Strong convergence rates of a fully discrete scheme for the Cahn-Hilliard-Cook equation

Abstract

The first aim of this paper is to examine existence, uniqueness and regularity for the Cahn-Hilliard-Cook (CHC) equation in space dimension d ≤ 3. By applying a spectral Galerkin method to the infinite dimensional equation, we elaborate the well-posedness and regularity of the finite dimensional approximate problem. The key idea lies in transforming the stochastic problem with additive noise into an equivalent random equation. The regularity of the solution to the equivalent random equation is obtained, in one dimension, with the aid of the Gagliardo-Nirenberg inequality and done in two and three dimensions, by the energy argument. Further, the approximate solution is shown to be strongly convergent to the unique mild solution of the original CHC equation, whose spatio-temporal regularity can be attained by similar arguments. In addition, a fully discrete approximation of such problem is investigated, performed by the spectral Galerkin method in space and the backward Euler method in time. The previously obtained regularity results of the problem help us to identify strong convergence rates of the fully discrete scheme.

Weak error estimates of fully-discrete schemes for the stochastic Cahn-Hilliard equation

“…In this section, we first describe in Section 3.1 the method employed for the spatial discretization: a spectral Galerkin approximation method is introduced. Our first main result is Theorem 3.1, which gives a weak order of convergence Γ in terms of λ N when the approximation parameter N goes to 8 -whereas the strong order of convergence is known to be equal to Γ{2, see for instance [21,45] and (52) below. We then describe in Section 3.2 the fully discrete method: a tamed exponential Euler scheme is employed for the temporal discretization.…”

“…We are now in position to state a well-posedness result for mild solutions of the stochastic evolution equation (45) dXptq `A`A Xptq `F pXptqq ˘dt " dW Q ptq with initial value Xp0q " x 0 . To indicate that the initial value of Xp0q is equal to x 0 , the notation E x 0 r¨s is used in the sequel.…”

“…Under Assumption 1-piq (space-time white noise case in dimension d " 1), the strong moment bounds (47) follow from [21, Proposition 1] for instance. Under Assumption 1-piiq (trace-class noise case in dimensions d " 1, 2, 3), the strong moment bounds (47) follow from [45,Theorem 3.7] and (44) for instance. The increment bounds (48) are then obtained by standard techniques.…”

“…In the last decade, numerical analysis of such stochastic phase-field models has been a very active field of research, see, e.g., [38,44,25,37,16,27,9,4,26] and references therein. Progress on the convergence aspects of numerical approximations for stochastic phase-field model has been made recently, including strong and weak convergence rate analysis for the stochastic Allen-Cahn equation (see, e.g., [3,7,19,46,10,41,42,48,13,22,6]) and the strong convergence rate analysis for the stochastic Cahn-Hilliard equation (see, e.g., [47,20,1,21,31,14,45]).…”

“…One of the major difficulty we need to deal with is the fact that the nonlinearity AF in (2) does not satisfy a monotonicity property in H (whereas the nonlinearity F in the Allen-Cahn case is one-sided Lipschitz continuous). To overcome this issue, we study energy estimates in L 2 and H ´1 norms, which are similar to the tools used in [14,45] for instance to prove strong error estimates. The proofs of Theorem 3.1 and 3.3 are then based on standard arguments which need to combine carefully the moment bounds on the numerical scheme and the regularity results on the Kolmogorov equation to obtain the expected result, that the weak order is twice the strong order.…”