Least-Squares Spectral Element Method on a Staggered Grid

This paper describes a mimetic spectral element formulation for the Poisson equation on quadrilateral elements. Two dual grids are employed to represent the two first order equations. The discrete Hodge operator, which connects variables on these two grids, is the derived Hodge operator obtained from the wedge product and the inner-product. The gradient operator is not discretized directly, but derived from the symmetry relation between gradient and divergence on dual meshes, as proposed by Hyman et al., [5], thus ensuring a symmetric discrete Laplace operator. The resulting scheme is a staggered spectral element scheme, similar to the staggering proposed by Kopriva and Kolias, [6]. Different integration schemes are used for the various terms involved. This scheme is equivalent to a least-squares formulation which minimizes the difference between the dual velocity representations. This generates the discrete Hodge-star operator. The discretization error of this schemes equals the interpolation error.