High-Order Mimetic Finite Differences for Anisotropic Elliptic Equations

Abstract Fractured geologic media can yield anisotropies in solute and heat diffusion due to the formation of changing fluid network connectivity in a rock matrix. In this paper we model Steady-state anisotropic heat diffusion as an elliptic partial differential equation with a symmetric positive definite second rank thermal conductivity tensor. We model diffusive flux as a non-diagonal symmetric tensor, which can potentially have jump discontinuities that are not aligned with the coordinate axis. The presence of jump discontinuities due to joints and faults in a rock matrix impose difficulties on existing, well-established numerical schemes that model diffusive transport. In our scheme, we model diffusive flux using mimetic finite difference operators, which are discrete analogs of the classical continuous differential operators. We introduce a 2nd- and 4th-order mimetic formulation for computing anisotropic fluxes. Numerical results demonstrate our formulation yields a substantial improvement compared to similar mimetic schemes.