Mixed Methods for Two-Phase Darcy--Stokes Mixtures of Partially Melted Materials with Regions of Zero Porosity

Abstract

The Earth's mantle (or, e.g., a glacier) involves a deformable solid matrix phase within which a second phase, a fluid, may form due to melting processes. The system is modeled as a dual-continuum mixture, with at each point of space the solid matrix being governed by a Stokes flow and the fluid melt, if it exists, being governed by Darcy's law. This system is mathematically degenerate when the porosity (volume fraction of fluid) vanishes. Assuming the porosity is given, we develop a mixed variational framework for the mechanics of the system by carefully scaling the Darcy variables by powers of the porosity. We prove that the variational problem is well-posed, even when there are regions of one and two phases. We then develop an accurate mixed finite element method for solving this Darcy--Stokes system and prove a convergence result. Numerical results are presented that illustrate and verify the convergence of the method.