Download paper from the SISC website here.

Abstract

Numerical resolution of exterior Helmholtz problems requires some approach to do- main 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. How- ever, it seems to work in general unstructured geometry, and Galerkin finite element discretization leads to convergence under the usual mesh constraints imposed by G˚arding-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

Presentations

I presented ideas from this paper in a talk at FEniCS 2021.

Bibtex

@article{kirbyKlocknerSepanski2021,
author = {Kirby, Robert C. and Klöckner, Andreas and Sepanski, Ben},
title = {Finite Elements for Helmholtz Equations with a Nonlocal Boundary Condition},
journal = {SIAM Journal on Scientific Computing},
volume = {43},
number = {3},
pages = {A1671-A1691},
year = {2021},
doi = {10.1137/20M1368100},
URL = {https://doi.org/10.1137/20M1368100},
eprint = {https://doi.org/10.1137/20M1368100}
}