An optimally blended finite-spectral element scheme with minimal dispersion for Maxwell equations

We study the dispersive properties of the time harmonic Maxwell equations for optimally blended finite-spectral element scheme using tensor product elements defined on rectangular grid in d  -dimensions. We prove and give analytical expressions for the discrete dispersion relations for this scheme. We find that for a rectangular grid (a) the analytical expressions for the discrete dispersion error in higher dimensions can be obtained using one dimensional discrete dispersion error expressions; (b) the optimum value of the blending parameter is p/(p+1)p/(p+1) for all p∈Np∈N and for any number of spatial dimensions; (c) analytical expressions for the discrete dispersion relations for finite element and spectral element schemes can be obtained when the value of blending parameter is chosen to be 0 and 1 respectively; (d) the optimally blended scheme guarantees two additional orders of accuracy compared with standard finite element and spectral element schemes; and (e) the absolute accuracy of the optimally blended scheme is O(p-2)O(p-2) and O(p-1)O(p-1) times better than that of the pure finite element and spectral element schemes respectively.