Ultra-weak formulation of a hypersingular integral
equation on polygons and DPG method with optimal test functions
Norbert Heuer and Felipe Pinochet
SIAM J. Numer. Anal. 52 (6), 2703-2721, 2014.
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 L2.
Theoretical results are confirmed by numerical experiments with uniform and adaptively refined meshes.
In the preprint version
arXiv:1309.1697
we also consider the case of an open polygon (interval). There, we analyze optimal test functions
and present numerical experiments for this case.