Decompositions of high-frequency Helmholtz solutions via functional calculus, and application to the finite element method

no code implementations • 25 Feb 2021 • David Lafontaine, Euan A. Spence, Jared Wunsch

These results allow us to prove new frequency-explicit convergence results for (i) the $hp$-finite-element method applied to the variable coefficient Helmholtz equation in the exterior of a Dirichlet obstacle, when the obstacle and coefficients are analytic, and (ii) the $h$-finite-element method applied to the Helmholtz penetrable-obstacle transmission problem.

Analysis of PDEs Numerical Analysis Numerical Analysis

Local absorbing boundary conditions on fixed domains give order-one errors for high-frequency waves

no code implementations • 6 Jan 2021 • Jeffrey Galkowski, David Lafontaine, Euan A. Spence

We consider approximating the solution of the Helmholtz exterior Dirichlet problem for a nontrapping obstacle, with boundary data coming from plane-wave incidence, by the solution of the corresponding boundary value problem where the exterior domain is truncated and a local absorbing boundary condition coming from a Pad\'e approximation (of arbitrary order) of the Dirichlet-to-Neumann map is imposed on the artificial boundary (recall that the simplest such boundary condition is the impedance boundary condition).

Numerical Analysis Numerical Analysis Analysis of PDEs 35J05, 65N99