. The eigenvalue problem can be solved efficiently for polynomial orders up to one thousand. Together with the reduced size of the linear system, this makes it possible to work with very high order basis functions, and consequently high frequency waves can be resolved on a coarse mesh. The hp-refinement is used to obtain an accurate solution for a minimal number of degrees of freedom. The effectiveness of our approach is demonstrated with numerical examples using polynomials of the degree up to one thousand.
ISSN:0950-0340
1362-3044
DOI:10.1080/09500340.2010.527067