Computational Geosciences

, Volume 23, Issue 5, pp 1141–1160 | Cite as

A direct mixed–enriched Galerkin method on quadrilaterals for two-phase Darcy flow

  • Todd ArbogastEmail author
  • Zhen Tao
Original Paper


We develop a locally conservative, finite element method for the simulation of two-phase flow on quadrilateral meshes that minimize the number of degrees of freedom (DoFs) subject to accuracy requirements and the DoF continuity constraints. We use a mixed finite element method (MFEM) for the flow problem and an enriched Galerkin method (EG) for the transport, stabilized with an entropy viscosity. Standard elements for MFEM lose accuracy on quadrilaterals, so we use the newly developed AC elements which have our desired properties. Standard tensor product spaces used in EG have many excess DoFs, so we would like to use the minimal DoF serendipity elements. However, the standard elements lose accuracy on quadrilaterals, so we use the newly developed direct serendipity elements. We use the Hoteit-Firoozabadi formulation, which requires a capillary flux. We compute this in a novel way that does not break down when one of the saturations degenerate to its residual value. Extension to three dimensions is described. Numerical tests show that accurate results are obtained.


Mixed method Enriched Galerkin AC elements Serendipity elements Capillary flux Entropy viscosity 

Mathematics Subject Classification (2010)

79S05 65M60 65M08 


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Texas at AustinAustinUSA
  2. 2.Institute for Computational Engineering and SciencesAustinUSA
  3. 3.Institute for Computational Engineering and SciencesUniversity of Texas at AustinAustinUSA

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