Finite elements for Helmholtz equations with a nonlocal boundary condition

TL;DR

This paper introduces an exact nonlocal boundary condition for Helmholtz equations that improves domain truncation in exterior problems, compatible with finite element methods and efficient computational techniques.

Contribution

It proposes a novel nonlocal boundary condition based on layer potentials, applicable to unstructured geometries, and demonstrates its convergence and computational efficiency.

Findings

Convergence achieved with standard mesh constraints.

Nonlocal boundary conditions can be efficiently approximated by fast multipole methods.

Linear systems can be preconditioned using local transmission boundary operators.

Abstract

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 G{\aa}rding-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.