Abstract

Discontinuous Petrov-Galerkin boundary elements

Norbert Heuer and Michael Karkulik
Numer. Math. 135 (4), 1011-1043, 2017.

Generalizing the framework of an ultra-weak formulation for a hypersingular integral equation on closed polygons in [ N. Heuer, F. Pinochet, SIAM J. Numer. Anal. 52 (6), 2703-2721, 2014], 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 L²-norm. Some numerical experiments confirm expected convergence rates.