Solution of Contaminant Transport Equations Using an Orthogonal-Upstream Finite Element Scheme

摘要

An orthogonal-upstream weighting finite element scheme, which will result in a matrix amenable to successive over-relaxation (SOR) solution strategies, is presented for approximating the contaminant transport equation in subsurface media. This scheme differs from the standard Galerkin and non-orthogonal-upstream weighting schemes in that the set of weighting functions is required to be orthogonal to the set of basis functions. These weighting functions are referred to as orthogonal-upstream weighting functions and are developed for line, quadrilateral, and triangular elements. Numerical results have been obtained for two examples and are compared with results obtained with analytical solutions and the Galerkin and/or non-orthogonal-upstream weighting schemes. It is found that the direct elimination solutions of the orthogonal-upstream finite element equations yield results comparable to those obtained by the direct solution of the Galerkin and/or non-orthogonal-upstream weighting finite element equations. The SOR computations of the orthogonal-upstream finite element scheme generate convergent solutions for all cases. In contrast, the SOR calculations of the Galerkin and/or non-orthogonal-upstream finite element schemes result in convergent solutions for dispersion-dominant cases, but produce divergent results for advection-dominant cases. For small problems, when the central processing unit (CPU) memory is not a consideration, the direct elimination solutions of the Galerkin and non-orthogonal-upstream finite element methods are the most efficient schemes. For large problems, when SOR iteration must be employed to solve the matrix equation owing to the unavailability of CPU memory, the orthogonal-upstream weighting scheme provides the only alternative because it is the only scheme giving convergent SOR computations for all Peclet numbers. 14 references, 4 figures, 2 tables. (ERA citation 09:038452)