hp-version time domain boundary elements for the wave equation on quasi-uniform meshes

Abstract

Solutions to the wave equation in the exterior of a polyhedral domain or a screen in R 3 exhibit singular behavior from the edges and corners. We present quasi-optimal hp-explicit estimates for the approximation of the Dirichlet and Neumann traces of these solutions for uniform time steps and (globally) quasi-uniform meshes on the boundary. The results are applied to an hp-version of the time domain boundary element method. Numerical examples confirm the theoretical results for the Dirichlet problem both for screens and polyhedral domains.This article initiates the study of high-order boundary elements in the time domain. For elliptic problems, p-and hp-versions of the finite element method give rise to fast approximations of both smooth solutions and geometric singularities. These methods converge to the solution by increasing the polynomial degree p of the elements, possibly in combination with reducing the mesh size h of the quasi-uniform mesh. They were first investigated in the group of Babuska [3,4,17,18]. See [49] for a comprehensive analysis for 2d problems.The analogous p-and hp-versions of the boundary element method go back to [2,53,54]. More recent optimal convergence results for boundary elements on screens and polyhedral surfaces covering 3d problems have been obtained, for example, in [9,10,11,12,13].Boundary element methods for time dependent problems have recently become of interest [48]. In this article we introduce a space-time hp-version of the time domain boundary element method for the wave equation with non-homogeneous Dirichlet or Neumann boundary conditions.

On a preconditioner for time domain boundary element methods

“…The error of the approximate solutions obtained from the preconditioner as an independent solver turns out to be small compared to the discretization error, and it decreases as the mesh size h tends to 0, provided the CFL ratio h ∆t is fixed. However, the rigorous numerical analysis of the surprisingly good stability properties remains open, while higher-order extrapolation for ansatz and test functions of higher polynomial degree is the content of current work [12]. For the non-polynomial basis functions of a partition of unity method with a large time step the preconditioner may still reduce the number of necessary GMRES iterations.…”

Boundary element methods for the wave equation based on hierarchical matrices and adaptive cross approximation

“…In the second step, we apply the MACA from Sect. 5 to each block of the partition and obtain low-rank factorisations of the form (26). Eventually, we end up with a hierarchical tensor approximation, which reads…”

“…To this end, the latter is partitioned either into a tensor grid or into an unstructured grid made of tetrahedral finite elements [41]. For that reason, space-time methods feature an inherent flexibility, including adaptive refinement in both time and space simultaneously as well as the ability to capture moving geometries [25,26,55]. However, the computational costs are high due to the increase in dimensionality and the calculation of the retarded potentials is far from trivial [56].…”

A time–dependent FEM-BEM coupling method for fluid–structure interaction in 3d

“…We now consider the discretization of the single layer potential. For the ansatz function we choose (16) and as test function we choose (18).…”