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In this paper we develop a plane wave type method for discretization ofhomogeneous Helmholtz equations with variable wave numbers. In the proposedmethod, local basis functions (on each element) are constructed by thegeometric optics ansatz such that they approximately satisfy a homogeneousHelmholtz equation without boundary condition. More precisely, each basisfunction is expressed as the product of an exponential plane wave function anda polynomial function, where the phase function in the exponential functionapproximately satisfies the eikonal equation and the polynomial factor isrecursively determined by transport equations associated with the consideredHelmholtz equation. We prove that the resulting plane wave spaces have highorder $h$-approximations as the standard plane wave spaces (which are availableonly to the case with constant wave number). We apply the proposed plane wavespaces to the discretization of nonhomogeneous Helmholtz equations withvariable wave numbers and establish the corresponding error estimates of theirfinite element solutions. We report some numerical results to illustrate theefficiency of the proposed method.

A geometric optics ansatz-based plane wave method for two dimensional Helmholtz equations with variable wave numbers