Finite elements for Helmholtz equations with a nonlocal boundary condition

Numerical resolution of exterior Helmholtz problems requires some approach to domain truncation. As an alternative to approximate nonreflecting boundary conditions and invocation of the Dirichlet-to-Neumann map, we introduce a new, nonlocal boundary condition. This condition is exact and requires the evaluation of layer potentials involving the free-space Green's function. However, it seems to work in general unstructured geometry, and Galerkin finite element discretization leads to convergence under the usual mesh constraints imposed by Gärding-type inequalities. The nonlocal boundary conditions are readily approximated by fast multipole methods, and the resulting linear system can be preconditioned by the purely local operator involving transmission boundary conditions.