Optimally Blended Spectral Finite Elements Scheme for Maxwell’s Equation

Abstract

We study the dispersive properties of the Maxwell equations for optimally blended nite-spectral element scheme using tensor product elements de ned on rectangular grid in d-dimensions. We prove and give analytical expressions for the discrete dispersion relations for Maxwell equations. We nd that for a rectangular grid (a) the analytical expressions for the discrete dispersion relation in higher dimensions can be obtained using one dimensional discrete dispersion error expressions; (b) the optimum value of the blending parameter for Maxwell equations is p p + 1 8p 2 N for any number of spatial dimensions; (c) we get analytical expressions for the discrete dispersion relations for nite element and spectral element schemes when the value of the blending parameter is chosen to be 0 and 1 respectively; (d) the absolute accuracy of the optimally blended scheme is O(pð€€€2) and O(pð€€€1) times better than that of the pure nite and spectral element schemes respectively.