An orthogonal‐upstream weighting (OUW) finite element scheme, which will result in a matrix amenable to point successive overrelaxation (SOR) solution strategies, is presented for approximating the aquifer contaminant‐transport equation. This scheme differs from the standard Galerkin weighting (GW) and nonorthogonal‐upstream weighting (NUW) 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 OUW functions and are developed for line, quadrilateral, and triangular elements. Numerical results have been obtained for two examples and are compared with results obtained through analytical solutions (AS) and the Galerkin and/or NUW schemes. The direct elimination solutions of the OUW finite element equations yield results comparable to those obtained by the direct solution of the Galerkin and/or NUW finite element equations. The SOR computations of the OUW finite element scheme generate convergent solutions for all cases. In contrast, the SOR calculations of the Galerkin and/or NUW 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 NUW finite element methods are the most efficient schemes. For large problems, when one wish to use point SOR iteration to solve the matrix equation because of CPU memory limitations, the OUW scheme provides an alternative because it gives convergent SOR computations for all Peclet numbers. Even when CPU memory is not a consideration, one may still prefer OUW to Galerkin or NUW because of its susceptibility to the SOR iteration for all grid Peclet numbers, especially for large problems when the SOR iteration will be much more efficient than the direct elimination solution.