# Finite Elements for Helmholtz equations with a nonlocal boundary condition

Published in SIAM Journal on Scientific Computing, 2021

Kirby, Robert C. and Klöckner, Andreas and Sepanski, Benjamin.(2021). "Finite Elements for Helmholtz Equations with a Nonlocal Boundary Condition." SIAM Journal on Scientific Computing, 43(3), A1671-A1691.

# 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}
}