Models of groundwater flow and solute transport may be affected by numerical error, leading to quantitative and qualitative changes in behavior. In this paper we compare and combine three methods of assessing the extent of numerical error: grid refinement, mathematical analysis, and benchmark test problems. In particular, we assess the popular solute transport code SUTRA <Voss, 1984> as being a typical finite element code. Our numerical analysis suggests that SUTRA incorporates a numerical dispersion error and that its mass-lumped numerical scheme increases the numerical error. This is confirmed using a Gaussian test problem. A modified SUTRA code, in which the numerical dispersion is calculated and subtracted, produces better results. The much more challenging Elder problem <Elder, 1967; Voss and Souza, 1987> is then considered. Calculation of its numerical dispersion coefficients and numerical stability show that the Elder problem is prone to error. We confirm that Elder problem results are extremely sensitive to the simulation method used.