Galerkin Differences for acoustic and elastic wave equations in two space dimensions

The Galerkin Difference method, a finite element method built using standard Galerkin projection techniques but employing nonstandard basis functions, was originally developed for one space dimension in [1]. Here the method is extended to two space dimensions using a tensor product construction. Theoretical and computational evidence shows the method behaves as expected for the acoustic wave equation. For the elastic wave equation, the approximations are found to be at least as accurate as predicted, but with a free surface the scheme may exhibit an unexpected superconvergence. Extension to curvilinear mapped grids is also considered for acoustics. In all cases, the use of a tensor product construction allows for efficient solution of the linear system involving the mass matrix, which implies optimal linear time solutions with respect to the number of degrees of freedom.