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Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions
Author(s) -
Deraemaeker Arnaud,
Babuška Ivo,
Bouillard Philippe
Publication year - 1999
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/(sici)1097-0207(19991010)46:4<471::aid-nme684>3.0.co;2-6
Subject(s) - finite element method , helmholtz equation , mathematics , galerkin method , helmholtz free energy , dispersion (optics) , square (algebra) , polygon mesh , mathematical analysis , topology (electrical circuits) , dimension (graph theory) , geometry , boundary value problem , physics , structural engineering , engineering , combinatorics , quantum mechanics , optics
For high wave numbers, the Helmholtz equation suffers the so‐called ‘pollution effect’. This effect is directly related to the dispersion. A method to measure the dispersion on any numerical method related to the classical Galerkin FEM is presented. This method does not require to compute the numerical solution of the problem and is extremely fast. Numerical results on the classical Galerkin FEM ( p ‐method) is compared to modified methods presented in the literature. A study of the influence of the topology triangles is also carried out. The efficiency of the different methods is compared. The numerical results in two of the mesh and for square elements show that the high order elements control the dispersion well. The most effective modified method is the QSFEM [1,2] but it is also very complicated in the general setting. The residual‐free bubble [3,4] is effective in one dimension but not in higher dimensions. The least‐square method [1,5] approach lowers the dispersion but relatively little. The results for triangular meshes show that the best topology is the ‘criss‐cross’ pattern. Copyright © 1999 John Wiley & Sons, Ltd.