A Hybrid Numerical Asymptotic Method for Scattering Problems

We develop a hybrid numerical asymptotic method for the Helmholtz equation. The method is a Galerkin finite element method in which the space of trial solutions is spanned by asymptotically derived basis functions. The basis functions are very “efficient” in representing the solution because each is the product of a smooth amplitude and an oscillatory phase factor, like the asymptotic solution. The phase is determined a priori by solving the eiconal equation using the ray method, while the smooth amplitude is represented by piecewise polynomials. The number of unknowns necessary to achieve a given accuracy with this new basis is dramatically smaller than the number necessary with a standard method, and it is virtually independent of the wavenumber k. We apply the method to the problems of scattering from a parabola and from a circle and compare the results with those of a standard finite element method.