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.
Cite
@article{arxiv.1408.5374,
title = {Discontinuous Petrov-Galerkin boundary elements},
author = {Norbert Heuer and Michael Karkulik},
journal= {arXiv preprint arXiv:1408.5374},
year = {2014}
}