Origin of vector parasites in numerical Maxwell solutions

Dispersion relations are derived for conventional finite-element (FE) and finite-difference (FD) approximations for four versions of the Maxwell equations in the plane: the double-curl equation; the vector Helmholtz equation; the penalty equation; and the primitive, coupled Maxwell curl equations. Comparison with their analytic counterparts reveals the presence and origin of vector parasites. For the double-curl and penalty methods, the dispersion relations are double-valued, admitting an extra, spurious dispersion surface of real-valued wavenumbers. As a result, low wavenumbers support well-resolved and poorly resolved vector parasites. The Helmholtz schemes have monotonic, single-valued dispersion relations for divergence-free physical modes. Specification of divergence-free boundary conditions is sufficient to guarantee the absence of parasites. The primitive schemes have single-valued but folded (nonmonotonic) dispersion relations, supporting poorly resolved vector parasites at low wavenumbers. Use of a staggered finite-difference grid eliminates these parasites and results in a dispersion relation identical to that for the Helmholtz scheme. In cases where vector parasites arise, the same essential weakness in the discretized form of either the first or cross-derivative is responsible