2 research outputs found
Ultra-weak formulation of a hypersingular integral equation on polygons and DPG method with optimal test functions
We present an ultra-weak formulation of a hypersingular integral equation on
closed polygons and prove its well-posedness and equivalence with the standard
variational formulation. Based on this ultra-weak formulation we present a
discontinuous Petrov-Galerkin method with optimal test functions and prove its
quasi-optimal convergence in . Theoretical results are confirmed by
numerical experiments on an open curve with uniform and adaptively refined
meshes.Comment: 23 pages, 2 figure
Discontinuous Petrov-Galerkin boundary elements
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