  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 on an open curve with uniform and adaptively refined
meshes.
