Mass Conserving Schemes for Saturated Groundwater Flow

Local mass conservation is of great importance for accurate simulation of fluid flows, especially if the resulting velocity field is to be coupled to a transport equation. Previous work on simulation of saturated groundwater flows have used both mixed finite element methods and discontinuous Galerkin methods to achieve local mass conservation. However, legacy code using a continuous Galerkin finite element scheme must be rewritten to incorporate either of these methods. Recent research on so called grad-div stabilization schemes has shown they are capable of significantly improving mass conservation in Taylor-Hood approximations of the Stokes and Navier-Stokes equations. Grad-div stabilization schemes introduce one additional term into the governing equation. Another approach currently receiving a good deal of attention for approximating conservative solutions to the Navier-Stokes equations is the use of Scott-Vogelius approximating elements. These elements (i) require a barycenter refinement of a regular mesh, and (ii) generate an approximation which is point-wise mass conservative. In this paper, we compare standard discretization schemes for the saturated flow equation with a grad-div stabilized scheme and a Scott-Vogelius scheme. We provide a comparison of the different schemes, including the degrees of freedom associated with each method, local mass conservation errors for each method, and convergence results.