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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 $L^2$. Theoretical results are confirmed by numerical experiments with uniform and adaptively refined meshes.

Ultra-weak Formulation of a Hypersingular Integral Equation on Polygons and DPG Method with Optimal Test Functions