Fully discrete IPDG–HMM for multiscale Richards equation of unsaturated flow in porous media

In this paper, we propose a fully discrete method for the multiscale Richards equation which describes the flow transport in unsaturated heterogeneous porous media. Under the framework of heterogeneous multiscale method (HMM), a fully discrete interior penalty discontinuous Galerkin finite element method (IPDG–FEM) is applied over a macro-scale mesh. The fully discrete method means that it takes into account not only the fully macro-scale discretization but also the fully micro-scale discretization for local cell problems. Error estimates between the numerical solution and the solution of homogenized problem are derived under the assumption that the permeability is periodic. Numerical experiments with periodic and random permeability are carried out both for Gardener model and van Genuchten–Mualem model of Richards equation to show the efficiency and accuracy of the proposed method.