Plane Wave Discontinuous Galerkin Methods: Exponential Convergence of the \(hp\)-Version
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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.
KeywordsHelmholtz equation Approximation by plane waves Trefftz-discontinuous Galerkin method \(hp\)-version A priori convergence analysis Exponential convergence
Mathematical Subject Classification65N30 65N15 35J05
The authors are grateful to Markus Melenk for advice on how to establish the analytic regularity result reported in Theorem 2.3. They also wish to thank Monique Dauge and Euan A. Spence for their help in strengthening some of the results. They also appreciate the valuable suggestions of the reviewers, which led to substantial enhancements compared to the first version of the manuscript: the wavenumber dependence in Proposition 2.1 and Lemma 4.5 is improved, the bounds in Sect. 5 are sharper, and the proof of Theorem 6.5 is simplified. Ilaria Perugia acknowledges support of the Italian Ministry of Education, University and Research (MIUR) through the project PRIN-2012HBLYE4.
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