# A multilevel Monte Carlo finite element method for the stochastic Cahn–Hilliard–Cook equation

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## Abstract

In this paper, we employ the multilevel Monte Carlo finite element method to solve the stochastic Cahn–Hilliard–Cook equation. The Ciarlet–Raviart mixed finite element method is applied to solve the fourth-order equation. In order to estimate the mild solution, we use finite elements for space discretization and the semi-implicit Euler–Maruyama method in time. For the stochastic scheme, we use the multilevel method to decrease the computational cost (compared to the Monte Carlo method). We implement the method to solve three specific numerical examples (both two- and three dimensional) and study the effect of different noise measures.

## Keywords

Multilevel Monte Carlo Finite element Cahn–Hilliard–Cook equation Euler–Maruyama method Time discretization## Mathematics Subject Classification

35R60 60H15 65M60## 1 Introduction

The Cahn–Hilliard equation is a robust mathematical model for describing different phase separation phenomena, from co-polymer systems to lipid membranes. The equation is used to model binary metal alloys [1], polymers [2] as well as cell proliferation and adhesion [3]. In material science, when a binary alloy is sufficiently cooled down, we observe a partial nucleation or spinodal decomposition, i.e., the material quickly becomes inhomogeneous. In fact, after a few seconds, material coarsening will be happened [4]. In polymer solutions and blends, the phase separation process is a dynamic process that one phase stable solution separates into two equilibrium phases upon changes in temperature, pressure, concentration, or even flow fields [5]. In these cases, the spinodal decomposition is described by the Cahn–Hilliard model [6].

The equation is a nonlinear partial differential equation of fourth-order in space and first order in time for which an analytical treatment is not possible. There are several numerical techniques to solve the equation including the finite element method (FEM) [7], isogeometric analysis based on finite element method [8], multigrid finite element [9], conservative nonlinear multigrid method [10], least squares spectral element method [11], Monte Carlo methods [12], radial basis functions (RBF) [13] and meshless local collocation methods [14]. A finite element error analysis of the equation is given in [15]. Adaptive finite elements can also be applied to solve the equation using residuals based a posteriori estimates [16, 17].

A difficulty of the numerical analysis of the Cahn–Hilliard equation is the discretization of the fourth-order operator. Here, after converting the fourth-order equation into a system of two second-order equations (by introducing an auxiliary variable) and writing the variational formulation, the Ciarlet–Raviart mixed finite element method is used for the spatial discretization. The method has been implemented for the damped Boussinesq equation by the authors [18] and they considered the convergence rate and the stability for the semi-discretization scheme and the fully discretized method. For the Cahn–Hilliard equation, the technique was used in [19, 20] for the space discretization.

The stochastic Cahn–Hilliard equation was first considered by Cook [21]. The system allows for considering thermal fluctuations directly in terms of the Cahn–Hilliard–Cook (CHC) equation by a conserved noise source term. The thermal fluctuations play an essential role in the early stage of phase dynamics such as initial dynamics of nucleation [22, 23]. Some authors, such as Binder [24] and Pego [25], have expressed the belief that only the stochastic version can correctly describe the whole decomposition process in a binary alloy [26]. In [27], as another numerical approach, the authors employed the direct meshless local Petrov–Galerkin (DMLPG) to solve the stochastic Cahn-Hilliard-Cook and stochastic Swift-Hohenberg equations.

Multilevel Monte Carlo (MLMC) [28] is a simple and efficient computational technique to estimate the expected value of a random process. Using the method enables us to decrease the computational costs noticeably. The multilevel method was implemented to solve the stochastic elliptic equations, e.g., the drift-diffusion-Poisson system with uniformly distributed random variables [29] and quasi-random points [30]. In [31], the convergence and complexity of the MLMC using Galerkin discretizations in space and a Euler–Maruyama discretization in time for the parabolic equations were explained in details. The technique was used in [32] for solving parabolic (heat equation) and hyperbolic (advection equation) driven by additive Wiener noise

Generally, for the time-dependent stochastic problems, the total error consists of the spatial error (due to the finite element method), the time discretization error (due to the Euler–Maruyama technique) and the statistical error (number of samples). We already know that for the space discretization, fine meshes are needed (specifically for the curved surfaces) which lead to the higher computational complexity. The multilevel Monte Carlo method uses hierarchies of meshes for time and space approximations in the sense that the number of samples and mesh sizes (as well as time steps) on the different levels are chosen such that the errors are equilibrated. For the stochastic Cahn–Hilliard–Cook equation, we strive to determine an optimal hierarchy of meshes, number of samples and time intervals which minimize the total computational work. As a result, we give a-priori estimates on the explained error contributions. In this paper, we use the MLMC-FEM for the fourth-order stochastic equations and calculate the mild solution of the Cahn–Hilliard–Cook equation. In fact, we estimate the total computational error according to the three error contributions. Then, we strive to minimize the computational complexity with respect to a given error tolerance. This procedure is compared to the Monte Carlo method.

The rest of the paper is organized as follows. In Sect. 2, we explain the Cahn–Hilliard and the Cahn–Hilliard–Cook equations with their boundary conditions. Then, we describe how the Ciarlet–Raviart mixed finite element can be used to convert the stochastic equation to a system of second-order equations. In Sect. 3, we demonstrate the implementation of the MLMC-FEM for the time-dependent stochastic equations. In Sect. 4, we give three numerical examples according to two different initial conditions. The solutions of the stochastic equation (the concentration) and the optimization (the optimal hierarchies) are given in this section. Finally, the conclusions are drawn in Sect. 5.

## 2 Cahn–Hilliard–Cook equation

*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

*F*(

*u*) and the interfacial energy (the second term). A popular example of a nonlinear function is

### 2.1 Ciarlet–Raviart mixed finite element

### 2.2 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:

*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

*W*(

*t*) is cylindrical) [34]. Therefore, the Cahn–Hilliard–Cook equation has a continuous 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]

*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]

*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

*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

## 3 Multilevel Monte Carlo finite element method

*T*] and the finite element method for the space discretization. In order to obtain the mean square error (MSE) of \(\varepsilon \), we require \(\delta t=\mathcal {O}(\varepsilon )\) (for the time discretization). The Monte Carlo error (statistical error) is \(\mathcal {O}(1/\sqrt{M})\) (where

*M*is the number of samples) which yields \(M^{-1}=\mathcal {O}(\varepsilon ^{2})\). Using a finite element scheme also gives rise to \(\mathcal {O}(\varepsilon ^{-d/\alpha })\), where \(\alpha \) is the convergence rate of the discretization error. Therefore, by taking

*M*samples, \(T/\delta t\) time steps and

*h*as the mesh size, we have the following total cost

*L*. First, for a given Hilbert space \((H, \Vert \cdot \Vert _H)\) the space, \(L^2(\Omega ;H)\) is defined to be the space of all measurable functions \(\mathcal {Y}:\Psi \rightarrow H\) such that

### Lemma 1

### Lemma 2

*X*be the solution of (20) and \(X_{\ell ,\zeta }\) be the sequence of discrete mild solution (i.e., the solution of (23)). Then, there is a constant

*C*(

*T*) such that for all \(\ell ,\zeta \in \mathbb {N}_0\), we have

The optimal hierarchies of MLMC-FEM with respect to different prescribed errors

\(\varepsilon \) | \(h_0\) | | \(M_0\) | \(M_1\) | \(M_2\) | \(M_3\) | \(M_4\) |
---|---|---|---|---|---|---|---|

0.100 | 0.764 | 1.458 | 73 | 50 | 15 | – | – |

0.050 | 0.615 | 1.568 | 280 | 138 | 31 | – | – |

0.020 | 0.461 | 1.726 | 1672 | 524 | 85 | – | – |

0.010 | 0.370 | 1.856 | 6474 | 1445 | 184 | – | – |

0.005 | 0.580 | 1.990 | 187,700 | 46,448 | 4615 | 459 | – |

The optimal values of MC-FEM with respect to different prescribed errors

\(\varepsilon \) | 0.1 | 0.05 | 0.02 | 0.01 | 0.005 |
---|---|---|---|---|---|

| 0.108 | 0.052 | 0.020 | 0.010 | 0.005 |

| 3 | 12 | 73 | 289 | 1 152 |

*L*, we estimate the optimal hierarchies of \(\left( h_\ell ,M_\ell ,L\right) _{\ell =0}^{\ell =L}\) such that

## 4 Numerical results

In this section, we present three numerical examples of the stochastic Cahn–Hilliard–Cook equations where in all cases the optimal MLMC-FEM is used to obtain the solution. Due to the fact that the examples are real-world problems, their exact solutions are not given. The simulations are performed using MATLAB 2017b software on an Intel Core i7 machine with \(32\,\mathrm {GB}\) of memory. In all examples, \(\epsilon =0.01\) is used and the constant mobility \(M=0.25\) is applied. For the nonlinear term (i.e., \(F'(u)\)), we use Newton’s method where several iterations are needed to reach the stopping tolerance (here \(TOL=1\times 10^{-8}\)). In each iteration, the built-in direct solver is employed to solve the linearized system.

### 4.1 A 2D example

As the first example, we take \(u_0=0.25+0.1{\omega }\) as the initial condition. The random variable \({\omega }\) is uniformly distributed between 0 and 1. The computational geometry (\(\Omega \)) is a circle with radius \(r=2\) and zero center point. As the first step, we try to solve the optimization problems (for MLMC-FEM and MC-FEM). This will enable us to find the optimal number of samples and mesh sizes. As explained in Sect. 2.1, we use the Ciarlet–Raviart mixed finite element with P1 elements. The estimation of the exponent \(\alpha \) is crucial, however, it relates to the order of polynomials. Due to the fact that the exact solution of the stochastic equation is not available, we calculate the error between different mesh sizes and \(h=0.01\) (as the reference solution) at \(T=5\). Figure 1 depicts \(\alpha \) with respect to different mesh sizes (here uniform refinement). The simulations show \(\alpha =0.952\), \(C_1=0.51\), where the exponent agrees very well with the order of P1 finite element technique (linear elements). The rest of the coefficients is estimated as \(C_2=0.066\), \(C_3=0.223\). Now we are ready to solve the optimization problem (47) with respect to the aforementioned parameters. In order to solve the optimization problem, we apply interior method where the details of the technique are given in [29]. The optimal hierarchies of the MLMC-FEM are shown in Table 1.

As the next step, in order to compare the efficiency of the multilevel setting with the Monte Carlo simulation, we solve the optimization given in (48). Again, the optimal mesh size and the optimal number of samples are given in Table 2 where the same convergence rate (\(\alpha \)) is used. Finally, we draw a comparison between MLMC-FEM and MC-FEM which is shown in Fig. 2. The results indicate that the multilevel method costs approximately \(\mathcal {O}(\varepsilon ^{-3.27})\) and the computational work of Monte Carlo sampling is \(\mathcal {O}(\varepsilon ^{-5.1})\). The comparison indicates noticeably the efficiency of the MLMC-FEM.

Finally, we compare the evolution of the concentration \(\mathbb {E}[u(T)]\) at different times (from \(T=0.2\) to \(T=15\)) (with \(\sigma =0.1\)) where the obtained results are depicted in Fig. 3. It is shown that from \(T=0.2\) to \(T=1\), a slow coarsening happens. Here, we use \(\varepsilon =0.05\) in the sense that the solution at the last level (here \(L=2\)) is shown in the figure (see Table 1 for the optimal mesh size and number of samples). In order to study the noise effect we solve the deterministic equation with the same mesh size as illustrated in Fig. 4.

### 4.2 The 3D examples

*x*,

*y*,

*z*) is a point on the cube. The same procedure for solving the optimization problem can be used, however, due to the three-dimensional structure we set \(d=3\). We should note that as \(\zeta =2\ell \), we define the optimal time interval as \(\delta t^\zeta \simeq h_\ell ^2\). First, we consider the effect of the noise intensify measure in the sense that the deterministic case (\(\sigma =0\)) is compared with the stochastic equation (\(\sigma =0.1,~0.2,~0.3\)). The results are shown in Fig. 5 at \(T=1\). Clearly, higher \(\sigma \) affects the concentration mostly. Similar to the 2D example, we consider the effect of time on the concentration. The results are shown in Fig. 6 for different times from \(T=0.1\) to \(T=10\). It illustrates that the initial homogeneous phase quickly segregates (at \(T=0.1\)), however, after the segregation the domain starts to slowly coarsen in time.

In the next step, we use the Monte Carlo finite element method to compare the effect of the number of grids. Here, two mesh sizes, i.e., \(h=0.5\) (with \(137^3\) nodes) and \(h=0.1\) (with \(6651^3\) nodes) are employed and the results are shown in Figs. 7 and 8 for stochastic and deterministic cases, respectively. We solved the CHC equation with \(\sigma =0.15\) and compared its solution with the deterministic case (\(\sigma =0\)) at time \(T=5\). It shows that the mesh size does not considerably affect the solution.

## 5 Conclusions

In this paper, we considered the Cahn–Hilliard and Cahn–Hilliard–Cook equations as forth-order time-dependent equations. As the first step, after defining an auxiliary variable, we converted the equation into a system of second-order time-dependent problems. Then, we presented a variational formulation for the system and used the Ciarlet–Raviart mixed finite element method. Afterwards, we rewrote the equation as a stochastic ODE in order to estimate its mild solution *u*(*t*).

We have already shown that for the stochastic time-dependent problems, the computational cost of the Monte Carlo finite element method is \(\mathcal {O}(\varepsilon ^{-(2+1+d/\alpha )})\). Applying the multilevel technique for this problem reduces noticeably the computational costs. In a two-dimensional problem, the optimal hierarchies \(\left( h_\ell ,r_\ell ,M_\ell ,\delta t^\ell \right) _{\ell =0}^{\ell =L}\) reduce the complexity to \(\mathcal {O}(\varepsilon ^{-3.27})\) as certified in numerical example.

We showed three numerical examples with two specific initial conditions. We estimated the solution of stochastic/deterministic Cahn–Hilliard equation for different time intervals. As a result, we demonstrated distinctive coarsening and phase separation. For the stochastic equation, we studied the effect of noise measure, for showing that more \(\sigma \) intensifies the noisy concentration.

## Notes

### Acknowledgements

Open access funding provided by Austrian Science Fund (FWF). The first and the last authors acknowledge support by FWF (Austrian Science Fund) START Project No. Y660 *PDE Models for Nanotechnology*. The second author also acknowledges support by FWF Project No. P28367-N35.

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