You have just completed your registration at OpenAire.
Before you can login to the site, you will need to activate your account.
An e-mail will be sent to you with the proper instructions.
Important!
Please note that this site is currently undergoing Beta testing.
Any new content you create is not guaranteed to be present to the final version
of the site upon release.
We consider the two-dimensional Helmholtz equation with constant coefficients on a domain with piecewise analytic boundary, modelling the scattering of acoustic waves at a sound-soft obstacle. Our discretisation relies on the Trefftz-discontinuous Galerkin approach with plane wave basis functions on meshes with very general element shapes, geometrically graded towards domain corners. We prove exponential convergence of the discrete solution in terms of number of unknowns.
[1] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, vol. 55 of National Bureau of Standards Applied Mathematics Series, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964.
[2] I. Babuˇska and B. Q. Guo, The h-p version of the finite element method for domains with curved boundaries, SIAM J. Numer. Anal., 25 (1988), 837-861.
[3] I. Babuˇska and J. M. Melenk, The partition of unity method, Internat. J. Numer. Methods Engrg., 40 (1997), 727-758.
[4] T. Betcke, S. N. Chandler-Wilde, I. G. Graham, S. Langdon, and M. Lindner, Condition number estimates for combined potential integral operators in acoustics and their boundary element discretisation., Numer. Methods Partial Differ. Equations, 27 (2011), 31-69.
[5] S. C. Brenner and L. R. Scott, Mathematical theory of finite element methods, 3rd ed., Texts Appl. Math., Springer-Verlag, New York, 2008.
[6] A. Buffa and P. Monk, Error estimates for the ultra weak variational formulation of the Helmholtz equation, M2AN, Math. Model. Numer. Anal., 42 (2008), 925-940.
[7] O. Cessenat and 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.
[8] M. Dauge, Elliptic boundary value problems on corner domains, vol. 1341 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1988. Smoothness and asymptotics of solutions.
[11] G. Gabard, Discontinuous Galerkin methods with plane waves for time-harmonic problems, J. Comput. Phys., 225 (2007), 1961-1984.
[12] M. Gander, I. Graham, and E. Spence, Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumber-independent convergence is guaranteed?, Numerische Mathematik, (2015). DOI:10.1007/s00211-015-0700-2.
[13] C. J. Gittelson, R. Hiptmair, and I. Perugia, Plane wave discontinuous Galerkin methods: analysis of the h-version, M2AN Math. Model. Numer. Anal., 43 (2009), 297- 332.
[14] P. Grisvard, Singularities in boundary value problems, vol. 22 of Recherches en Math´ematiques Appliqu´ees [Research in Applied Mathematics], Masson, Paris; SpringerVerlag, Berlin, 1992.
[21] P. Houston, C. Schwab, and E. Su¨li, Discontinuous hp-finite element methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal., 39 (2002), 2133-2163.
[22] T. Huttunen, P. Monk, and J. P. Kaipio, Computational aspects of the ultra-weak variational formulation, J. Comput. Phys., 182 (2002), 27-46.
[23] T. Luostari, T. Huttunen, and P. Monk, Improvements for the ultra weak variational formulation, Internat. J. Numer. Methods Engrg., 94 (2013), 598-624.
[24] W. McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000.
[25] J. M. Melenk, On Generalized Finite Element Methods, PhD thesis, University of Maryland, 1995.
[26] J. M. Melenk, hp-finite element methods for singular perturbations, vol. 1796 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2002.
[27] J. M. Melenk, On approximation in meshless methods, in Frontiers of numerical analysis, Universitext, Springer, Berlin, 2005, pp. 65-141.
[28] J. M. Melenk, A. Parsania, and S. Sauter, General DG-methods for highly indefinite Helmholtz problems, Journal of Scientific Computing, (2013), 1-46.
[32] , Vekua theory for the Helmholtz operator, Z. Angew. Math. Phys., 62 (2011), 779- [36] D. Scho¨tzau, C. Schwab, and T. P. Wihler, hp-dGFEM for Second-Order Elliptic Problems in Polyhedra I: Stability on Geometric Meshes, SIAM J. Numer. Anal., 51 (2013), 1610-1633.
[37] , hp-DGFEM for Second Order Elliptic Problems in Polyhedra II: Exponential Convergence, SIAM J. Numer. Anal., 51 (2013), 2005-2035.
[38] C. Schwab, p- and hp-Finite Element Methods. Theory and Applications in Solid and Fluid Mechanics, Numerical Mathematics and Scientific Computation, Clarendon Press, Oxford, 1998.
[39] E. A. Spence, Wavenumber-explicit bounds in time-harmonic acoustic scattering, SIAM J. Math. Anal., 46 (2014), 2987-3024.
[40] I. N. Vekua, New methods for solving elliptic equations, North Holland, 1967.