Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-22T20:44:44.572Z Has data issue: false hasContentIssue false
  • English
  • Français

The Ultra Weak Variational Formulation Using Bessel Basis Functions

Published online by Cambridge University Press:  20 August 2015

Teemu Luostari ,
Tomi Huttunen  and
Peter Monk
Teemu Luostari*
Affiliation:
Department of Applied Physics, University of Eastern Finland, P.O. Box 1627, FI-70211 Kuopio, Finland
Tomi Huttunen*
Affiliation:
Department of Applied Physics, University of Eastern Finland, P.O. Box 1627, FI-70211 Kuopio, Finland
Peter Monk*
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA
*
Corresponding author.Email:[email protected]
Email address:[email protected]
Email address:[email protected]
Get access

Abstract

We investigate the ultra weak variational formulation (UWVF) of the 2-D Helmholtz equation using a new choice of basis functions. Traditionally the UWVF basis functions are chosen to be plane waves. Here, we instead use first kind Bessel functions. We compare the performance of the two bases. Moreover, we show that it is possible to use coupled plane wave and Bessel bases in the same mesh. As test cases we shall consider propagating plane and evanescent waves in a rectangular domain and a singular 2-D Helmholtz problem in an L-shaped domain.

Keywords

02.60.Cb02.60.Lj02.70.Dh43.20.-fThe ultra weak variational formulationHelmholtz problemplane wave basisBessel basisnon-polynomial basis
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 2 , February 2012 , pp. 400 - 414
Copyright
Copyright © Global Science Press Limited 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Arnold, D., Brezzi, F., Cockburn, B. and Marini, L., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 42(6) (2002), 1749–1779.Google Scholar
[2]Badics, Z. and Matsumoto, Y., Trefftz discontinuous Galerkin methods for time-harmonic electromagnetic and ultrasound transmission problems, Int. J. Appl. Electromagn. Mech., 28(1-2) (2008), 17–24.Google Scholar
[3]Barnett, A. H. and Betcke, T., An exponentially convergent nonpolynomial finite element method for time-harmonic scattering from polygons, submitted.Google Scholar
[4]Buffa, A. and Monk, P., Error estimates for the ultra weak variational formulation of the Helmholtz equation, ESAIM: M2AN Math. Model. Numer. Anal., 42(6) (2008), 925–940.CrossRefGoogle Scholar
[5]Carayol, Q. and Collino, F., Error estimates in the fast multipole method for scattering problems, part 1: truncation of the Jacobi-Anger series, ESAIM: Math. Model. Numer. Anal., 38(2) (2004), 371–394.Google Scholar
[6]Cessenat, O., Application d’une Nouvelle Formulation Variationnelle aux Équations D’ondes Harmoniques, Probl’emes de Helmholtz 2D et de Maxwell 3D, PhD thesis, Université Paris IX Dauphine, 1996.Google Scholar
[7]Cessenat, O. and Després, B., Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem, SIAM J. Numer. Anal., 35(1) (1998), 255–299.Google Scholar
[8]Cessenat, O. and Després, B., Using plane waves as base functions for solving time harmonic equations with the ultra weak variational formulation, J. Comput. Acoust., 11(2) (2003), 227–238.Google Scholar
[9]Farhat, C., Harariand, I.Franca, L. P., A discontinuous enrichment method, Comput. Methods Appl. Mech. Eng., 190(48) (2001), 6455–6479.Google Scholar
[10]Farhat, C., Harari, I. and Hetmaniuk, U., A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime, Comput. Methods Appl. Mech. Eng., 192(11-12) (2003), 1389–1429.CrossRefGoogle Scholar
[11]Gabard, G., Discontinuous Galerkin methods with plane waves for time-harmonic problems, J. Comput. Phys., 225(2) (2007), 1961–1984.CrossRefGoogle Scholar
[12]Gamallo, P. and Astley, R. J., Comparison of two Trefftz-type methods: the ultraweak vari-ational formulation and the least-squares method, for solving shortwave 2-D Helmholtz problems, Int. J. Numer. Methods Eng., 71(4) (2007), 406–432.CrossRefGoogle Scholar
[13]Gabard, G., Gamallo, P. and Huttunen, T., A comparison of Trefftz, discontinuous Galerkin, ultra-weak and least-square methods for wave problems, submitted.Google Scholar
[14]Gittelson, C., Hiptmair, R. and Perugia, I., Plane wave discontinuous Galerkin methods: analysis of the h-version, ESAIM: M2AN Math. Model. Numer. Anal., 43(2) (2009), 297–331.Google Scholar
[15]Hiptmair, R., Moiola, A. and Perugia, I., Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version, SAM-ETH Zurich Report 2009-20, 2009-20, 1-20.Google Scholar
[16]Huttunen, T., Gamallo, P. and Astley, R. J., Comparison of two wave element methods for the Helmholtz problem, Commun. Numer. Methods Eng., 25(1) (2009), 35–52.CrossRefGoogle Scholar
[17]Huttunen, T., Malinen, M., Kaipio, J. P., White, P. J. and Hynynen, K., A full-wave Helmholtz model for continuous-wave ultrasound transmission, IEEE T. Ultrason. Ferr., 52(3) (2005), 397–409.CrossRefGoogle ScholarPubMed
[18]Huttunen, T., Malinen, M. and Monk, P., Solving Maxwell’s equations using the ultra weak variational formulation, J. Comput. Phys., 223(2) (2007), 731–758.CrossRefGoogle Scholar
[19]Huttunen, T., Monk, P., Collino, F. and Kaipio, J. P., The ultra weak variational formulation for elastic wave problems, SIAM J. Sci. Comput., 25(5) (2004), 1717–1742.CrossRefGoogle Scholar
[20]Huttunen, T., Monk, P. and Kaipio, J. P., Computational aspects of the ultra-weak variational formulation, J. Comput. Phys., 182(1) (2002), 27–46.Google Scholar
[21]Huttunen, T., Seppälä, E. T., Kirkeby, O., Kärkkäinen, A. and Kärkkäinen, L., Simulation of the transfer function for a head-and-torso model over the entire audible frequency range, J. Comput. Acoust., 15(4) (2007), 429–448.CrossRefGoogle Scholar
[22]Loeser, M. and Witzigmann, B., The ultra weak variational formulation applied to radiation problems with macroscopic sources in inhomogeneous domains, IEEE J. Sel. Top. Quant., 15(4) (2009), 1144–1155.Google Scholar
[23]Melenk, J. M. and Babuška, , The partition of unity finite element method: basic theory and applications, Comput. Methods Appl. Mech. Eng., 139(1-4) (1996), 289–314.Google Scholar
[24]Monk, P. and Wang, D. Q., A least-squares method for the Helmholtz equation, Comput. Methods Appl. Mech. Eng., 175(1-2) (1999), 121–136.Google Scholar
[25]Monk, P., Schöberl, J. and Sinwel, A., Hybridizing Raviart-Thomas elements for the Helmholtz equation, submitted RICAM REPORT 2008-22.Google Scholar
[26]Perrey-Debain, E., Plane wave decomposition in the unit disc: convergence estimates and computational aspects, J. Comput. Appl. Math., 193 (2006), 140–156.CrossRefGoogle Scholar