Abstract: We present non-conforming variants of the boundary element method, for the case of hypersingular integral operators. These operators act on trace spaces of $H^1$ where the standard trace operator is not well defined. On the other hand, conforming basis functions must be continuous. These facts are contradictory so that there is no standard continuous variational formulation of hypersingular boundary integral equations which corresponds to a non-conforming setting. On the discrete level, however, functions are more regular and traces are well-defined so that non-conforming boundary elements can be weakly coupled.
We analyze the two extreme cases of Crouzeix-Raviart elements (being non-conforming on the element level) and domain decomposition (being non-conforming on a sub-domain level). For the latter case we consider a mortar coupling and a Nitsche method. For all cases we prove (almost) quasi-optimal convergence and illustrate this by numerical experiments.
This is joint work with Franz Chouly (Besancon, France), Gabriel N. Gatica (Concepcion, Chile), Martin Healey (Brunel, UK), and Francisco-Javier Sayas (Delaware, USA).