Convergence analysis of a fully discrete finite difference scheme for the Cahn-Hilliard-Hele-Shaw equation

We present an error analysis for an unconditionally energy stable, fully discrete finite difference scheme for the Cahn-Hilliard-Hele-Shaw equation, a modified Cahn-Hilliard equation coupled with the Darcy flow law. The scheme, proposed in [47], is based on the idea of convex splitting. In this paper, we rigorously prove first order convergence in time and second order convergence in space. Instead of the (discrete) Ls (0, T ;L 2 h) ∩ L 2 s(0, T ;H 2 h) error estimate – which would represent the typical approach – we provide a discrete Ls (0, T ;H 1 h) ∩ L 2 s(0, T ;H 3 h) error estimate for the phase variable, which allows us to treat the nonlinear convection term in a straightforward way. Our convergence is unconditional, in the sense that the time step s is in no ways constrained by the mesh spacing h. This is accomplished with the help of an Ls(0, T ;H 3 h) bound of the numerical approximation of the phase variable. To facilitate both the stability and convergence analyses, we establish a finite difference analog of a Gagliardo-Nirenberg type inequality.