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An Efficient Lumped Mixed Hybrid Finite Element Formulation for Variably Saturated Groundwater Flow

Laboratoire d'Hydrologie et Géochimie de Strasbourg, Université de Strasbourg, CNRS/UMR 7517, 1 rue Blessig, F-67084 Strasbourg, France

Correspondence: ^* Corresponding author (benjamin.belfort{at}unistra.fr).

Received for publication 11 July 2008. Accurate numerical simulation of infiltration in the vadose zone remains a challenge, especially when very sharp fronts are modeled. In this work, we used the mixed hybrid finite element (MHFE) method, which allows a simultaneous approximation of both pressure head and velocity and can handle general irregular grids with highly heterogeneous permeability. For many problems dealing with unsaturated water flow, however, the MHFE solutions exhibit significant unphysical oscillations. To avoid this phenomenon, we developed an efficient mass-lumping scheme with the MHFE method for solving the mixed form of the Richards equation. In this work, the standard and lumped MHFE formulations were compared and the ability of the lumped formulation to reduce unphysical oscillations was determined for one- and two-dimensional infiltration problems. Theoretical analysis based on the M-matrix property, which guarantees the discrete maximum principle, and practical test cases performed in this study underline the advantage of using an acute triangulation to completely remove the unphysical oscillations. Indeed, contrary to the standard approach, the lumped formulation satisfies the M-matrix property without any constraint on the time step size to be used.

Abbreviations: MHFE, mixed hybrid finite element • RE, Richards equation • RT0, lowest order Raviart–Thomas method