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Boundary elements with mesh refinements for the wave equation. (English) Zbl 1407.65172

This remarkable long paper (for a publication in Numer. Math.) studies the solution of the initial value problem for the wave equation outside a polyhedral screen \(\Gamma \subset \mathbb{R}^3\) with inhomogeneous Dirichlet or Neumann boundary conditions on \(\Gamma\). The given problem is formulated as a time-dependent boundary integral equation on \(\Gamma\) with either the single or double layer operator (involving the hypersingular operator in the weak formulation). The aim of the authors is to obtain a precise asymptotic description of the singularities of the solution near edges and corners. For the discretization, tensor products of piecewise polynomial functions on a \(\beta\)-graded spatial mesh and a uniform in time mesh are used. The graded spatial meshes are shown to be suitable to recover the optimal approximation rates known for smooth solutions.
The paper is technically involved. About 12 pages are used for introducing the main concepts, the definitions and basic properties needed and the citation of relevant results from other sources. On another 12 pages follow the main results and corresponding proofs concerning the asymptotic expansion of solutions and the numerical approximation. The final 19 pages are dedicated to extensive numerical experiments and a specific application to traffic noise in form of the horn effect. The list of references comprises 35 items among them 13 coauthored by E. P. Stephan.

MSC:

65M38 Boundary element methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
76Q05 Hydro- and aero-acoustics
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
35C20 Asymptotic expansions of solutions to PDEs
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References:

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