Error analysis of a mixed finite element method for a Cahn–Hilliard–Hele–Shaw system - Liu, Yuan; Chen, Wenbin; Wang, Cheng; Wise, Steven M. (Numerische Mathematik, 2017)
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 ℓ ∞ ( 0 , T ; L 2 ) ∩ ℓ 2 ( 0 , T ; H 2 ) error estimate, we perform a discrete ℓ ∞ ( 0 , T ; H 1 ) ∩ ℓ 2 ( 0 , T ; H 3 ) error estimate for the phase variable, through an L 2 inner product with the numerical error function associated with the chemical potential. As a result, an unconditional convergence (for the time step 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 Δ h of the numerical solution, such that Δ h ϕ ∈ S h , for every ϕ ∈ S h , where S h is the finite element space.