Boundary element methods with weakly imposed boundary conditions

Abstract: We consider boundary element methods where the Calder\'on projector is used for the system matrix and boundary conditions are weakly imposed using a particular variational boundary operator designed using techniques from augmented Lagrangian methods. Regardless of the boundary conditions, both the primal trace variable and the flux are approximated. We focus on the imposition of Dirichlet, mixed Dirichlet--Neumann, and Robin conditions. A salient feature of the Robin condition is that the conditioning of the system is robust also for stiff boundary conditions. The theory is illustrated by a series of numerical examples.