Error analysis of a mixed finite element method for a Cahn-Hilliard-Hele-Shaw system

We present and analyze a mixed finite element numerical scheme for the Cahn-Hilliard-Hele-Shaw equation, a modified Cahn-Hilliard equation coupled with the Darcy flow law. This numerical scheme was first reported in Feng and Wise (SIAM J Numer Anal 50:1320-1343, 2012), with the weak convergence to a weak solution proven. In this article, we provide an optimal rate error analysis. A convex splitting approach is taken in the temporal discretization, which in turn leads to the unique solvability and unconditional energy stability. Instead of the more standard error estimate, we perform a discrete error estimate for the phase variable, through an inner product with the numerical error function associated with the chemical potential. As a result, an unconditional convergence (for the time step in terms of the spatial resolution h) is derived. The nonlinear analysis is accomplished with the help of a discrete Gagliardo-Nirenberg type inequality in the finite element space, gotten by introducing a discrete Laplacian of the numerical solution, such that , for every , where is the finite element space.