Mixed finite element methods and higher-order temporal approximations

The accurate numerical approximation of subsurface flow and transport processes in heterogeneous aquifers remains difficult. A necessary step in this task is the accurate representation of fluid velocity fields. In the recent past, mixed finite element methods have been investigated, since they provide velocity approximations that both conserve mass over individual mesh elements and are continuous across element interfaces. But, little work has been done to this point for fully three-dimensional problems. Furthermore, the existing temporal discretizations have been restricted to traditional first- and second-order approximations. In this work, we consider a fully three-dimensional mixed-hybrid finite element spatial discretization together with an adaptive higher-order time discretization applied to single-phase groundwater flow in heterogeneous porous media. We compare the adaptive higher-order temporal approximation, which is robust and provides formal error control, to traditional lower-order methods for accuracy and efficiency for a set of problems. The numerical experiments demonstrate that this approach provides several benefits with negligible computational overhead.