Achieving local mass conservation when using continuous Galerkin finite element methods to solve solute transport equations with spatially variable coefficients in a transient state

Abstract

Mass balance checks are often used to evaluate the abilities of numerical simulators and assess the accuracy of the solutions but are often insufficient. In addition, there is a need to calculate the difference in mass fluxes, through downstream subdomain analysis, to determine whether remediation technologies are suitable for upstream contaminated sites. The Galerkin finite element method (GFEM) has been commonly considered locally nonconservative, whereas the finite difference method (FDM) is a local mass conservation method. In this study, we proved that the GFEM has local mass balance equations on “mass conserved elements” (constructed by connecting lines that perpendicularly cross element faces at midpoints) in solute transport problems with spatially variable velocities and dispersion coefficients, identical to discretized balance equations in the FDM. Furthermore, we found that locally conservative GFEMs with two types of approaches involving spatially variable coefficients (i.e., a linear combination approach and an average approach) give rise to two types of postprocessing techniques for calculating local mass fluxes. These are equivalent to an area-weighted method and first-order Taylor series method, respectively. Finally, to illustrate the practical applicability of these GFEMs, we used the proposed postprocessing approaches to resolve practical problems involving computation of local mass fluxes, based on information directly obtainable from finite element solutions of concentration distribution. Our findings revealed that the local mass balance errors were negligibly small, regardless of numerical concentration accuracy.