Nonlocal UFL: Finite elements for Helmholtz equations with a nonlocal boundary condition


Presentation of my undergraduate research at the FEniCS 2021 Conference.

Slides, abstract, and presentation time linked here. Presented my work as an undergraduate at the Mathematics Department at Baylor University. This work was joint with Dr. Robert Kirby at the Mathematics Department of Baylor University and Dr. Andreas Klöckner.


Numerical resolution of exterior Helmholtz problems require some approach to domain truncation. As an alternative to approximate nonreflecting boundary conditions and invocation of the Dirichlet-to-Neumann map, we introduce new, nonlocal boundary conditions. These conditions are exact and require the evaluation of layer potentials involving Green’s functions. The nonlocal boundary conditions are readily approximated by fast multipole methods, and the resulting linear system can be preconditioned by the purely local operator. Integration of the layer potential evaluation library pytential with the new external operator feature of Firedrake allows us to express these boundary conditions in UFL.


Many of the ideas in this talk come from a paper in submission: 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.