A multiscale finite element method for the Helmholtz equation Original Research Article

It is well known that when the standard Galerkin method is applied to the Helmholtz equation it exhibits an error in the wavenumber and the solution does not, therefore, preserve the phase characteristics of the exact solution. Improvements on the Galerkin method, including Galerkin least-squares (GLS) methods, have been proposed. However, these approaches rely on a dispersion analysis of the underlying difference stencils in order to reduce error in the solution. In this paper we propose a multiscale finite element for the Helmholtz equation. The method employs a multiscale variational formulation which leads to a subgrid model in which subgrid scales are incorporated analytically through appropriate Greenʹs functions. It is shown that entirely new and accurate methods emerge and that GLS methods can be obtained as special cases of the more general subgrid model.