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A Numerical Study on the Weak Galerkin Method for the Helmholtz Equation

Published online by Cambridge University Press:  03 June 2015

Lin Mu ,
Junping Wang ,
Xiu Ye  and
Shan Zhao
Lin Mu*
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
Junping Wang*
Affiliation:
Division of Mathematical Sciences, National Science Foundation, Arlington, VA 22230, USA
Xiu Ye*
Affiliation:
Department of Mathematics and Statistics, University of Arkansas at Little Rock, Little Rock, AR 72204, USA
Shan Zhao*
Affiliation:
Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA
*
Email:[email protected]
Email:[email protected]
Email:[email protected]
Corresponding author.Email:[email protected]
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Abstract

A weak Galerkin (WG) method is introduced and numerically tested for the Helmholtz equation. This method is flexible by using discontinuous piecewise polynomials and retains the mass conservation property. At the same time, the WG finite element formulation is symmetric and parameter free. Several test scenarios are designed for a numerical investigation on the accuracy, convergence, and robustness of the WG method in both inhomogeneous and homogeneous media over convex and non-convex domains. Challenging problems with high wave numbers are also examined. Our numerical experiments indicate that the weak Galerkin is a finite element technique that is easy to implement, and provides very accurate and robust numerical solutions for the Helmholtz problem with high wave numbers.

Keywords

65N1565N3076D0735B4535J50Galerkin finite element methodsdiscrete gradientHelmholtz equationlarge wave numbersweak Galerkin
Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 5 , May 2014 , pp. 1461 - 1479
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Ainsworth, M., Discrete dispersion relation for hp-version finite element approximation at high wave number, SIAM J. Numer. Anal., 42 (2004), 553–575.Google Scholar
[2]Ainsworth, M. and Wajid, H. A., Dispersive and dissipative behavior of the spectral element method, SIAM J. Numer. Anal., 47 (2009), 3910–3937.Google Scholar
[3]Arnold, D., Brezzi, F., Cockburn, B. and Marini, L. D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), 1749–1779.Google Scholar
[4]Babuška, I., Ihlenburg, F., Paik, E. T. and Sauter, S. A., A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution, Comput. Methods Appl. Mech. Eng., 128 (1995), 325–359.CrossRefGoogle Scholar
[5]Babuška, I. and Sauter, S. A., Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave number?, SIAM J. Numer. Anal., 34 (1997), 2392-2423, reprinted in SIAM Rev., 42 (2000), 451–484.CrossRefGoogle Scholar
[6]Bao, G., Wei, G. W. and Zhao, S., Numerical solution of the Helmholtz equation with high wavenubers, Int. J. Numer. Meth. Eng., 59 (2004), 389–408.Google Scholar
[7]Brezzi, F. and Fortin, M., Mixed and Hybrid Finite Elements, Springer-Verlag, New York, 1991.CrossRefGoogle Scholar
[8]Cessenat, O. and Despres, B., Application of the ultra-weak variational formulation of elliptic PDEs to the 2-dimensional Helmholtz problem, SIAM J. Numer. Anal., 35 (1998), 255–299.CrossRefGoogle Scholar
[9]Cessenat, O. and Despres, B., Using plane waves as base functions for solving time harmonic equations with the ultra weak varational formulation, J. Comput. Acoustics, 11 (2003), 227– 238.Google Scholar
[10]Chandler-Wilde, S. N. and Langdon, S., A Galerkin boundary element method for high frequency scattering by convex polygons, SIAM J. Numer. Anal., 45 (2007), 610–640.Google Scholar
[11]Chen, Z., Baker, N. A. and Wei, G. W., Differentialgeometry based solvation model I: Eulerian formulation, J. Comput. Phys., 229 (2010), 8231–8258.CrossRefGoogle Scholar
[12]Chung, E. T. and Engquist, B., Optimal discontinuous Galerkin methods for wave propagation, SIAM J. Numer. Anal., 44 (2006), 2131–2158.Google Scholar
[13]Ciarlet, P. G., The Finite Element Method for Elliptic Problems, North-Holland, New York, 1978.Google Scholar
[14]Cockburn, B., Dong, B. and Guzman, J., A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems, Math. Comput., 77 (2008), 1887–1916.Google Scholar
[15]Cockburn, B., Gopalakrishnan, J. and Lazarov, R., Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second-order elliptic problems, SIAM J. Numer. Anal., 47 (2009), 1319–1365.Google Scholar
[16]Harhat, C., Harari, I. and Hetmaniuk, U., A discontinuous Galerkin methodwith Lagrange multiplies for the solution of Helmholtz problems in the mid-frequency regime, Comput. Methods Appl. Mech. Eng., 192 (2003), 1389–1419.Google Scholar
[17]Harhat, C., Tezaur, R. and Weidemann-Goiran, P., Higher-order extensions of a discontinuous Galerkin method for mid-frequency Helmholtz problems, Int. J. Numer. Meth. Eng., 61 (2004), 1938–1956.Google Scholar
[18]Feng, X. and Wu, H., Discontinuous Galerkin methods for the Helmholtz equation with large wave number, SIAM J. Numer. Anal., 47 (2009), 2872–2896.CrossRefGoogle Scholar
[19]Giladi, E., Asymptotically derived boundary elements for the Helmholtz equation in high frequencies, J. Comput. Appl. Math., 198 (2007), 52–74.Google Scholar
[20]Griesmaier, R. and Monk, P., Error analysis for a hybridizable discontinuous Galerkin method for the Helmholtz equation, J. Sci. Comput., 49 (2011), 291–310.CrossRefGoogle Scholar
[21]Heikkola, E., Monkola, S., Pennanen, A. and Rossi, T., Controllability method for the Helmholtz equation with higher-order discretizations, J. Comput. Phys., 225 (2007), 1553–1576.Google Scholar
[22]Ihlenburg, F. and Babuška, I., Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation, Int. J. Numer. Methods Eng., 38 (1995), 3745–3774.Google Scholar
[23]Ihlenburg, F. and Babuška, I., Finite element solution of the Helmholtz equation with high wavenumber, part I: the h-version of the FEM, Comput. Math. Appl., 30 (1995), 9–37.Google Scholar
[24]Ihlenburg, F. and Babuška, I., Finite element solution of the Helmholtz equation with high wavenumber, part II: the h-p-version of the FEM, SIAM J. Numer. Anal., 34 (1997), 315–358.Google Scholar
[25]Langdon, S. and Chandler-Wilde, S. N., A wavenumber independent boundary element method for an accoustic scattering problem, SIAM J. Numer. Anal., 43 (2006), 2450–2477.Google Scholar
[26]Melenk, J. M. and Babuška, I., The partition of unity finite element method: basic theory and applications, Comput. Methods Appl. Mech. Eng., 139 (1996), 289–314.Google Scholar
[27]Monk, P. and Wang, D. Q., A least-squares method for the Helmholtz equation, Comput. Methods Appl. Mech. Eng., 175 (1999), 121–136.CrossRefGoogle Scholar
[28]Mu, L., Wang, J. and Ye, X., A Weak Galerkin Finite Element Method with Polynomial Reduction, 2014, availabe at: (arXiv:1304.6481).Google Scholar
[29]Mu, L., Wang, J. and Ye, X., Weak Galerkin Finite Element Methods for the Biharmonic Equation on Polytopal Meshes, Numer. Meth. Part. D. E., to appear, 2014, available at: (arXiv:1303.0927).Google Scholar
[30]Shao, Z. H., Wei, G. W. and Zhao, S., DSC time-domain solution of Maxwell’s equations, J. Comput. Phys., 189 (2003), 427–453.Google Scholar
[31]Shen, J. and Wang, L.-L., Spectral approximation of the Helmholtz equation with high wave numbers, SIAM J. Numer. Anal., 43 (2005), 623–644.Google Scholar
[32]Shen, J. and Wang, L.-L., Analysis of a spectral-Galerkin approximation to the Helmhotlz equation in exterior domains, SIAM J. Numer. Anal., 45 (2007), 1954–1978.Google Scholar
[33]Wang, J. and Ye, X., A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math., 241 (2013), 103–115.Google Scholar
[34]Wang, J. and Ye, X., A Weak Galerkin mixed finite element method for second-order elliptic problems, Math. Comput., to appear, (2013), available at: (arXiv:1202.3655).Google Scholar
[35]Zhao, S., High order matched interface and boundary methods for the Helmholtz equation in media with arbitrarily curved interfaces, J. Comput. Phys., 229 (2010), 3155–3170.Google Scholar
[36]Zhao, S., Pseudo-time coupled nonlinear models for biomolecular surface representation and solvation analysis, Int. J. Numer. Methods Biomedical Eng., 27 (2011), 1964–1981.CrossRefGoogle Scholar
[37]Zienkiewicz, O. C., Achievements and some unsolved problems of the finite element method, Int. J. Numer. Methods Eng., 47 (2000), 9–28.Google Scholar
[38]Wang, J. and Ye, X., A weak Galerkin finite element method for the Stokes equations, 2014, available at: (arXiv:1302.2707).Google Scholar
[39]Mu, L., Wang, J., Ye, X., and Zhang, S., A weak Galerkin finite element method for the Maxwell equations, 2014, available at: (arXiv:1312.2309).Google Scholar