Finite elements for Helmholtz equations with a nonlocal boundary condition

Robert C. Kirby, Andreas Klöckner, Ben Sepanski

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)A1671-A1691
JournalSIAM Journal on Scientific Computing
Issue number3
StatePublished - May 2021
Externally publishedYes


  • Boundary conditions
  • Finite element method
  • Helmholtz
  • Layer potentials

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


Dive into the research topics of 'Finite elements for Helmholtz equations with a nonlocal boundary condition'. Together they form a unique fingerprint.

Cite this