NDLI: A discontinuous Petrov–Galerkin (DPG) formulation for the FEM applied to the Helmholtz equation for high wavenumbers

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A discontinuous Petrov–Galerkin (DPG) formulation for the FEM applied to the Helmholtz equation for high wavenumbers

Content Provider	Springer Nature Link
Author	do Carmo, E. G. D. Monteiro, C. S. G. Mansur, W. J. Dias, R.
Copyright Year	2016
Abstract	Pollution error is a well known source of inaccuracies in continuous or discontinuous FE approaches to solve the Helmholtz equation. This topic is widely studied in a large number of papers, e.g., (Ihlenburg and Babuška, Comput Math Appl 30(9):9–37, 1995; Ihlenburg, Finite element analysis of acoustic scattering applied mathematical sciences, vol 132. Springer, New York, 1998). Robust methodologies for structured square meshes have been developed in recent years. This work seeks to develop a methodology based on Petrov–Galerkin discontinuous formulation, to minimize phase error for Helmholtz equation for both structured and unstructured meshes. A Petrov–Galerkin finite element formulation is introduced for the Helmholtz problem in two dimensions using polynomial weighting functions. At each node of the triangular mesh, a global basis function for the weighting space is obtained, adding bilinear $$C^0$$ Lagrangian weighting function linear combinations. The optimal weighting functions, with the same support of the corresponding global test functions, are obtained after computing the coefficients of these linear combinations attending to optimal criteria. This is done numerically through a preprocessing technique that is naturally applied to non-uniform and unstructured meshes. In particular, for uniform meshes a quasi optimal interior stencil of the same order of the quasi-stabilized finite element method stencil derived by (Babuška et al., Comput Meth Appl Mech Eng 128:325–359, 1995) is obtained. The numerical results indicate a better performance in relation to the classic discontinuous Galerkin method.
Starting Page	1529
Ending Page	1544
Page Count	16
File Format	PDF
ISSN	16785878
Journal	Journal of the Brazilian Society of Mechanical Sciences and Engineering
Volume Number	39
Issue Number	5
e-ISSN	18063691
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2016-11-26
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Discontinuous Galerkin Discontinuous Petrov–Galerkin (DPG) Discontinuous FEM Helmholtz equation Phase error Mechanical Engineering
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Industrial and Manufacturing Engineering Automotive Engineering Mechanical Engineering Aerospace Engineering

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