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Discontinuous Galerkin Methods for Acoustic Wave Propagation in Polygons

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Abstract

We analyze space semi-discretizations of linear, second-order wave equations by discontinuous Galerkin methods in polygonal domains where solutions exhibit singular behavior near corners. To resolve these singularities, we consider two families of locally refined meshes: graded meshes and bisection refinement meshes. We prove that for appropriately chosen refinement parameters, optimal asymptotic rates of convergence with respect to the total number of degrees of freedom are obtained, both in the energy norm errors and the \(\mathcal {L}^2\)-norm errors. The theoretical convergence orders are confirmed in a series of numerical experiments which also indicate that analogous results hold for incompatible data which is not covered by the currently available regularity theory.

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Authors and Affiliations

  1. Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, 8092, Zurich, Switzerland

    Fabian Müller & Christoph Schwab

  2. Mathematics Department, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada

    Dominik Schötzau

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  1. Fabian Müller
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  3. Christoph Schwab
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Corresponding author

Correspondence to Dominik Schötzau.

Additional information

Research supported by the Swiss National Science Foundation under Grant No. SNF 200021_149819/1 and by the Natural Sciences and Engineering Research Council of Canada (NSERC). This paper benefitted from helpful discussions at the Mathematical Research Institute Oberwolfach (MFO) during meeting No. 1711, March 13–17, 2017.

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Müller, F., Schötzau, D. & Schwab, C. Discontinuous Galerkin Methods for Acoustic Wave Propagation in Polygons. J Sci Comput 77, 1909–1935 (2018). https://doi.org/10.1007/s10915-018-0706-x

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  • DOI: https://doi.org/10.1007/s10915-018-0706-x

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