We propose and analyze a numerical method to solve an elliptic transmission problem in full space. The method consists of a variational formulation involving standard boundary integral operators on the coupling interface and an ultra-weak formulation
in the interior. To guarantee the discrete inf-sup condition, the system is discretized by the DPG method with optimal test functions. We prove that principal unknowns are approximated quasi-optimally. Numerical experiments for problems with smooth and singular solutions confirm optimal convergence orders.
For the non-conforming Crouzeix-Raviart boundary elements from [Heuer, Sayas: Crouzeix-Raviart boundary elements, Numer. Math. 112, 2009], we develop and analyze a posteriori error estimators based on the $h-h/2$ methodology. We discuss the optimal r
ate of convergence for uniform mesh refinement, and present a numerical experiment with singular data where our adaptive algorithm recovers the optimal rate while uniform mesh refinement is sub-optimal. We also discuss the case of reduced regularity by standard geometric singularities to conjecture that, in this situation, non-uniformly refined meshes are not superior to quasi-uniform meshes for Crouzeix-Raviart boundary elements.
