A fully implicit edge/face centered discontinuous Galerkin / mixed finite element scheme for the advection-dispersion equation

Mixed Finite Element (MFE) method is a robust numerical technique for solving elliptic and parabolic partial differential equations (PDEs). However, MFE can generate solutions with strong unphysical oscillations and/or large numerical diffusion for hyperbolic type PDEs. For its part, Discontinuous Galerkin (DG) finite element method is well adapted to solve hyperbolic systems and can accurately reproduce solutions involving sharp fronts. Therefore, the combination of DG and MFE is a good strategy for solving hyperbolic/parabolic problems such as advection - diffusion/dispersion equations. The classical formulation of the two methods is based on operator and time splitting allowing for separate solutions to advection with an explicit scheme and to dispersion with an implicit scheme. However, this kind of approach has the following drawbacks: (i) it lacks efficiency, as two systems with different unknowns are solved at each time step, (ii) it induces errors generated by the splitting, (iii) it can be CPU wise-expensive because of the CFL constraint, and (iv) it cannot be employed for steady-state transport simulations. To overcome these difficulties, we develop in this work a fully implicit edge/face centered DG-MFE formulation where the two methods share the same unknowns. In this formulation, the DG method is developed on lumping regions associated with the mesh edges/faces instead of mesh elements. Thus, the traces of concentration at mesh edges/faces, which are the Degrees Of Freedom (DOF) of the hybrid-MFE, are also part of the DOFs of the DG. The temporal discretization is based on the Crank-Nicolson method for both advection and dispersion. Numerical tests are performed to validate the new scheme by comparison against an analytical solution and to show its ability to handle steady-state transport simulations. The procedure is developed for 2D triangular meshes but can easily be extended to other 2D and 3D shape elements.