Discontinuous Petrov-Galerkin boundary elements (Q522235)

The authors generalize the framework of an ultra-weak formulation for a hypersingular integral equation on open and closed polyhedral surfaces. Previously, Heuer and Pinochet studied the case of closed polygons. The general ultra-weak setting is developed in fractional-order Sobolev spaces. Additionally, well-posedness and equivalence with the traditional formulation is proven. Next, a discontinuous Petrov-Galerkin method using optimal test functions is established and its quasi-optimal convergence in related Sobolev norms is shown. The special case for closed surfaces results in quasi-optimal converergence in the \(L^2\)-norm. Finally, numerical results are presented that confirm the theroretical convergence rates.