Boundary element methods with weakly imposed boundary conditions

TL;DR

This paper introduces a boundary element method that employs the Calderón projector and weakly imposes boundary conditions via a variational operator, ensuring robustness and accurate approximation of boundary variables.

Contribution

It presents a novel approach combining Calderón projectors with augmented Lagrangian techniques for weak boundary condition enforcement in boundary element methods.

Findings

Robust conditioning for Robin boundary conditions.

Accurate approximation of primal trace and flux variables.

Numerical examples demonstrating method effectiveness.

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.