On a preconditioner for time domain boundary element methods

Abstract

We propose a time stepping scheme for the space-time systems obtained from Galerkin time-domain boundary element methods for the wave equation. Based on extrapolation, the method proves stable, becomes exact for increasing degrees of freedom and can be used either as a preconditioner, or as an efficient standalone solver for scattering problems with smooth solutions. It also significantly reduces the number of GMRES iterations for screen problems, with less regularity, and we explore its limitations for enriched methods based on non-polynomial approximation spaces.

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

“…The resulting discretization of the Poincaré-Steklov operator has been tested in [15,16], and corresponding results are obtained for more natural discretizations with piecewise constant λ h,△t . Piecewise linear and higher order test functions are considered in [19]. As shown in Appendix B, this discretization of (5) leads to a time-stepping scheme, which solves a system of the following structure in each time step ≥ 3:…”

“…The resulting discretization of the Poincaré-Steklov operator has been tested in [15,16], and corresponding results are obtained for more natural discretizations with piecewise constant λ h,△t . Piecewise linear and higher order test functions are considered in [19].…”

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

“…In the literature decompositions like (25) and (26) in polar coordinates are also expressed in Cartesian coordinates or a mixture of Cartesian and polar coordinates. The Appendix of [14] shows the equivalence of these descriptions, also remarked in Corollary 4 of [44].…”

“…Part a) is the content of Theorem 3.5 in [11] and its extension to hp in Theorem 5.1 of [13], whereas part b) follows from Theorem 3.4 in [12] and its extension to hp in Theorem 5.1 of [14]. Concerning the edge-vertex singularities of the Dirichlet trace u| Γ , we restrict ourselves to u ev 1 in (25). We bound the corresponding approximation error by .…”

“…The system can be solved with an approximate time stepping scheme, respectively a space-time preconditioned GMRES method [25]. Note that the common, but non-conforming MOT time stepping schemes are based on piecewise constant test functions in time.…”