摘要

We consider the linear degenerate elliptic system of two first order equations u = -a(phi)(del p-g) and del.(b(phi)u) + phi p = phi(1/2)f, where a and b satisfy a(0) = b(0) = 0 and are otherwise positive, and the porosity phi >= 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, phi represents the phase variable separating the solid one-phase (phi = 0) and fluid-solid two-phase (phi > 0) regions. The equations should remain well-posed as phi vanishes so that the free boundary between the one-and two-phase regions need not be found explicitly. Two main problems arise. First, as phi vanishes, one equation is lost. Second, it is shown by stability or energy bounds for the solution that the pressure p is not controlled outside the support of phi . After an appropriate scaling of the pressure and velocity, we obtain a mixed system for which we can show existence and uniqueness of a solution over the entire domain, regardless of where phi vanishes. The key is to define the appropriate Hilbert space containing the velocity, which must have a well defined scaled divergence and normal trace. We then develop for the scaled problem a mixed finite element method based on lowest order Raviart-Thomas elements which is stable and has an optimal convergence rate for sufficiently smooth solutions. We show some numerical results that verify the optimal rates of convergence for sufficiently regular solutions.

  • 出版日期2016