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 th e asymptotic solution. The phase is determined a priori by solving the eico nal equation using the ray method, while the smooth amplitude is represente d by piecewise polynomials. The number of unknowns necessary to achieve a g iven accuracy with this new basis is dramatically smaller than the number n ecessary with a standard method, and it is virtually independent of the wav enumber k. We apply the method to the problems of scattering from a parabol a and from a circle and compare the results with those of a standard finite element method. (C) 2001 Elsevier Science.