Discontinuous Petrov–Galerkin boundary elements

Abstract

Generalizing the framework of an ultra-weak formulation for a hypersingular integral equation on closed polygons in [N. Heuer, F. Pinochet, arXiv 1309.1697 (to appear in SIAM J. Numer. Anal.)], we study the case of a hypersingular integral equation on open and closed polyhedral surfaces. We develop a general ultra-weak setting in fractional-order Sobolev spaces and prove its well-posedness and equivalence with the traditional formulation. Based on the ultra-weak formulation, we establish a discontinuous Petrov-Galerkin method with optimal test functions and prove its quasi-optimal convergence in related Sobolev norms. For closed surfaces, this general result implies quasi-optimal convergence in the L 2 -norm. Some numerical experiments confirm expected convergence rates.

DPG Method with Optimal Test Functions for a Fractional Advection Diffusion Equation

“…This is particularly important for fractional-order problems where inner products are defined by double integrals so that global calculations are prohibitively costly. Let us also mention that there is DPG-technology available for hypersingular integral equations [22,21]. Hypersingular operators are of order one with energy spaces of order 1/2.…”

On the coupling of DPG and BEM

“…That is, they cannot be represented equivalently as broken norms over elements and, therefore, there are no straightforward efficient techniques for the calculation of optimal test functions in these cases. Perhaps surprisingly, in [26,24] we found an ultra-weak variational formulation for hypersingular operators that is well posed in integer-order Sobolev spaces as long as the underlying polygon (in two dimensions) or surface (in three dimensions) is closed. Then, also the DPG framework with optimal test functions goes through without complications (this is different for open curves and surfaces).…”

DPG method with optimal test functions for a transmission problem

“…Until recently the DPG method with optimal test functions has been studied only for problems on bounded domains. In [24], we considered boundary value and screen problems of Neumann type which can be reduced to a hypersingular boundary integral equation. This includes the case of a PDE on an unbounded domain.…”