Description:
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 towork in general unstructured geometry, and Galerkin finite element discretization leads to convergence under some mesh constraintsimposed by Garding-type inequalities. We will sketch this method, show how we have integrated Firedrake with pytential (a fast-multipole code), and discuss what issues this raises for extending Firedrake to work with more general nonlocal operators.