Boundary element methods with weakly imposed boundary conditions
This work provides a new variational framework for boundary element methods that simplifies the treatment of various boundary conditions, which is relevant for computational mechanics and engineering applications.
The authors develop boundary element methods that weakly impose boundary conditions using an augmented Lagrangian approach, enabling robust approximation of both primal and flux variables for Dirichlet, mixed, and Robin conditions. The method maintains system conditioning even for stiff Robin conditions, as demonstrated through numerical examples.
We consider boundary element methods where the Calderón 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.