A Cell-Centered Finite Difference Method for a Degenerate Elliptic Equation Arising from Two-Phase Mixtures

We consider the linear degenerate elliptic system of two first order equations u = −d(φ)(∇p − g) and ∇ · u + φp = φf , where d satisfies d(0) = 0 and is otherwise positive, and the porosity φ ≥ 0 may be zero on a set of positive measure. This model equation has a similar degeneracy to that arising in the equations describing the mechanical system modeling the dynamics of partially melted materials, e.g., in the Earth’s mantle and in polar ice sheets and glaciers. In the context of mixture theory, φ represents the phase variable separating the solid one-phase (φ = 0) and fluid-solid two phase (φ > 0) regions. After an appropriate scaling of the pressure and velocity, we obtain a well-posed mixed system, and we develop a cell-centered finite difference method based on lowest order RaviartThomas elements. The scheme is both stable and locally mass conservative. We present numerical results that show optimal rates of convergence and that superconvergence is attained for sufficiently regular solutions.