A Two-Steps Method Based on Plane Wave for Nonhomogeneous Helmholtz Equations in Inhomogeneous Media

摘要

In this paper we are concerned with numerical methods for nonhomogeneous Helmholtz equations in inhomogeneous media.We try to extend the plane wave method,which has been widely used to the discretization of homogeneous Helmholtz equations with constant wave numbers,to solving nonhomogeneous Helmholtz equations with piecewise constant wave numbers.To this end,we propose a combination between the plane wave discontinuous Galerkin (PWDG) method and the high order element discontinuous Galerkin (HODG) method for the underlying Helmholtz equations.In this composite methods,we need only to solve a series of local Helmholtz equations by the HODG method and solve a locally homogeneous Helmholtz equation with element-wise constant wave number by the PWDG method.In particular,we first consider non-matching grids so that the generated approximations have better performance for the current case.Numerical experiments show that the proposed methods are very effective to the tested Helmholtz equations.