publication . Article . 2013

Trefftz discontinuous Galerkin methods for acoustic scattering on locally refined meshes,

Ralf Hiptmair; Andrea Moiola; Ilaria Perugia;
Closed Access
  • Published: 01 Jan 2013
  • Country: United Kingdom
Abstract
We extend the a priori error analysis of Trefftz discontinuous Galerkin methods for time-harmonic wave propagation problems developed in previous papers to acoustic scattering problems and locally refined meshes. To this aim, we prove refined regularity and stability results with explicit dependence of the stability constant on the wave number for non-convex domains with non-connected boundaries. Moreover, we devise a new choice of numerical flux parameters for which we can prove L2-error estimates in the case of locally refined meshes near the scatterer. This is the setting needed to develop a complete hp-convergence analysis.
Subjects
free text keywords: Applied Mathematics, Numerical Analysis, Computational Mathematics, Wave propagation, Scattering, Mathematical optimization, A priori and a posteriori, Polygon mesh, Discontinuous Galerkin method, Numerical flux, Mathematical analysis, Mathematics
Funded by
SNSF| Computational Wave Propagation
Project
  • Funder: Swiss National Science Foundation (SNSF)
  • Project Code: PBEZP2_137294
  • Funding stream: Careers;Fellowships | Fellowships for prospective researchers
37 references, page 1 of 3

[1] D.N. Arnold, F. Brezzi, B. Cockburn, L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2002) 1749-1779. [OpenAIRE]

[2] I. Babuˇska, M. Suri, The p and h-p versions of the finite element method, basic principles and properties, SIAM Rev. 36 (1994) 578-632.

[3] A.H. Barnett, T. Betcke, An exponentially convergent nonpolynomial finite element method for time-harmonic scattering from polygons, SIAM J. Sci. Comput. 32 (2010) 1417-1441. [OpenAIRE]

[4] S.C. Brenner, L.R. Scott, Mathematical theory of finite element methods, 3rd ed., Texts Appl. Math., Springer-Verlag, New York, 2007.

[5] A. Buffa, P. Monk, Error estimates for the ultra weak variational formulation of the Helmholtz equation, M2AN, Math. Model. Numer. Anal. 42 (2008) 925-940.

[6] P. Castillo, B. Cockburn, I. Perugia, D. Scho¨tzau, An a priori error analysis of the local discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal. 38 (2000) 1676-1706. [OpenAIRE]

[7] O. Cessenat, B. Despr´es, Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz equation, SIAM J. Numer. Anal. 35 (1998) 255-299. [OpenAIRE]

[8] O. Cessenat, B. Despr´es, Using plane waves as base functions for solving time harmonic equations with the ultra weak variational formulation, J. Comput. Acoust. 11 (2003) 227-238.

[9] S.N. Chandler-Wilde, P. Monk, Wave-number-explicit bounds in timeharmonic scattering, SIAM J. Math. Anal. 39 (2008) 1428-1455.

[10] D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, volume 93 of Applied Mathematical Sciences, Springer, Heidelberg, 2nd edition, 1998.

[11] P. Cummings, X. Feng, Sharp regularity coefficient estimates for complex-valued acoustic and elastic Helmholtz equations, Math. Models Methods Appl. Sci. 16 (2006) 139-160.

[12] S. Esterhazy, J. Melenk, On stability of discretizations of the Helmholtz equation, in: I. Graham, T. Hou, O. Lakkis, R. Scheichl (Eds.), Numerical Analysis of Multiscale Problems, volume 83 of Lecture Notes in Computational Science and Engineering, Springer Verlag, 2011, pp. 285-324. [OpenAIRE]

[13] X.B. Feng, H.J. Wu, hp-Discontinuous Galerkin methods for the Helmholtz equation with large wave number, Math. Comp. 80 (2011) 1997-2024.

[14] C.J. Gittelson, R. Hiptmair, I. Perugia, Plane wave discontinuous Galerkin methods: analysis of the h-version, M2AN Math. Model. Numer. Anal. 43 (2009) 297-332.

[15] U. Hetmaniuk, Stability estimates for a class of Helmholtz problems, Commun. Math. Sci. 5 (2007) 665-678. [OpenAIRE]

37 references, page 1 of 3
Powered by OpenAIRE Research Graph
Any information missing or wrong?Report an Issue