Optimal Strong Rates of Convergence for a Space-Time Discretization of the Stochastic Allen–Cahn Equation with Multiplicative Noise

1 Introduction

Let (ℍ,(⋅,⋅)ℍ) be a separable Hilbert space, and let 𝕍 be a reflexive Banach space such that 𝕍↪ℍ↪𝕍′ constitutes a Gelfand triple. The main motivation for this work is to identify the structural properties for the drift operator of the nonlinear SPDE

which allow to construct a space-time discretization of (1.1) for which optimal strong rates of convergence may be shown. Relevant works in this direction are [5, 6], where both, σ and 𝒜 are required to be Lipschitz. The Lipschitz assumption for the drift operator 𝒜:𝕍→𝕍′ does not hold for many nonlinear SPDEs including the stochastic Navier–Stokes equation, or the stochastic version of general phase field models (including (1.2)) below for example. A usual strategy for a related numerical analysis is then to truncate nonlinearities (see e.g. [9]), or to quantify the mean square error on large subsets Ωk,h:=Ωk∩Ωh⊂Ω. As an example, the following estimate for a (time-implicit, finite element based) space-time discretization of the 2D stochastic Navier–Stokes equation with solution {𝐔m:m≥0} was obtained in [2],

for all η∈(0,12), where Ωk⊂Ω (respectively Ωh⊂Ω) is such that ℙ⁢[Ω∖Ωk]→0 for k→0 (respectively ℙ⁢[Ω∖Ωh]→0 for h→0). We also mention the work [8] which studies a spatial discretization of the stochastic Cahn–Hilliard equation.

Let 𝒪⊂ℝd, d∈{1,2,3}, be a bounded Lipschitz domain. We consider the stochastic Allen–Cahn equation with multiplicative noise, where the process X:Ω×[0,T]×𝒪¯→ℝ solves

where W≡{Wt:0≤t≤T} is an ℝ-valued Wiener process which is defined on the given filtered probability space 𝔓≡(Ω,ℱ,𝔽,ℙ); however, it is easily possible to generalize the analysis below to a trace class Q-Wiener process. Obviously, the drift operator 𝒜⁢(y)=Δ⁢y-(|y|2-1)⁢y is only locally Lipschitz, but is the negative Gâteaux differential of 𝒥⁢(y)=12⁢∥∇⁡y∥𝕃22+14⁢∥|y|2-1∥𝕃22 and satisfies the weak monotonicity property

for some K>0; see Section 2 for the notation. Our goal is a (variational) error analysis for the structure preserving finite element based space-time discretization (2.5) which accounts for this structural property, avoiding arguments that exploit only the locally Lipschitz property of 𝒜 to arrive at optimal strong error estimate.

The existing literature (see e.g. [8]) for estimating the numerical strong error on problem (1.2) mainly uses the involved linear semigroup theory; the authors have considered the additive colored noise case, in which they have benefited from it by using the stochastic convolution, and then used a truncation of the nonlinear drift operator 𝒜 to prove a rate of convergence for a (spatial) semi-discretization on sets of probability close to 1 without exploiting the weak monotonicity of 𝒜. In contrast, property (1.3) and variational arguments were used in the recent work [10], where strong error estimates for both, semi-discrete (in time) and fully-discrete schemes for (1.2) were obtained, which are of sub-optimal order 𝒪⁢(k+h2-δ6) for the fully discrete scheme in the case d=3. In [10], a standard implicit discretization of (1.2) was considered for which it is not clear to obtain uniform bounds for arbitrary higher moments of the solution of the fully discrete scheme, thus leading to sub-optimal convergence rates above. In this work, we consider the modified scheme (2.5) for (1.2), and derive optimal strong numerical error estimate.

The subsequent analysis for scheme (2.5) is split into two steps to independently address errors due to the temporal and spatial discretization. First we exploit the variational solution concept for (1.2) and the semi-linear structure of 𝒜⁢(y)=-𝒥′⁢(y) to derive uniform bounds for the arbitrarily higher moments of the solution of (1.2) in strong norms; these bounds may then be used to bound increments of the solution of (1.2) in Lemma 3.2. The second ingradient to achieve optimal error bounds is a temporal discretization which inherits the structural properties of (1.2); scheme (4.1) is constructed to allow for bounds of arbitrary moments of {𝒥⁢(Xj):0≤j≤J} in Lemma 4.1, which then settles the error bounds in Theorem 4.2 by using property (1.3) to effectively handle the nonlinear terms. We recover the asymptotic rate 12 which is known for SPDEs of the form (1.1) when 𝒜 is linear elliptic. It is interesting to compare the present error analysis for the SPDE (1.2) with the one in [7] for a general SODE with polynomial drift (see [7, Assumptions 3.1, 4.1, 4.2]) which also exploits the weak monotonicity of the drift.

The temporal semi-discretization was studied as a first step rather than spatial discretization to inherit bounds in strong norms which are needed for a complete error analysis of the problem. The second part of the error analysis is then on the structure preserving finite element based fully discrete scheme (2.5), for which we first verify the uniform bounds of arbitrary moments of {𝒥⁢(Yj):0≤j≤J} (cf. Lemma 5.1). It is worth mentioning that, if {Yj:0≤j≤J} is a solution to a standard space-time discretization which involves the nonlinearity 𝒜⁢(Yj)=-𝒥′⁢(Yj), then only basic uniform bounds may be obtained (see [10, Lemma 2.5]), as opposed to those in Lemma 5.1. Next to it, we use again (1.3) for the drift, in combination with well-known approximation results for a finite element discretization to show that the error part due to spatial discretization is of order 𝒪⁢(k+h) where k>0 is the time discretization parameter and h>0 is the space discretization parameter (see Theorem 5.2). In this context, we mention the numerical analysis in [11] for an extended model of (1.2), where the uniform bounds for the exponential moments next to arbitrary moments in stronger norms are obtained for the solution of a semi-discretization in space in the case d=1 (see [11, Propositions 4.2, 4.3]); those bounds, together with a monotonicity argument are then used to properly address the nonlinear effects in the error analysis and arrive at the (lower) strong rate 12 for the p-th mean convergence of the numerical solution.

2 Technical Framework and Main Result

Throughout this paper, we use the letter C>0 to denote various generic constants. Let 𝒪≡(0,R)d, 1≤d≤3, with R∈(0,∞) be a cube in ℝd. Let us denote Γj=∂𝒪∩{xj=0} and Γj+d=∂𝒪∩{xj=R} for j=1,…,d. Problem (1.2) is then supplemented by the space-periodic boundary condition

Let (𝕃perp,∥⋅∥𝕃p) respectively (𝕎perm,p,∥⋅∥𝕎m,p) denote the Lebesgue respectively Sobolev space of R-periodic functions φ∈𝕎locm,p⁢(ℝd). Recall that functions in 𝕎perm,2 may be characterized by their Fourier series expansion, i.e.,

Below, we set ψ⁢(x)=14⁢∥|x|2-1∥𝕃22 for x∈𝕃2. Throughout this article, we make the following assumption on σ:ℝ→ℝ.

The following estimate for the strong solution is well known (p≥1),

3 Stochastic Allen–Cahn Equation: The Continuous Case

In this section, we derive uniform bounds of arbitrary moments for the strong solution X≡{Xt:t∈[0,T]} of (1.2) and using these uniform bounds, we estimate the expectation of the increment ∥Xt-Xs∥𝕃22 in terms of |t-s|.

The following estimate may be shown by a standard Galerkin method which employs a (finite) sequence of (𝕎per1,2-orthonormal) eigenfunctions {wj:1≤j≤N} of the inversely compact, self-adjoint isomorphic operator I-Δ:𝕎per2,2→𝕃per2, the use of Itô’s formula to the functional y↦𝒥⁢(y):=12⁢∥∇⁡y∥𝕃22+ψ⁢(y), and the final passage to the limit (see e.g. [4]),

for which we require the improved regularity property x∈𝕎per1,2. Thanks to Assumption A.1, we see that σ satisfies the following estimates:

1. There exists a constant C>0 such that

   (3.2)∥∇⁡σ⁢(ξ)∥𝕃22+(D2⁢ψ⁢(ξ)⁢σ⁢(ξ),σ⁢(ξ))𝕃2≤C⁢(1+𝒥⁢(ξ)) for all ⁢ξ∈𝕎per1,2.

2. There exist constants K2,K3,K4>0 and L2,L3,L4>0 such that for all ξ∈𝕎per2,2,

   (3.3)∥Δ⁢σ⁢(ξ)∥𝕃22≤{K2⁢∥Δ⁢ξ∥𝕃22+K3⁢∥∇⁡ξ∥𝕃22⁢∥Δ⁢ξ∥𝕃22+K4⁢∥∇⁡ξ∥𝕃24if ⁢d=2,L2⁢∥Δ⁢ξ∥𝕃22+L3⁢∥∇⁡Δ⁢ξ∥𝕃232⁢∥∇⁡ξ∥𝕃252+L4⁢∥∇⁡ξ∥𝕃24if ⁢d=3.

The following result is to bound the increments Xt-Xs of the solutions of (1.2) in terms of |t-s|α for some α>0; its proof uses Lemma 3.1 in particular.