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

Abstract

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 $\ell ^\infty (0,T;L^2) \cap \ell ^2 (0,T; H^2)$ error estimate, we perform a discrete $\ell ^\infty (0,T; H^1) \cap \ell ^2 (0,T; H^3 )$ 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 $\tau$ 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 $\Delta _h$ of the numerical solution, such that $\Delta _h \phi \in S_h$, for every $\phi \in S_h$, where $S_h$ is the finite element space.

Appendix 1: Discrete Gronwall inequality

We need the following discrete Gronwall inequality, cited in [26, 30]:

Lemma 4.1

Fix $T>0$, and suppose $\left\{ a_m\right\} _{m=1}^M$, $\left\{ b_m\right\} _{m=1}^M$ and $\left\{ c_m\right\} _{m=1}^{M-1}$ are non-negative sequences such that $\tau \sum _{m=1}^{M-1} c_m \le C_1$, where $C_1$ is independent of $\tau$ and M, and $M\cdot \tau = T$. Suppose that, for all $\tau >0$,

$$\begin{aligned} a_M + \tau \sum _{m=1}^{M} b_m \le C_2 +\tau \sum _{m=1}^{M-1} a_m c_m , \end{aligned}$$

(4.1)

where $C_2>0$ is a constant independent of $\tau$ and M. Then, for all $\tau >0$,

$$\begin{aligned} a_M +\tau \sum _{m=1}^{M} b_m \le C_2 \exp \left( \tau \sum _{m=1}^{M-1}c_m \right) \le C_2\exp (C_1). \end{aligned}$$

(4.2)

Note that the sum on the right-hand-side of (4.1) must be explicit.

Lemma 4.2

Suppose $\left\{ a_m\right\} _{m=1}^M$ and $\left\{ b_m\right\} _{m=0}^M$ are sequences such that $b_0=0$. Define, for any integer m, $1\le m\le M$,

$$\begin{aligned} I_m := \sum _{j=1}^m a_j b_j(b_j-b_{j-1}). \end{aligned}$$

(4.3)

Then the following identity is valid:

$$\begin{aligned} I_m =\ -\frac{1}{2} \sum _{j=1}^m(a_j-a_{j-1})b_{j-1}^2 + \frac{1}{2}\sum _{j=1}^{m} a_j (b_j - b_{j-1})^2 +\frac{1}{2}a_mb_m^2 . \end{aligned}$$

(4.4)