Abstract
Generalizing the framework of an ultra-weak formulation for a hypersingular integral equation on closed polygons in Heuer and 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 $$L^2$$L2-norm. Some numerical experiments confirm expected convergence rates.
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