Direct Conservative Domain in the Continuous Galerkin Method for Groundwater Models

The continuous Galerkin finite element method is commonly considered locally nonconservative because a single element with fluxes computed directly from its potential distribution is unable to conserve its mass and fluxes across edges that are discontinuous. Some literature sources have demonstrated that the continuous Galerkin method can be locally conservative with postprocessed fluxes. This paper proposes the concept of a direct conservative domain (DCD), which could conserve mass when fluxes are computed directly from the potential distribution. Also presented here is a method for modifying the advection fluxes to obtain different conservative domains from the DCDs. Furthermore, DCDs are used to analyze the local conservation of several postprocessing algorithms, for which DCDs provide the theoretical basis. The local conservation of DCDs and the proposed method are illustrated and verified by using a hypothetical 2-D model.