The discontinuous enrichment method for medium-frequency Helmholtz problems with a spatially variable wavenumber

Numerical dispersion, or what is often referred to as the pollution effect, presents a challenge to an efficient finite element discretization of the Helmholtz equation in the medium frequency regime. To alleviate this effect and improve the unsatisfactory pre-asymptotic convergence of the classical Galerkin finite element method based on piecewise polynomial basis functions, several discretization methods based on plane wave bases have been proposed. Among them is the discontinuous enrichment method that has been shown to offer superior performance to the classical Galerkin finite element method for a number of constant wavenumber Helmholtz problems and has also outperformed two representative methods that use plane waves – the partition of unity and the ultra-weak variation formulation methods. In this paper, the discontinuous enrichment method is extended to the variable wavenumber Helmholtz equation. To this effect, the concept of enrichment functions based on free-space solutions of the homogeneous form of the governing differential equation is enlarged to include free-space solutions of approximations of this equation obtained in this case by successive Taylor series expansions of the wavenumber around a reference point. This leads to plane wave enrichment functions based on the piece-wise constant approximation of the wavenumber, and to Airy wave enrichment functions. Several elements based on these enrichment functions are constructed and evaluated on benchmark problems modeling sound-hard scattering by a disk submerged in an acoustic fluid where the speed of sound varies in space. All these elements are shown to outperform by a substantial margin their continuous polynomial counterparts.