A technique for improving the accuracy of quadrangular mixed finite elements for Darcy’s flow on heterogeneous domains

Abstract

The mixed hybrid finite element (MHFE) method is well suited for the resolution of the Darcy’s flow on anisotropic and heterogeneous domains. However, the accuracy of the method can be altered for highly distorted meshes with non-convex quadrangles. Moreover, the standard quadrangular MHFE method can lead to non-physical oscillations for transient simulations since it does not respect the maximum principle. In this work, we derive a new realization of the MHFE method on general convex or non-convex quadrangular elements. Each element is fictitiously divided into triangles. The mass balance and flux law are then discretized over each triangle and aggregated to eliminate interior degrees of freedom at the quadrangular element level. The method is combined with the mass lumping procedure for triangles to improve the monotonicity of the discretization. The material properties as well as the pressure and the divergence of the flux are allowed to vary inside the quadrangular element to better describe heterogeneous domains. The obtained matrix is symmetric and positive definite and has the same stencil than the standard approach. The numerical experiments show the performances of this formulation compared to the standard one for heterogeneous domains and non-convex elements. An example is also provided for transient flow simulations where the unphysical oscillations are avoided with the new approach.