Space-time discontinuous Galerkin approximation of acoustic waves with point singularities
| Autor: | Ilaria Perugia, Christoph Schwab, Pratyuksh Bansal, Andrea Moiola |
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| Rok vydání: | 2020 |
| Předmět: |
Discontinuous Galerkin method
Space-time discretization Wave equation A priori error analysis Corner singularities Local refined meshes H-convergence Sparse-tensor approximation Applied Mathematics General Mathematics Space time Mathematical analysis Degrees of freedom (statistics) 010103 numerical & computational mathematics Numerical Analysis (math.NA) 01 natural sciences 010101 applied mathematics Computational Mathematics Rate of convergence Convergence (routing) Piecewise FOS: Mathematics Gravitational singularity Mathematics - Numerical Analysis 0101 mathematics 65M60 65M15 Mathematics |
| Zdroj: | IMA Journal of Numerical Analysis, 41 (3) |
| DOI: | 10.48550/arxiv.2002.11575 |
| Popis: | We develop a convergence theory of space-time discretizations for the linear, 2nd-order wave equation in polygonal domains $\Omega\subset\mathbb{R}^2$, possibly occupied by piecewise homogeneous media with different propagation speeds. Building on an unconditionally stable space-time DG formulation developed in [Moiola, Perugia 2018], we (a) prove optimal convergence rates for the space-time scheme with local isotropic corner mesh refinement on the spatial domain, and (b) demonstrate numerically optimal convergence rates of a suitable \emph{sparse} space-time version of the DG scheme. The latter scheme is based on the so-called \emph{combination formula}, in conjunction with a family of anisotropic space-time DG-discretizations. It results in optimal-order convergent schemes, also in domains with corners, with a number of degrees of freedom that scales essentially like the DG solution of one stationary elliptic problem in $\Omega$ on the finest spatial grid. Numerical experiments for both smooth and singular solutions support convergence rate optimality on spatially refined meshes of the full and sparse space-time DG schemes. Comment: 38 pages, 8 figures |
| Databáze: | OpenAIRE |
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