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 $$\ell ^\infty (0,T;L^2) \cap \ell ^2 (0,T; H^2)$$ℓ∞(0,T;L2)∩ℓ2(0,T;H2) error estimate, we perform a discrete $$\ell ^\infty (0,T; H^1) \cap \ell ^2 (0,T; H^3 )$$ℓ∞(0,T;H1)∩ℓ2(0,T;H3) error estimate for the phase variable, through an $$L^2$$L2 inner product with the numerical error function associated with the chemical potential. As a result, an unconditional convergence (for the time step $$\tau $$τ 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 $$\Delta _h$$Δh of the numerical solution, such that $$\Delta _h \phi \in S_h$$Δhϕ∈Sh, for every $$\phi \in S_h$$ϕ∈Sh, where $$S_h$$Sh is the finite element space.