Spatial Finite-Element Lumping Techniques for the Quadrilateral Mesh S_n Equations in X-Y Geometry

We consider two general finite-element lumping techniques for the S_n equations with discontinuous finite-element spatial discretization and apply them to quadrilateral meshes in x-y geometry. One technique is designed to ensure a conservative approximation and is referred to as conservation preserving (CP). The other technique is designed to preserve the exact solution whenever it is contained within the trial space and is referred to as solution preserving (SP). These techniques are applied in x-y geometry on structured nonorthogonal grids using the bilinear-discontinuous finite-element approximation. The schemes are both theoretically analyzed and computationally tested. Analysis shows that the two lumping schemes are equivalent on parallelogram meshes. Computational results indicate that both techniques perform extremely well on smooth quadrilateral meshes. On nonsmooth meshes, the preserving technique retains its excellent performance while the CP technique degrades. The reasons for this degradation are discussed. Although the SP scheme has proven to be generally effective on quadrilateral meshes in x-y geometry, it is not expected to be effective for quadrilaterals in r-z geometry or for hexahedra in three-dimensional Cartesian geometry. Thus, a full lumping procedure for general nonorthogonal meshes that possesses all of the desired properties has yet to be found. For reasons that are discussed, it appears unlikely that such a procedure exists.