Locally conservative, stabilized finite element methods for variably saturated flow

Standard Galerkin finite element methods for variably saturated groundwater flow have several deficiencies. For instance, local oscillations can appear around sharp infiltration fronts without the use of mass-lumping, and velocity fields obtained from differentiation of pressure fields are discontinuous at element boundaries. Here, we consider conforming finite element discretizations based on a multiscale formulation along with recently developed, local postprocessing schemes. The resulting approach maintains the basic flexibility and appeal of traditional finite element methods, while controlling nonphysical oscillations and producing element-wise mass-conservative velocity fields. Accuracy and efficiency of the proposed schemes are evaluated through a series of steady-state and transient variably saturated groundwater flow problems in homogeneous as well as heterogeneous domains. The schemes are formulated for a generic nonlinear advection–diffusion equation and are thus applicable to many other flow models.