J. W. Cahn and J. E. Hilliard proposed the Cahn–Hilliard (CH) equation. The equation is a mathematical physics model that describes the process of phase separation. The CH equation is as follows
$$\begin{aligned} \frac{du}{dt} = M \Delta \,( F^{\prime }(u) -\epsilon ^2\Delta u) \qquad \text {in} ~\Omega \times [0,T], \end{aligned}$$
(1)
with the Neumann boundary conditions
$$\begin{aligned} \frac{\partial u}{\partial \nu }=0,\qquad \frac{\partial \left( -\epsilon ^2\Delta u+F^\prime (u)\right) }{\partial \nu }=0\qquad \text {on}~\partial \Omega \times [0,T]. \end{aligned}$$
(2)
We consider the initial condition at \(t=0\) as
$$\begin{aligned} u(\varvec{x},0)=u_0(\varvec{x}) \qquad \text {for}~ \varvec{x}\in \Omega , \end{aligned}$$
(3)
where \(\nu \) denotes the unit outward normal of the boundary and \(\Omega \) is a bounded domain in \(\mathbb {R}^d ~(d=1,2,3)\). The solution u is a rescaled density of atoms or concentration of one of the material components where, in the most applications \(u\in [-1,1]\). We should note that M is the mobility (here a constant) and the variable \(\epsilon \) is a positive constant. The equation arises from the Ginzburg–Landau free energy
$$\begin{aligned} \mathcal {L}(u)=\int _{\Omega }\left( F(u)+\frac{\epsilon ^2}{2}|\nabla u|^2\right) ~\text {d}\varvec{x}. \end{aligned}$$
(4)
The above free energy includes the bulk energy F(u) and the interfacial energy (the second term). A popular example of a nonlinear function is
$$\begin{aligned} F(u)=\frac{1}{4}u^2(1-u)^2. \end{aligned}$$
(5)
The Cahn–Hilliard–Cook equation presents a more realistic model including the internal thermal fluctuations. It can be derived from (1) by adding the thermal noise as
$$\begin{aligned} \frac{du}{dt}&= M \Delta \,( F^{'}(u) -\epsilon ^2\Delta u)+\sigma \, \xi \qquad&\text {in} ~\Omega \times [0,T], \end{aligned}$$
(6a)
$$\begin{aligned} \frac{\partial u}{\partial \nu }&=0,\quad \frac{\partial \Delta u}{\partial \nu }=0&\text {on} ~\partial \Omega \times [0,T], \end{aligned}$$
(6b)
where \(\xi \) indicates the colored noise (here white noise) and \(\sigma \) is the noise intensify measure.
Ciarlet–Raviart mixed finite element
To construct a mixed finite element approximation of the Cahn–Hilliard–Cook equation, we first find its weak formulation. For this purpose, we define the auxiliary variable
$$\begin{aligned} \gamma :=-M\Delta u+F'(u). \end{aligned}$$
(7)
Therefore, the Cahn–Hilliard–Cook equation can be rewritten in the form
$$\begin{aligned} \gamma&=-M\Delta u+F'(u), \end{aligned}$$
(8a)
$$\begin{aligned} d u&=\nabla \cdot (M\nabla \gamma )+\sigma \, dW, \end{aligned}$$
(8b)
$$\begin{aligned} \frac{\partial u}{\partial \nu }&=\frac{\partial \gamma }{\partial \nu }=0. \end{aligned}$$
(8c)
The weak formulation of (8) is given by seeking \((u,\gamma )\in H^1_{*}(\Omega )\times H^1_{*}(\Omega )\) such that
$$\begin{aligned} (\gamma ,\chi )_\Omega&=\left( M\nabla u,\nabla \chi \right) _{\Omega }\nonumber \\&\quad +\left( F'(u),\chi \right) _{\Omega }\qquad&\forall \,\chi \in H^1_{*}(\Omega ), \end{aligned}$$
(9a)
$$\begin{aligned} \left( d u,\psi \right) _{\Omega }&=-\left( M\nabla \gamma ,\nabla \psi \right) _{\Omega }\nonumber \\&\quad +\sigma \left( d W,\psi \right) _{\Omega }&\forall \,\psi \in H^1_*(\Omega ), \end{aligned}$$
(9b)
where
$$\begin{aligned} H_*^1(\Omega )=\left\{ u\in H^1_*(\Omega )~|\int _{\Omega } u~ \text {d}x=0 \right\} . \end{aligned}$$
(10)
Now let \(\tau _h\) be a family of triangulations of \(\Omega \) into a finite number of elements (simplex) such that
$$\begin{aligned} h=\max _{k\in \tau _h} \text {diam}(k). \end{aligned}$$
(11)
We assume that each element has at least one face on \(\partial \Omega \) and \(k_1,k_2\in \tau _h\) have only one common vertex or a whole edge. Now we define
$$\begin{aligned} M_h&:=\left\{ v\in C(\Omega )|~v|_k\in \mathbb {P}_n,~n\ge 1~\forall k\tau _h \right\} , \end{aligned}$$
(12)
$$\begin{aligned} N_h&:=M\cap H_*^1(\Omega ), \end{aligned}$$
(13)
and \(\mathbb {P}_n\) is the space of all polynomials of degree at most \(n\ge 1\). The semi-discrete Galerkin approximation of the solutions (9a)–(9b) may be defined as a pair of approximations \((u_h,\gamma _h)\in N_h \times M_h\) for which the equalities
$$\begin{aligned} (\gamma _h,\chi _{_h})_\Omega&=\left( M\nabla u_h,\nabla \chi _h\right) _\Omega \nonumber \\&\quad +\left( F^\prime (u_h),\chi _h\right) _\Omega \quad&\forall \,\chi _{_h}\in M_h, \end{aligned}$$
(14a)
$$\begin{aligned} (du_h,\psi _h)_\Omega&=-\left( M\nabla \gamma _h,\nabla \psi _h\right) _\Omega \nonumber \\&\quad +(dW,\psi _h)_\Omega&\forall \, \psi _h\in N_h, \end{aligned}$$
(14b)
hold.
Full discretization scheme
In this section we apply a fully discretize scheme based the mild solution of (8). In order to obtain the fully discretized scheme, we first rewrite the variational formulation of (9) as follows:
Find \((u,\gamma )\in H^1_{*}(\Omega )\times H^1_{*}(\Omega )\) such that
$$\begin{aligned}&(\gamma ,\chi )_\Omega \nonumber \\&\quad =(M\nabla u,\nabla \chi )_{\Omega }+(F^\prime (u),\chi )_{\Omega }\qquad \forall ~\chi \in H^1_*(\Omega ), \end{aligned}$$
(15a)
$$\begin{aligned}&(u(t),\psi )_\Omega -(u_0(t),\psi )_\Omega \nonumber \\&\quad =-\int _{0}^{t}(M\nabla \gamma ,\nabla \psi )_\Omega {+}\sigma (W(t),\psi )_\Omega \qquad \forall ~\psi \in H^1_*(\Omega ). \end{aligned}$$
(15b)
The mixed finite element formulation of (15) is defined by \((u_h(t),\gamma _h(t))\in N_h\times M_h\) such that
$$\begin{aligned}&(\gamma _h,\chi _h)_\Omega \nonumber \\&\quad =(M\nabla u_h,\nabla \chi _h)_{\Omega }+(F^\prime (u_h),\chi _h)_{\Omega }\qquad \quad \forall ~\chi _h\in M_h, \end{aligned}$$
(16a)
$$\begin{aligned}&(u_h(t),\psi _h)_\Omega -(u_0(t),\psi _h)_\Omega \nonumber \\&\quad =-\int _{0}^{t}(M\nabla \gamma _h,\nabla \psi _h)_\Omega \nonumber \\&\qquad +\sigma (W(t),\psi _h)_\Omega \quad \forall ~\psi _h \in N_h\quad t\in (0,T]. \end{aligned}$$
(16b)
Now we can rewrite (8) in the following abstract evolution equation
$$\begin{aligned} dX(t)+\left( A^2 X+AF(X)\right) dt&=\sigma d W(t) \qquad t\in (0,T], \end{aligned}$$
(17)
$$\begin{aligned} X(0)&=X_0, \end{aligned}$$
(18)
where A is the negative Neumann Laplacian considered as an unbounded operator in the Hilbert space \(H=L_2(\Omega )\), which is the generator of an analytic semigroup \((S(t),~t\ge 0)\) on H [33]. The initial value \(X_0\) is deterministic and W is a cylindrical Wiener process in H (i.e., the spatial derivative of a space–time white noise) with respect to a filtered probability space \((\Psi ,\mathcal {F},\mathbb {P}, \{F_t\}_{t\ge 0})\) defined as
$$\begin{aligned} W(t)=\sum _{j,k=1}^{\infty } \mu _{j,k}^{\frac{1}{2}}\,\beta _{j,k}(t)\,\sin (j\pi x) \sin (k\pi y). \end{aligned}$$
(19)
Here, \(\left\{ \beta _{j,k}\right\} _{j,k\in \mathbb {N}}\) indicates a family of real-valued, identically distributed independent Brownian motions and \(\left\{ \mu _{j,k}\right\} _{j,k\in \mathbb {N}}\) denote the eigenvalues (here, \(\mu _{j,k}=1\) since W(t) is cylindrical) [34]. Therefore, the Cahn–Hilliard–Cook equation has a continuous mild solution
$$\begin{aligned} X(t)&=S(t)X_0+\int _{0}^{t}A S(t-s)F(X(s))~\text {d}s\nonumber \\&\quad +\sigma \int _{0}^{t} S(t-s)~\text {d}W(s), \end{aligned}$$
(20)
where \(t\in [0,T]\), \(X:[0,T]\times \Omega \rightarrow H\) and \(S(t)=\text {e}^{-tA^2}\) used as the analytic semigroup generated by \(-A^2\). The existence of the mild solution X was shown in [35]. Considering \(\Vert X_0\Vert _{L^2(\Omega ,H)}\le +\infty \), for all \(t\in [0,T]\) the solution X satisfies [31]
$$\begin{aligned} \Vert X(t)\Vert _{L^2(\Omega ,H)}\le C(T) \left( 1+\Vert X_0\Vert _{L^2(\Omega ,H)}\right) , \end{aligned}$$
(21)
where C is a constant which depends on T. Also, for \(0\le s<t\le T\), there exists a constant C(T) such that the mild solution satisfies the inequality [31]
$$\begin{aligned} \Vert X(t)-X(s)\Vert _{L^2(\Omega ,H)}\le C(T)\sqrt{t-s}\left( 1+\Vert X_0\Vert _{L^2(\Omega ,H)}\right) . \end{aligned}$$
(22)
In order to estimate the mild solution we use finite elements for space discretization and the semi-implicit Euler–Maruyama scheme in time direction. Let us assume that \(V_\ell \) (\(\ell \in \mathbb {N}_0\)) is a nested family of finite element subsequences of H with refinement level \(\ell >0\) and refinement size \(h_\ell ~(\ell \in \mathbb {N}_0\)). Defining the analytic semigroup \(S_{\ell }=\text {e}^{-tA_\ell ^2}\), for \(t\in T\), the semidiscrete problem (20) has the form
$$\begin{aligned} X_\ell (t)&=\text {e}^{-tA_\ell ^2}X_\ell (0)+\int _{0}^{t}A_\ell \text {e}^{-(t-s)A_\ell ^2} F(X_\ell (s))~\text {d}s\nonumber \\&\quad +\sigma \int _{0}^{t}\text {e}^{-(t-s)A_\ell ^2} ~\text {d}W(s). \end{aligned}$$
(23)
For the time direction, we approximate the time discretization with step sizes \(\delta t^\zeta =Tr^{-\zeta }\) where \(r>1\). Therefore, for \(\zeta \in \mathbb {N}_0\), we define the sequence
$$\begin{aligned} \Theta ^\zeta :=\left\{ t_k^\zeta =Tr^{-\zeta }k=\delta t^\zeta k,~k=0,\ldots ,r^\zeta \right\} \end{aligned}$$
(24)
of equidistant time discretization. In the computational geometry \((\Omega )\), we estimate the mild solution X, with a finite element discretization. In other words, we suppose that the domain can be partitioned into quasi-uniform triangles or tetrahedra such that sequences \(\{\tau _{h_{\ell }} \}_{{\ell }=0}^{\infty }\) of regular meshes are obtained. For any \( \ell \ge 0\), we denote the mesh size of \(\tau _{h_{\ell }}\) by
$$\begin{aligned} h_{\ell } := \max _{K\in \tau _{h_\ell }} {\text {diam}} K. \end{aligned}$$
Uniform refinement of the mesh can be achieved by regular subdivision. This results in the mesh sizes
$$\begin{aligned} h_\ell := r^{-\ell } h_0, \end{aligned}$$
(25)
where \(h_0\) denotes the mesh size of the coarsest triangulation and \(r>1\) is independent of \(\ell \).