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Computational Methods in Applied Mathematics

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Volume 18, Issue 2

Issues

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

Ananta K. MajeeORCID iD: http://orcid.org/0000-0002-1507-5772 / Andreas Prohl
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  • Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
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Published Online: 2017-07-18 | DOI: https://doi.org/10.1515/cmam-2017-0023

Abstract

The stochastic Allen–Cahn equation with multiplicative noise involves the nonlinear drift operator 𝒜⁢(x)=Δ⁢x-(|x|2-1)⁢x. We use the fact that 𝒜⁢(x)=-𝒥′⁢(x) satisfies a weak monotonicity property to deduce uniform bounds in strong norms for solutions of the temporal, as well as of the spatio-temporal discretization of the problem. This weak monotonicity property then allows for the estimate

sup1≤j≤J⁡𝔼⁢[∥Xtj-Yj∥𝕃22]≤Cδ⁢(k1-δ+h2)

for all small δ>0, where X is the strong variational solution of the stochastic Allen–Cahn equation, while {Yj:0≤j≤J} solves a structure preserving finite element based space-time discretization of the problem on a temporal mesh {tj:1≤j≤J} of size k>0 which covers [0,T].

Keywords: Stochastic Allen–Cahn Equation; Monotone Operator; Variational Solution; Strong Rate ofConvergence

MSC 2010: 45K05; 46S50; 49L20; 49L25; 91A23; 93E20

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

dXt=𝒜(Xt)dt+σ(Xt)dWt (t>0),X(0)=x∈ℍ,(1.1)

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],

𝔼[χΩk,hmax1≤m≤M∥𝐮(tm)-𝐔m∥𝕃22]≤C(kη-ε+kh-ε+h2-ε) (ε>0)

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

dXt-(ΔXt-(|Xt|2-1)Xt)dt=σ(Xt)dWt (t>0),X0=x,(1.2)

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

〈𝒜⁢(y1)-𝒜⁢(y2),y1-y2〉(𝕎per1,2)*×𝕎per1,2≤K⁢∥y1-y2∥𝕃22-∥∇⁡(y1-y2)∥𝕃22 for all ⁢y1,y2∈𝕎per1,2(1.3)

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

X|Γj=X|Γj+d (1≤j≤d).

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.,

𝕎perm,2⁢(𝒪)={φ:ℝd→ℝ:φ⁢(x)=∑k∈ℤdck⁢exp⁡(2⁢i⁢π⁢〈k,x〉R),c¯k=c-k,∑k∈ℤd|k|2⁢m⁢|ck|2<∞}.

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.

Assumption A.1.

We have that σ, σ′, and σ′′ are bounded. Moreover, σ is Lipschitz continuous, i.e., there exists a constant K1>0 such that

∥σ⁢(u)-σ⁢(v)∥𝕃22≤K1⁢∥u-v∥𝕃22 for all ⁢u,v∈𝕃per2.(2.1)

Definition 2.1 (Strong Variational Solution).

Fix T∈(0,∞), and x∈𝕃per2. A 𝕎per1,2-valued 𝔽-adapted stochastic process X≡{Xt:t∈[0,T]} is called a strong variational solution of (1.2) if X∈L2⁢(Ω;C⁢([0,T];𝕃per2)) satisfies ℙ-a.s. for all t∈[0,T] that

(Xt,ϕ)𝕃2+∫0t{(∇⁡Xs,∇⁡ϕ)𝕃2+(D⁢ψ⁢(Xs),ϕ)𝕃2}⁢ds=(x,ϕ)𝕃2+∫0t(σ⁢(Xs),ϕ)𝕃2⁢dWs for all ⁢ϕ∈𝕎per1,2.(2.2)

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

supt∈[0,T]⁡𝔼⁢[1p⁢(∥Xt∥𝕃2p-∥x∥𝕃2p)+∫0T∥Xs∥𝕃2p-2⁢(∥∇⁡Xs∥𝕃22+∥Xs∥𝕃44)⁢ds]≤C.(2.3)

2.1 Fully Discrete Scheme

Let us introduce some notation needed to define the structure preserving finite element based fully discrete scheme. Let 0=t0<t1<…<tJ be a uniform partition of [0,T] of size k=TJ. Let 𝒯h be a quasi-uniform triangulation of the domain 𝒪. We consider the 𝕎per1,2-conforming finite element space (cf. [1]) 𝕍h⊂𝕎per1,2 such that

𝕍h={ϕ∈C⁢(𝒪¯;ℝ):ϕ|K∈𝒫1⁢(K)⁢ for all ⁢K∈𝒯h},

where 𝒫1⁢(K) is the space of ℝ-valued functions on K which are polynomials of degree less or equal to 1. We may then consider the space-time discretization of (1.2): Let Y0=𝒫𝕃2⁢x∈𝕍h, where 𝒫𝕃2:𝕃per2→𝕍h denotes the 𝕃per2-orthogonal projection, i.e., for all g∈𝕃per2

(g-𝒫𝕃2⁢g,ϕ)𝕃2=0 for all ⁢ϕ∈𝕍h.(2.4)

Find the {ℱtj:0≤j≤J}-adapted 𝕍h-valued process {Yj:0≤j≤J} such that ℙ-almost surely

(Yj-Yj-1,ϕ)𝕃2+k⁢[(∇⁡Yj,∇⁡ϕ)𝕃2+(f⁢(Yj,Yj-1),ϕ)𝕃2]=Δj⁢W⁢(σ⁢(Yj-1),ϕ)𝕃2 for all ⁢ϕ∈𝕍h,(2.5)

where

Δj⁢W:=W⁢(tj)-W⁢(tj-1)∼𝒩⁢(0,k),and f⁢(y,z)=(|y|2-1)⁢y+z2.(2.6)

Solvability for k<1 is again immediate via the Brouwer fixed point theorem.

We are now in a position to state the main result of this article.

Main Theorem.

Let Assumption A.1 hold and x∈Wper2,2. For every δ>0, there exist constants 0<Cδ<∞, independent of the discretized parameters k,h>0, and k0=k0⁢(T,x)>0 such that for all k≤k0 sufficiently small, there holds

sup1≤j≤J⁡𝔼⁢[∥Xtj-Yj∥𝕃22]+k⁢∑j=1J𝔼⁢[∥∇⁡(Xtj-Yj)∥𝕃22]≤Cδ⁢(k1-δ+h2),

where {Xt:t∈[0,T]} solves (1.2) while {Yj:0≤j≤J} solves (2.5).

The proof is detailed in Sections 3, 4 and 5 and uses the semi-linear structure of 𝒜 along with the weak monotonicity property (1.3). We first consider a semi-discrete (in time) scheme (4.1) of problem (1.2) and derive the error estimate between the strong solution X of (1.2) and the discretized solution {Xj:0≤j≤J} of (4.1), see Theorem 4.2. Again, using uniform bounds for higher moments of the solutions {Xj} and {Yj}, we derive the error estimate of Xj and Yj in strong norm, cf. Theorem 5.2. Putting things together then settles the main theorem.

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]),

supt∈[0,T]⁡𝔼⁢[𝒥⁢(Xt)+∫0t∥Δ⁢Xs-D⁢ψ⁢(Xs)∥𝕃22⁢ds]≤C⁢(1+𝔼⁢[𝒥⁢(x)]),(3.1)

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

  • (a)

    There exists a constant C>0 such that

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

  • (b)

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

    ∥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.(3.3)

Lemma 3.1.

Let Assumption A.1 hold and p∈N. Then there exists a constant C≡C⁢(∥x∥W1,2,p,T)>0 such that

supt∈[0,T]⁡𝔼⁢[(𝒥⁢(Xt))p]+𝔼⁢[∫0T∥∇⁡Xs∥𝕃22⁢(p-1)⁢∥Δ⁢Xs∥𝕃22⁢ds]≤C.(i)

Suppose in addition that x∈Wper2,2. Then there exists a constant C≡C⁢(∥x∥W2,2,T)>0 such that

supt∈[0,T]⁡𝔼⁢[∥Δ⁢Xt∥𝕃22]+𝔼⁢[∫0T∥∇⁡Δ⁢Xs∥𝕃22⁢ds]≤C.(ii)

Proof.

(i) We proceed formally. Note that

𝒥⁢(ξ):=12⁢∥∇⁡ξ∥𝕃22+ψ⁢(ξ) and -𝒜⁢(ξ)≡𝒥′⁢(ξ)=-Δ⁢ξ+D⁢ψ⁢(ξ).

We use Itô’s formula for ξ↦g⁢(ξ):=(𝒥⁢(ξ))p:

D⁢g⁢(ξ)=-p⁢(𝒥⁢(ξ))p-1⁢𝒜⁢(ξ),D2⁢g⁢(ξ)=p⁢(p-1)⁢(𝒥⁢(ξ))p-2⁢𝒜⁢(ξ)⊗𝒜⁢(ξ)+p⁢(𝒥⁢(ξ))p-1⁢(-Δ+D2⁢ψ⁢(ξ)),

where a⊗b⋅c=a⁢(b,c)𝕃2 for all a,b,c∈𝕃2. By the Cauchy–Schwarz inequality, and (3.2), we have

𝔼⁢[(𝒥⁢(Xt))p-(𝒥⁢(x))p+p⁢∫0t(𝒥⁢(Xs))p-1⁢∥𝒜⁢(Xs)∥𝕃22⁢ds]=p2⁢𝔼⁢[∫0t{(p-1)⁢(𝒥⁢(Xs))p-2⁢(𝒜⁢(Xs),σ⁢(Xs))𝕃22+(𝒥⁢(Xs))p-1⁢([-Δ+D2⁢ψ⁢(Xs)]⁢σ⁢(Xs),σ⁢(Xs))𝕃2}⁢ds]≤p2⁢𝔼⁢[∫0t(p-1)⁢(𝒥⁢(Xs))p-2⁢(𝒜⁢(Xs),σ⁢(Xs))𝕃22⁢ds]+C⁢(p)⁢∫0t𝔼⁢[(𝒥⁢(Xs))p]⁢ds+C.

Since σ is bounded, by using Young’s inequality, we have

(𝒥⁢(Xs))p-2⁢(𝒜⁢(Xs),σ⁢(Xs))𝕃22≤C⁢(𝒥⁢(Xs))p-2⁢∥𝒜⁢(Xs)∥𝕃22≤θ⁢(𝒥⁢(Xs))p-1⁢∥𝒜⁢(Xs)∥𝕃22+C⁢(θ)⁢∥𝒜⁢(Xs)∥𝕃22.

We choose θ>0 such that p-p2⁢(p-1)⁢θ>0. With this choice of θ, by (3.1), we have for some constant C1=C1⁢(p)>0,

𝔼⁢[(𝒥⁢(Xt))p]+C1⁢(p)⁢𝔼⁢[∫0t(𝒥⁢(Xs))p-1⁢∥𝒜⁢(Xs)∥𝕃22⁢ds]≤𝔼⁢[(𝒥⁢(x))p]+C⁢(p)⁢∫0t𝔼⁢[(𝒥⁢(Xs))p]⁢ds+C.

We use Gronwall’s lemma to conclude that

supt∈[0,T]⁡𝔼⁢[(𝒥⁢(Xt))p]+𝔼⁢[∫0T(𝒥⁢(Xs))p-1⁢∥𝒜⁢(Xs)∥𝕃22⁢ds]≤C.(3.4)

In view of (2.3), and (3.4) it follows that

supt∈[0,T]⁡𝔼⁢[∥Xt∥𝕎1,2p]≤C for all ⁢p≥1.(3.5)

Note that

∥Δ⁢Xs∥𝕃22≤∥𝒜⁢(Xs)∥𝕃22+∥Xs∥𝕃66+∥Xs∥𝕃22≤∥𝒜⁢(Xs)∥𝕃22+C⁢∥Xs∥𝕎1,26+∥Xs∥𝕎1,22,

where we use the embedding 𝕎1,2↪𝕃6 for d≤3. Thanks to (3.4) and (3.5), together with Cauchy–Schwarz inequality and the above estimate, we see that

𝔼⁢[∫0T(𝒥⁢(Xs))p-1⁢∥Δ⁢Xs∥𝕃22⁢ds]≤C+𝔼⁢[∫0T(𝒥⁢(Xs))p-1⁢(∥Xs∥𝕎1,26+∥Xs∥𝕎1,22)⁢ds]≤C+T⁢supt∈[0,T]⁡𝔼⁢[(𝒥⁢(Xt))p]+T⁢supt∈[0,T]⁡𝔼⁢[∥Xt∥𝕎1,26⁢p+∥Xt∥𝕎1,22⁢p]≤C.(3.6)

One can combine (3.4) and (3.6) to conclude the assertion.

(ii) Use Itô’s formula for ξ↦g⁢(ξ)=12⁢∥Δ⁢ξ∥𝕃22. We compute its derivatives:

D⁢g⁢(ξ)=Δ2⁢ξ,D2⁢g⁢(ξ)=Δ2.

Note that by integration by parts

(Δ2⁢Xs,-Δ⁢Xs+|Xs|2⁢Xs)𝕃2≥12⁢[∥∇⁡Δ⁢Xs∥𝕃22-3⁢∥|Xs|2⁢∇⁡Xs∥𝕃22].(3.7)

Because of 𝕎1,2↪𝕃6 for d≤3 we estimate the last term through

∥|Xs|2⁢∇⁡Xs∥𝕃22≤∥Xs∥𝕃64⁢∥∇⁡Xs∥𝕃62≤C⁢∥Xs∥𝕎1,24⁢(∥∇⁡Xs∥𝕃22+∥Δ⁢Xs∥𝕃22)≤C⁢∥Xs∥𝕎1,26+C⁢(∥Xs∥𝕃22+∥∇⁡Xs∥𝕃22)2⁢∥Δ⁢Xs∥𝕃22≡C⁢∥Xs∥𝕎1,26+ℳ.(3.8)

Note that ∥Xs∥𝕃22≤C⁢(1+ψ⁢(Xs)), and therefore we see that

ℳ≤C⁢(1+(𝒥⁢(Xs))2)⁢∥Δ⁢Xs∥𝕃22.(3.9)

Inserting (3.8) and (3.9) into (3.7), we obtain

𝔼⁢[∥Δ⁢Xt∥𝕃22]+𝔼⁢[∫0t∥∇⁡Δ⁢Xs∥𝕃22⁢ds]≤𝔼⁢[Δ⁢x∥𝕃22]+C⁢𝔼⁢[∫0t∥Xs∥𝕎1,26⁢ds]+C⁢𝔼⁢[∫0T(𝒥⁢(Xs))2⁢∥Δ⁢Xs∥𝕃22⁢ds]+C⁢∫0t𝔼⁢[∥Δ⁢Xs∥𝕃22]⁢ds+C⁢𝔼⁢[∫0t∥Δ⁢σ⁢(Xs)∥𝕃22⁢ds].(3.10)

Let d=2. Then by (3.3), we see that

𝐆:=𝔼⁢[∫0t∥Δ⁢σ⁢(Xs)∥𝕃22⁢ds]≤K2⁢∫0t𝔼⁢[∥Δ⁢Xs∥𝕃22]⁢ds+K3⁢𝔼⁢[∫0T∥∇⁡Xs∥𝕃22⁢∥Δ⁢Xs∥𝕃22⁢ds]+K4⁢∫0T𝔼⁢[∥∇⁡Xs∥𝕃24]⁢ds.

One can combine the above estimate in (3.10) and use (3.5), (3.6) along with Gronwall’s lemma to conclude the assertion for d≤2.

Let d=3. Then, thanks to (3.3) and the Cauchy–Schwarz inequality, we have

𝐆≤12⁢𝔼⁢[∫0t∥∇⁡Δ⁢Xs∥𝕃22⁢ds]+C⁢𝔼⁢[∫0T∥∇⁡Xs∥𝕃210⁢ds]+L4⁢∫0T𝔼⁢[∥∇⁡Xs∥𝕃24]⁢ds+L2⁢∫0t𝔼⁢[∥Δ⁢Xs∥𝕃22]⁢ds.(3.11)

We combine (3.10) and (3.11) and use (3.5), (3.6) along with Gronwall’s lemma to conclude the assertion for d=3. This completes the proof. ∎

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.

Lemma 3.2.

Let Assumption A.1 hold and x∈Wper2,2. Then, for every 0≤s≤t≤T, there exists a constant C≡C⁢(p,T)>0 such that

  • (i)

    𝔼⁢[∥Xt-Xs∥𝕃2p]≤C⁢|t-s| (p≥2),

  • (ii)

    𝔼⁢[∥Xt-Xs∥𝕎1,22]≤C⁢|t-s|.

Proof.

(i) Fix s≥0. An application of Itô’s formula for u↦1p⁢∥u-β∥𝕃2p with β=Xs⁢(⋅,ω)∈ℝ to (1.2) yields, after taking expectation

𝔼⁢[1p⁢∥Xt-Xs∥𝕃2p+∫st∥Xζ-Xs∥𝕃2p-2⁢(𝒜⁢(Xζ)-𝒜⁢(Xs),Xζ-Xs)𝕃2⁢dζ]≤𝔼⁢[∫st∥Xζ-Xs∥𝕃2p-2⁢(-𝒜⁢(Xs),Xζ-Xs)𝕃2⁢dζ]+Cp⁢𝔼⁢[∫st∥Xζ-Xs∥𝕃2p-2⁢∥σ⁢(Xζ)∥𝕃22⁢dζ]≡A1+A2.

We use the weak monotonicity property (1.3) to bound from below the second term on the left-hand side,

≥𝔼⁢[∫st(∥Xζ-Xs∥𝕃2p-2⁢∥∇⁡(Xζ-Xs)∥𝕃22-C⁢∥Xζ-Xs∥𝕃2p)⁢dζ].

The integration by parts formula and Young’s inequality reveal that

(-𝒜⁢(Xs),Xζ-Xs)𝕃2≤∥∇⁡Xs∥𝕃2⁢∥∇⁡(Xζ-Xs)∥𝕃2+∥D⁢ψ⁢(Xs)∥𝕃2⁢∥Xζ-Xs∥𝕃2≤(∥Xs∥𝕃63+∥Xs∥𝕎1,2)⁢∥Xζ-Xs∥𝕎1,2.

Since 𝕎1,2↪𝕃6 for d≤3, by using Young’s inequality, we see that

A1≤C⁢𝔼⁢[∫st∥Xζ-Xs∥𝕃2p-2⁢(∥Xs∥𝕎1,2+∥Xs∥𝕃63)⁢∥Xζ-Xs∥𝕎1,2⁢dζ]≤𝔼⁢[12⁢∫st∥Xζ-Xs∥𝕃2p-2⁢(∥Xζ-Xs∥𝕃22+∥∇⁡(Xζ-Xs)∥𝕃22)⁢dζ]+12⁢𝔼⁢[∫st∥Xζ-Xs∥𝕃2p-2⁢(∥Xs∥𝕎1,22+∥Xs∥𝕎1,26)⁢dζ]≤12𝔼[∫st∥Xζ-Xs∥𝕃2p-2∥∇(Xζ-Xs)∥𝕃22dζ]+Cp∫st𝔼[∥Xζ-Xs∥𝕃2p]dζ]+C⁢|t-s|⁢supζ∈[s,t]⁡𝔼⁢[∥Xζ∥𝕎1,2p+∥Xζ∥𝕎1,23⁢p].

Again, thanks to the boundedness of σ and Young’s inequality, we see that

A2≤Cp⁢∫st𝔼⁢[∥Xζ-Xs∥𝕃2p]⁢dζ+C⁢|t-s|.

We combine all the above estimates and use (2.3) and Lemma 3.1, (i) along with Gronwall’s inequality to get the result.

(ii) We apply Itô’s formula to the function 12⁢|∇⁡Xt-β|2 for any β∈ℝd to (1.2), and then use β=∇⁡Xs for fixed 0<s≤t and integrate with respect to spatial variable. Thanks to Young’s inequality, and the boundedness of σ′,

12𝔼[∥∇(Xt-Xs)∥𝕃22]≤|∫st𝔼[(-Δ[Xζ-Xs],𝒜(Xζ))𝕃2]dζ|+C∫st𝔼[∥∇σ(Xζ)∥𝕃22]dζ≤C⁢|t-s|⁢supζ∈[s,t]⁡𝔼⁢[∥Δ⁢Xζ∥𝕃22]+C⁢∫st𝔼⁢[∥𝒜⁢(Xζ)∥𝕃22+∥∇⁡Xζ∥𝕃22]⁢dζ.

Notice that

∥𝒜⁢(Xζ)∥𝕃22≤∥Δ⁢Xζ∥𝕃22+C⁢(∥Xζ∥𝕃66+∥Xζ∥𝕃22).

From the above estimate, and Lemma 3.1, (ii), (3.5), and the embedding 𝕎1,2↪𝕃6 for d≤3, we conclude

𝔼⁢[∥∇⁡(Xt-Xs)∥𝕃22]≤C⁢|t-s|⁢supζ∈[s,t]⁡𝔼⁢[∥Δ⁢Xζ∥𝕃22+∥Xζ∥𝕎1,26+∥Xζ∥𝕎1,22]≤C⁢|t-s|.(3.12)

One can use (i) of Lemma 3.2 for p=2, and (3.12) to arrive at (ii). This finishes the proof. ∎

4 Semi-Discrete Scheme (in Time) and Its Bound

Let 0=t0<t1<…<tJ be an equi-distant partition of [0,T] of size k=TJ. The structure preserving time discrete version of (1.2) defines an {ℱtj:0≤j≤J}-adapted 𝕎per1,2-valued process {Xj:0≤j≤J} such that ℙ-almost surely and for all ϕ∈𝕎per1,2,

{(Xj-Xj-1,ϕ)𝕃2+k⁢[(∇⁡Xj,∇⁡ϕ)𝕃2+(f⁢(Xj,Xj-1),ϕ)𝕃2]=Δj⁢W⁢(σ⁢(Xj-1),ϕ)𝕃2,X0=x∈𝕃per2,(4.1)

where Δj⁢W and f are defined in (2.6). Solvability for k<1 easily follows from a coercivity property of the drift operator, and the Lipschitz continuity property (2.1) for the diffusion operator. Below, we denote again

𝒥⁢(Xj)=12⁢∥∇⁡Xj∥𝕃22+ψ⁢(Xj).

The proof of the following lemma evidences why D⁢ψ⁢(Xj) is substituted by f⁢(Xj,Xj-1) in (4.1) to recover uniform bounds for arbitrary higher moments of 𝒥⁢(Xj).

Lemma 4.1.

Suppose that x∈Wper1,2, and that Assumption A.1 holds. For every p=2r, r∈N*, there exists a constant C≡C⁢(p,T)>0 such that

max1≤j≤J𝔼[|𝒥(Xj)|p]+∑j=1J𝔼[∏ℓ=1r[[𝒥(Xj)]2ℓ-1+[𝒥(Xj-1)]2ℓ-1]×(∥∇(Xj-Xj-1)∥𝕃22+∥|Xj|2-|Xj-1|2∥𝕃22+k∥-ΔXj+f(Xj,Xj-1)∥𝕃22)]≤C.

Proof.

(1) Consider (4.1) for a fixed ω∈Ω and choose ϕ=-Δ⁢Xj⁢(ω)+f⁢(Xj,Xj-1)⁢(ω). Then one has ℙ-a.s.,

(Xj-Xj-1,-Δ⁢Xj+f⁢(Xj,Xj-1))𝕃2+k⁢∥-Δ⁢Xj+f⁢(Xj,Xj-1)∥𝕃22=ΔjW(∇σ(Xj-1),∇Xj)𝕃2+ΔjW(σ(Xj-1),f(Xj,Xj-1))𝕃2=:𝒜1+𝒜2.(4.2)

By using the identity (a-b)⁢a=12⁢(|a|2-|b|2+|a-b|2) for all a,b∈ℝ along with integration by parts formula, we calculate

(Xj-Xj-1,-Δ⁢Xj+f⁢(Xj,Xj-1))𝕃2=(∇⁡(Xj-Xj-1),∇⁡Xj)𝕃2+12⁢(|Xj|2-1,|Xj|2-1-(|Xj-1|2-1))𝕃2=12⁢(∥∇⁡Xj∥𝕃22-∥∇⁡Xj-1∥𝕃22+∥∇⁡(Xj-Xj-1)∥𝕃22)+14⁢(∥|Xj|2-1∥𝕃22-∥|Xj-1|2-1∥𝕃22+∥|Xj|2-|Xj-1|2∥𝕃22)=𝒥⁢(Xj)-𝒥⁢(Xj-1)+12⁢∥∇⁡(Xj-Xj-1)∥𝕃22+14⁢∥|Xj|2-|Xj-1|2∥𝕃22.(4.3)

Since σ′ is bounded, we observe that

𝒜1≤14⁢∥∇⁡(Xj-Xj-1)∥𝕃22+C⁢∥∇⁡Xj-1∥𝕃22⁢|Δj⁢W|2+(∇⁡σ⁢(Xj-1),∇⁡Xj-1)𝕃2⁢Δj⁢W≤14⁢∥∇⁡(Xj-Xj-1)∥𝕃22+C⁢𝒥⁢(Xj-1)⁢|Δj⁢W|2+(∇⁡σ⁢(Xj-1),∇⁡Xj-1)𝕃2⁢Δj⁢W.

We decompose 𝒜2 into the sum of two terms 𝒜2,1 and 𝒜2,2, where

{𝒜2,1=(σ⁢(Xj-1),(|Xj|2-|Xj-1|2)⁢Xj+Xj-12)𝕃2⁢Δj⁢W,𝒜2,2=(σ⁢(Xj-1),(|Xj-1|2-1)⁢Xj+Xj-12)𝕃2⁢Δj⁢W.

In view of Young’s inequality and the boundedness of σ, we have

𝒜2,1≤18⁢∥|Xj|2-|Xj-1|2∥𝕃22+C⁢(∥Xj-Xj-1∥𝕃22+∥Xj-1∥𝕃22)⁢|Δj⁢W|2,𝒜2,2=(σ⁢(Xj-1),(|Xj-1|2-1)⁢Xj-Xj-12)𝕃2⁢Δj⁢W+(σ⁢(Xj-1),(|Xj-1|2-1)⁢Xj-1)𝕃2⁢Δj⁢W≤∥Xj-Xj-1∥𝕃22+∥|Xj-1|2-1∥𝕃22⁢|Δj⁢W|2+(σ⁢(Xj-1),(|Xj-1|2-1)⁢Xj-1)𝕃2⁢Δj⁢W≤∥Xj-Xj-1∥𝕃22+C⁢𝒥⁢(Xj-1)⁢|Δj⁢W|2+(σ⁢(Xj-1),(|Xj-1|2-1)⁢Xj-1)𝕃2⁢Δj⁢W.

Next we estimate ∥Xj-Xj-1∥𝕃22 independently to bound the term 𝒜2,2. To do so, we choose as test function ϕ=(Xj-Xj-1)⁢(ω) in (4.1) and obtain

∥Xj-Xj-1∥𝕃22+k2⁢(∥∇⁡Xj∥𝕃22-∥∇⁡Xj-1∥𝕃22+∥∇⁡(Xj-Xj-1)∥𝕃22)+k2⁢(|Xj|2-1,|Xj|2-|Xj-1|2)𝕃2=(σ⁢(Xj-1),Xj-Xj-1)𝕃2⁢Δj⁢W.(4.4)

Note that

k2⁢(|Xj|2-1,|Xj|2-|Xj-1|2)𝕃2=k⁢(ψ⁢(Xj)-ψ⁢(Xj-1)+14⁢∥|Xj|2-|Xj-1|2∥𝕃22),(σ⁢(Xj-1),Xj-Xj-1)𝕃2⁢Δj⁢W≤12⁢∥Xj-Xj-1∥𝕃22+C⁢|Δj⁢W|2,

where in the last inequality we have used the boundedness property of σ. We use (4.5) in (4.4) to get

∥Xj-Xj-1∥𝕃22≤C⁢k⁢𝒥⁢(Xj-1)+C⁢|Δj⁢W|2.(4.6)

Again, since 𝒪 is a bounded domain, one has

∥Xj-1∥𝕃22=∫𝒪(|Xj-1|2-1)dx+|𝒪|≤C(1+14∥|Xj-1|2-1∥𝕃22)≤C(1+𝒥(Xj-1),(4.7)

where |𝒪| denotes the Lebesgue measure of 𝒪. Combining the above estimates and then those for 𝒜1 and 𝒜2 in (4.2), and then (4.3), we obtain after taking expectation

𝔼⁢[𝒥⁢(Xj)-𝒥⁢(Xj-1)]+14⁢𝔼⁢[∥∇⁡(Xj-Xj-1)∥𝕃22]+18⁢𝔼⁢[∥|Xj|2-|Xj-1|2∥𝕃22]+k⁢𝔼⁢[∥-Δ⁢Xj+f⁢(Xj,Xj-1)∥𝕃22]≤C⁢k⁢(1+𝔼⁢[𝒥⁢(Xj-1)]).

Summation over all time steps, and the discrete Gronwall’s lemma then establish the assertion for r=0.

(2) In order to validate the assertion for p=2r,r∈ℕ*, we proceed inductively and illustrate the argument for r=1. Recall that we have from before

𝒥⁢(Xj)-𝒥⁢(Xj-1)+14⁢∥∇⁡(Xj-Xj-1)∥𝕃22+18⁢∥|Xj|2-|Xj-1|2∥𝕃22+k⁢∥-Δ⁢Xj+f⁢(Xj,Xj-1)∥𝕃22≤C⁢𝒥⁢(Xj-1)⁢{k⁢(1+|Δj⁢W|2)+|Δj⁢W|2}+C⁢|Δj⁢W|2⁢(1+|Δj⁢W|2)+(∇⁡σ⁢(Xj-1),∇⁡Xj-1)𝕃2⁢Δj⁢W+(σ⁢(Xj-1),(|Xj-1|2-1)⁢Xj-1)𝕃2⁢Δj⁢W.(4.8)

To prove the assertion for r=1, one needs to multiply (4.8) by some quantity to produce a term like 𝒥2⁢(Xj)-𝒥2⁢(Xj-1)+α⁢|𝒥⁢(Xj)-𝒥⁢(Xj-1)|2 with α>0 on the left-hand side of the inequality in order to absorb related terms coming from the right-hand side of the inequality before discrete Gronwall’s lemma. Therefore, we multiply (4.8) with 𝒥⁢(Xj)+12⁢𝒥⁢(Xj-1) to get by binomial formula

34⁢(𝒥2⁢(Xj)-𝒥2⁢(Xj-1))+14⁢|𝒥⁢(Xj)-𝒥⁢(Xj-1)|2+12⁢(𝒥⁢(Xj)+𝒥⁢(Xj-1))⁢{14⁢∥∇⁡(Xj-Xj-1)∥𝕃22+18∥|Xj|2-|Xj-1|2∥𝕃22+k⁢∥-Δ⁢Xj+f⁢(Xj,Xj-1)∥𝕃22} ≤C⁢𝒥⁢(Xj-1)⁢(𝒥⁢(Xj)+12⁢𝒥⁢(Xj-1))⁢{k⁢(1+|Δj⁢W|2)+|Δj⁢W|2}+C⁢(𝒥⁢(Xj)+12⁢𝒥⁢(Xj-1))⁢|Δj⁢W|2⁢(1+|Δj⁢W|2)+(𝒥⁢(Xj)+12⁢𝒥⁢(Xj-1))⁢(σ⁢(Xj-1),(|Xj-1|2-1)⁢Xj-1)𝕃2⁢Δj⁢W+(𝒥⁢(Xj)+12⁢𝒥⁢(Xj-1))⁢(∇⁡σ⁢(Xj-1),∇⁡Xj-1)𝕃2⁢Δj⁢W:=𝒜3+𝒜4+𝒜5+𝒜6.(4.9)

By Young’s inequality, we have (θ1,θ2>0)

𝒜3≤θ1⁢|𝒥⁢(Xj)-𝒥⁢(Xj-1)|2+C⁢(θ1)⁢𝒥2⁢(Xj-1)⁢{k⁢(1+|Δj⁢W|2)+|Δj⁢W|2}2+C⁢𝒥2⁢(Xj-1)⁢{k⁢(1+|Δj⁢W|2)+|Δj⁢W|2},𝒜4≤θ2⁢|𝒥⁢(Xj)-𝒥⁢(Xj-1)|2+C⁢(θ2)⁢|Δj⁢W|4⁢(1+|Δj⁢W|4)+C⁢𝒥2⁢(Xj-1)⁢|Δj⁢W|4+C⁢(1+|Δj⁢W|4).

We can decompose 𝒜6 as

𝒜6=(𝒥(Xj)-𝒥(Xj-1))(∇σ(Xj-1),∇Xj-1)𝕃2ΔjW+32𝒥(Xj-1))(∇σ(Xj-1),∇Xj-1)𝕃2ΔjW:=𝒜6,1+𝒜6,2.

Note that 𝔼⁢[𝒜6,2]=0. By using Young’s inequality and the boundedness of σ′, we estimate 𝒜6,1,

𝒜6,1≤θ3⁢|𝒥⁢(Xj)-𝒥⁢(Xj-1)|2+C⁢(θ3)⁢∥∇⁡Xj-1∥𝕃24⁢|Δj⁢W|2≤θ3⁢|𝒥⁢(Xj)-𝒥⁢(Xj-1)|2+C⁢(θ3)⁢|Δj⁢W|2⁢(1+𝒥2⁢(Xj-1)).

Again, 𝒜5 can be written as 𝒜5,1+𝒜5,2 with 𝔼⁢[𝒜5,2]=0, where

𝒜5,1=(𝒥⁢(Xj)-𝒥⁢(Xj-1))⁢(σ⁢(Xj-1),(|Xj-1|2-1)⁢Xj-1)𝕃2⁢Δj⁢W.

Thanks to Young’s inequality, the boundedness of σ and (4.7) we get for θ4>0,

𝒜5,1≤θ4⁢|𝒥⁢(Xj)-𝒥⁢(Xj-1)|2+C⁢(θ4)⁢∥|Xj-1|2-1∥𝕃22⁢∥Xj-1∥𝕃22⁢|Δj⁢W|2≤θ4⁢|𝒥⁢(Xj)-𝒥⁢(Xj-1)|2+C⁢(θ4)⁢(𝒥2⁢(Xj-1)+∥Xj-1∥𝕃24)⁢|Δj⁢W|2≤θ4⁢|𝒥⁢(Xj)-𝒥⁢(Xj-1)|2+C⁢(θ4)⁢(1+𝒥2⁢(Xj-1))⁢|Δj⁢W|2.

We combine all the above estimates in (4.9), and choose θ1,…,θ4>0 with ∑i=14θi<14 to have, after taking expectation

𝔼⁢[𝒥2⁢(Xj)-𝒥2⁢(Xj-1)+C1⁢|𝒥⁢(Xj)-𝒥⁢(Xj-1)|2]+C2⁢𝔼⁢[(𝒥⁢(Xj)+𝒥⁢(Xj-1))⁢{∥∇⁡(Xj-Xj-1)∥𝕃22+∥|Xj|2-|Xj-1|2∥𝕃22+k⁢∥-Δ⁢Xj+f⁢(Xj,Xj-1)∥𝕃22}] ≤C3⁢(1+k)+C4⁢k⁢𝔼⁢[𝒥2⁢(Xj-1)].(4.10)

Summation over all time steps 0≤j≤J in (4.10), together with the discrete Gronwall’s lemma then validates the assertion of the theorem for r=1. This completes the proof. ∎

We employ the bounds for arbitrary moments of X in the strong norms in Lemma 3.1 (i), and a weak monotonicity argument to prove the following error estimate for the solution {Xj:0≤j≤J} of (4.1).

Theorem 4.2.

Assume that x∈Wper2,2, and Assumption A.1 holds true. Then, for every δ>0, there exist constants 0≤Cδ<∞ and k1=k1⁢(x,T)>0 such that for all k≤k1 sufficiently small,

sup0≤j≤J⁡𝔼⁢[∥Xtj-Xj∥𝕃22]+k⁢∑j=0J𝔼⁢[∥∇⁡(Xtj-Xj)∥𝕃22]≤Cδ⁢k1-δ,

where {Xt:t∈[0,T]} solves (2.2) while {Xj:0≤j≤J} solves (4.1).

The parameter δ>0 which appears in Theorem 4.2 is due to the non-Lipschitz drift in the problem and is caused by estimate (4.12) below.

Proof.

Consider (2.2) for the time interval [tj-1,tj], and denote ej:=Xtj-Xj. There holds ℙ-a.s. for all ϕ∈𝕎per1,2,

(ej-ej-1,ϕ)𝕃2+∫tj-1tj((∇⁡[Xtj-Xj],∇⁡ϕ)𝕃2+(D⁢ψ⁢(Xtj)-D⁢ψ⁢(Xj),ϕ)𝕃2)⁢ds=-∫tj-1tj(∇⁡[Xs-Xtj],∇⁡ϕ)𝕃2⁢ds-∫tj-1tj(D⁢ψ⁢(Xs)-D⁢ψ⁢(Xtj),ϕ)𝕃2⁢ds-12⁢∫tj-1tj((|Xj|2-1)⁢(Xj-Xj-1),ϕ)𝕃2⁢ds+∫tj-1tj(σ⁢(Xs)-σ⁢(Xtj-1),ϕ)𝕃2⁢dWs-∫tj-1tj(σ⁢(Xtj-1)-σ⁢(Xj-1),ϕ)𝕃2⁢dWs=:Ij+IIj+IIIj+IVj+Vj.(4.11)

The third term on the right-hand side attributes to the use of f⁢(Xj,Xj-1) instead of D⁢ψ⁢(Xj) in (4.1). Choose ϕ=ej⁢(ω), and apply expectation. By the weak monotonicity property (1.3) of the drift, the left-hand side of (4.11) is then bounded from below by

12⁢𝔼⁢[∥ej∥𝕃22-∥ej-1∥𝕃22+∥ej-ej-1∥𝕃22+2⁢k⁢(∥∇⁡ej∥𝕃22-∥ej∥𝕃22)].

Because of Young’s inequality and Lemma 3.2 (ii), we conclude

𝔼⁢[Ij]≤C⁢k2+k8⁢𝔼⁢[∥∇⁡ej∥𝕃22].

Next we bound 𝔼⁢[I⁢Ij]. For this purpose, we use the embedding 𝕎1,2↪𝕃6 for d≤3, the algebraic identity a3-b3=12⁢(a-b)⁢((a+b)2+a2+b2), and Young’s and Hölder’s inequalities in combination with Lemma 3.2 to estimate (δ>0)

𝔼⁢[I⁢Ij]≤12⁢∫tj-1tj𝔼⁢[∥Xs-Xtj∥𝕃2⁢∥(Xs+Xtj)2+Xs2+Xtj2∥𝕃3⁢∥ej∥𝕃6]⁢ds+∫tj-1tj𝔼⁢[∥Xs-Xtj∥𝕃2⁢∥ej∥𝕃2]⁢ds≤C⁢k⁢sups∈[tj-1,tj]⁡𝔼⁢[∥Xs-Xtj∥𝕃22⁢(∥Xtj∥𝕃64+∥Xs∥𝕃64)]+k8⁢𝔼⁢[∥ej∥𝕎1,22]+C⁢k2≤C⁢k⁢sups∈[tj-1,tj]⁡(𝔼⁢[∥Xs-Xtj∥𝕃22⁢(1+δ)])11+δ⁢(𝔼⁢[(∥Xtj∥𝕎1,24+∥Xs∥𝕎1,24)1+δδ])δ1+δ+k8⁢𝔼⁢[∥∇⁡ej∥𝕃22]+C⁢k⁢𝔼⁢[∥ej∥𝕃22]+C⁢k2.(4.12)

The leading factor is bounded by C⁢k11+δ by Lemma 3.2 (i), while the second factor may be bounded by Cδ due to (3.5). Thus we have

𝔼⁢[I⁢Ij]≤Cδ⁢k2+δ1+δ+C⁢k2+k8⁢𝔼⁢[∥∇⁡ej∥𝕃22]+C⁢k⁢𝔼⁢[∥ej∥𝕃22].

It is immediate to validate

|𝔼⁢[I⁢Vj]|+|𝔼⁢[Vj]|≤C⁢k2+18⁢𝔼⁢[∥ej-ej-1∥𝕃22]+C⁢k⁢𝔼⁢[∥ej-1∥𝕃22]

by adding and subtracting ej-1 in the second argument and proceeding as before, and Itô’s isometry in combination with (2.1) and Lemma 3.2 (i). Next we focus on the term I⁢I⁢Ij. In view of generalized Hölder’s inequality, and the embedding 𝕎1,2↪𝕃6 for d≤3,

𝔼⁢[I⁢I⁢Ij]≤12⁢𝔼⁢[∫tj-1tj∥ej∥𝕃6⁢∥Xj-Xj-1∥𝕃2⁢∥|Xj|2-1∥𝕃3⁢ds]≤k8⁢𝔼⁢[∥ej∥𝕃62]+C⁢k⁢𝔼⁢[∥Xj-Xj-1∥𝕃22⁢∥|Xj|2-1∥𝕃32]≤k8⁢𝔼⁢[∥ej∥𝕎1,22]+C⁢k⁢𝔼⁢[∥Xj-Xj-1∥𝕃22⁢(1+∥Xj∥𝕎1,24)]≡k8⁢𝔼⁢[∥ej∥𝕎1,22]+I⁢I⁢Ij,1.

In view of Lemma 4.1, we see that

supj⁡𝔼⁢[∥Xj∥𝕎1,2p]≤C for any ⁢p≥2,(4.13)

We use (4.6), Lemma 4.1 and (4.13) to estimate I⁢I⁢Ij,1,

I⁢I⁢Ij,1≤C⁢k⁢𝔼⁢[(k⁢𝒥⁢(Xj-1)+|Δj⁢W|2)⁢(1+∥Xj∥𝕎1,24)]≤C⁢k2⁢𝔼⁢[1+𝒥⁢(Xj-1)+(𝒥⁢(Xj-1))2+∥Xj∥𝕎1,28]+C⁢𝔼⁢[k⁢|Δj⁢W|2⁢∥Xj∥𝕎1,24]≤C⁢k2+C⁢k2⁢𝔼⁢[∥Xj∥𝕎1,28]+C⁢𝔼⁢[|Δj⁢W|4]≤C⁢k2,

and therefore we obtain

𝔼⁢[I⁢I⁢Ij]≤k8⁢𝔼⁢[∥∇⁡ej∥𝕃22]+C⁢k⁢𝔼⁢[∥ej∥𝕃22]+C⁢k2.

We combine all the above estimates to have

𝔼⁢[∥ej∥𝕃22-∥ej-1∥𝕃22+k⁢∥∇⁡ej∥𝕃22]≤C⁢k2+Cδ⁢k2+δ1+δ+C⁢k⁢(𝔼⁢[∥ej∥𝕃22+∥ej-1∥𝕃22]).(4.14)

Summation over all time steps 0≤j≤J in (4.14), together with the discrete (implicit form) Gronwall’s lemma then validates the assertion of the theorem. ∎

5 Space-Time Discretization and Strong Error Estimate

In this section, we first derive the uniform moment estimate for the discretized solution {Yj:0≤j≤J} of the structure preserving finite element based fully discrete scheme (2.5). Then by using these uniform bounds along with Lemma 4.1 we bound the error Ej:=Xj-Yj, where {Xj:0≤j≤J} solves (4.1).

We define the discrete Laplacian Δh:𝕍h→𝕍h by the variational identity

-(Δh⁢ϕh,ψh)𝕃2=(∇⁡ϕh,∇⁡ψh)𝕃2 for all ⁢ϕh,ψh∈𝕍h.

One can use the test function ϕ=-Δh⁢Yj+𝒫𝕃2⁢f⁢(Yj,Yj-1)∈𝕍h in (2.5) and proceed as in the proof of Lemma 4.1 along with (2.4), the 𝕎1,2- and 𝕃q-stabilities (1≤q≤∞) of the projection operator 𝒫𝕃2 (cf. [3]) to arrive at the following uniform moment estimates for {Yj:0≤j≤J}.

Lemma 5.1.

For every p=2r, r∈N*, there exists a constant C≡C⁢(p,T)>0 such that

max1≤j≤J𝔼[|𝒥(Yj)|p]+∑j=1J𝔼[∏ℓ=1r[[𝒥(Yj)]2ℓ-1+[𝒥(Yj-1)]2ℓ-1]×(∥∇(Yj-Yj-1)∥𝕃22+∥|Yj|2-|Yj-1|2∥𝕃22+k∥-ΔhYj+𝒫𝕃2f(Yj,Yj-1)∥𝕃22)]≤C,

provided E⁢[|J⁢(Y0)|p]≤C.

In view of Lemma 5.1, it follows that

sup0≤j≤J⁡𝔼⁢[∥Yj∥𝕎1,2p]≤C for all ⁢p≥2.(5.1)

We have the following theorem regarding the error Ej in strong norm.

Theorem 5.2.

Assume that x∈Wper2,2. Then, under Assumption A.1, there exist constants C>0, independent of the discretization parameters h,k>0, and k2≡k2⁢(T,x)>0 such that for all k≤k2 sufficiently small, there holds

sup0≤j≤J⁡𝔼⁢[∥Xj-Yj∥𝕃22]+k⁢∑j=0J𝔼⁢[∥∇⁡(Xj-Yj)∥𝕃22]≤C⁢(k+h2),

where {Xj:0≤j≤J} solves (4.1) while {Yj:0≤j≤J} solves (2.5).

Proof.

We subtract (2.5) from (4.1), and restrict to the test functions ϕ∈𝕍h. Choosing ϕ=𝒫𝕃2⁢Ej⁢(ω), and using (2.4), we obtain

12⁢𝔼⁢[∥𝒫𝕃2⁢Ej∥𝕃22-∥𝒫𝕃2⁢Ej-1∥𝕃22+∥𝒫𝕃2⁢[Ej-Ej-1]∥𝕃22]+k⁢𝔼⁢[(∇⁡Ej,∇⁡Ej)𝕃2+(D⁢ψ⁢(Xj)-D⁢ψ⁢(Yj),Ej)𝕃2] =k⁢𝔼⁢[(∇⁡Ej,∇⁡(Ej-𝒫𝕃2⁢Ej))𝕃2+(D⁢ψ⁢(Xj)-D⁢ψ⁢(Yj),Ej-𝒫𝕃2⁢Ej)𝕃2]+k⁢𝔼⁢[((|Xj|2-1)⁢Xj-Xj-12-(|Yj|2-1)⁢Yj-Yj-12,𝒫𝕃2⁢Ej)𝕃2]+𝔼⁢[(σ⁢(Xj-1)-σ⁢(Yj-1),𝒫𝕃2⁢[Ej-Ej-1])𝕃2⁢Δj⁢W] ≤k2⁢𝔼⁢[∥∇⁡Ej∥𝕃22]+k2⁢𝔼⁢[∥∇⁡(Xj-𝒫𝕃2⁢Xj)∥𝕃22]-k⁢𝔼⁢[∥Ej∥𝕃22]+k⁢𝔼⁢[∥𝒫𝕃2⁢Ej∥𝕃22]+|𝔼⁢[(|Xj|2⁢Xj-|Yj|2⁢Yj,Xj-𝒫𝕃2⁢Xj)𝕃2]|+k⁢𝔼⁢[((|Xj|2-1)⁢Xj-Xj-12-(|Yj|2-1)⁢Yj-Yj-12,𝒫𝕃2⁢Ej)𝕃2]+𝔼⁢[(σ⁢(Xj-1)-σ⁢(Yj-1),𝒫𝕃2⁢[Ej-Ej-1])𝕃2⁢Δj⁢W].

Note that the third term on the right-hand side of the first equality reflects that f⁢(Xj,Xj-1) is a perturbation of D⁢ψ⁢(Xj). By the weak monotonicity property (1.3), we see that

𝔼⁢[∥∇⁡Ej∥𝕃22-∥Ej∥𝕃22]≤𝔼⁢[(∇⁡Ej,∇⁡Ej)𝕃2+(D⁢ψ⁢(Xj)-D⁢ψ⁢(Yj),Ej)𝕃2],

and therefore we arrive at the following inequality

12⁢𝔼⁢[(∥𝒫𝕃2⁢Ej∥𝕃22-∥𝒫𝕃2⁢Ej-1∥𝕃22)+∥𝒫𝕃2⁢[Ej-Ej-1]∥𝕃22+k⁢∥∇⁡Ej∥𝕃22] ≤C⁢k⁢𝔼⁢[∥𝒫𝕃2⁢Ej∥𝕃22]+C⁢k⁢𝔼⁢[∥∇⁡(Xj-𝒫𝕃2⁢Xj)∥𝕃22]+C⁢k⁢|𝔼⁢[(|Xj|2⁢Xj-|Yj|2⁢Yj,Xj-𝒫𝕃2⁢Xj)𝕃2]|+k⁢𝔼⁢[((|Xj|2-1)⁢Xj-Xj-12-(|Yj|2-1)⁢Yj-Yj-12,𝒫𝕃2⁢Ej)𝕃2]+𝔼⁢[(σ⁢(Xj-1)-σ⁢(Yj-1),𝒫𝕃2⁢[Ej-Ej-1])𝕃2⁢Δj⁢W] =:Ck𝔼[∥𝒫𝕃2Ej∥𝕃22]+𝐁1,j+𝐁2,j+𝐁3,j+𝐁4,j.(5.2)

Note that, in view of Lemma 4.1, Young’s inequality and the embedding 𝕎1,2↪𝕃6 for d≤3

𝔼⁢[∥f⁢(Xj,Xj-1)∥𝕃22]≤C⁢𝔼⁢[∫𝒪(|Xj|4+1)⁢(|Xj|2+|Xj-1|2)⁢dx]≤C⁢𝔼⁢[∫𝒪(|Xj|6+|Xj-1|6+|Xj|2+|Xj-1|2)⁢dx]≤C⁢(1+supj⁡𝔼⁢[|𝒥⁢(Xj)|8]).

Thus using Lemma 4.1 and the estimate above, we see that

k⁢∑j=1J𝔼⁢[∥Δ⁢Xj∥𝕃22]≤k⁢∑j=1J𝔼⁢[∥-Δ⁢Xj+f⁢(Xj,Xj-1)∥𝕃22]+k⁢∑j=1J𝔼⁢[∥f⁢(Xj,Xj-1)∥𝕃22]≤C.(5.3)

Let us recall the following well-known properties of 𝒫𝕃2, see [1]:

{∥g-𝒫𝕃2⁢g∥𝕃2≤C⁢h⁢∥g∥𝕎1,2for all ⁢g∈𝕎1,2,∥g-𝒫𝕃2⁢g∥𝕃2+h⁢∥∇⁡[g-𝒫𝕃2⁢g]∥𝕃2≤C⁢h2⁢∥Δ⁢g∥𝕃2for all ⁢g∈𝕎2,2.(5.4)

We use (5.3) and (5.4) to infer that

∑j=1J𝐁1,j≤C⁢h2⁢∑j=1Jk⁢𝔼⁢[∥Δ⁢Xj∥𝕃22]≤C⁢h2.

Next we estimate ∑j=1J𝐁4,j. A simple approximation argument, (2.1), and (5.3) together with Young’s inequality lead to

∑j=1J𝐁4,j≤∑j=1J𝔼⁢[∥σ⁢(Xj-1)-σ⁢(Yj-1)∥𝕃2⁢∥𝒫𝕃2⁢[Ej-Ej-1]∥𝕃2⁢|Δj⁢W|]≤14⁢∑j=1J𝔼⁢[∥𝒫𝕃2⁢[Ej-Ej-1]∥𝕃22]+C⁢∑j=1Jk⁢𝔼⁢[∥Ej-1∥𝕃22]≤14⁢∑j=1J𝔼⁢[∥𝒫𝕃2⁢[Ej-Ej-1]∥𝕃22]+C⁢k⁢∑j=1J𝔼⁢[∥𝒫𝕃2⁢Ej-1∥𝕃22+∥Xj-1-𝒫𝕃2⁢Xj-1∥𝕃22]≤14⁢∑j=1J𝔼⁢[∥𝒫𝕃2⁢[Ej-Ej-1]∥𝕃22]+C⁢(h4+k⁢∑j=1J𝔼⁢[∥𝒫𝕃2⁢Ej-1∥𝕃22]).

We now bound the term 𝐁2,j. We use the algebraic formula given before (4.12), the embedding 𝕎1,2↪𝕃6 for d≤3, and a generalized Young’s inequality to have

𝐉2,j≤C⁢k⁢𝔼⁢[∥Ej∥𝕃6⁢(∥Xj∥𝕃62+∥Yj∥𝕃62)⁢∥Xj-𝒫𝕃2⁢Xj∥𝕃2]≤C⁢k⁢𝔼⁢[∥Ej∥𝕎1,2⁢(∥Xj∥𝕎1,22+∥Yj∥𝕎1,22)⁢∥Xj-𝒫𝕃2⁢Xj∥𝕃2]≤k8⁢𝔼⁢[∥∇⁡Ej∥𝕃22]+k8⁢𝔼⁢[∥Ej∥𝕃22]+C⁢k⁢𝔼⁢[(∥Xj∥𝕎1,24+∥Yj∥𝕎1,24)⁢∥Xj-𝒫𝕃2⁢Xj∥𝕃22]=:k8𝔼[∥∇Ej∥𝕃22]+𝐁2,j1+𝐁2,j2.

Thanks to (4.13) and (5.4), we note that

∑j=0J𝐁2,j1≤∑j=0Jk⁢𝔼⁢[∥𝒫𝕃2⁢Ej∥𝕃22]+C⁢h2⁢k⁢∑j=0J𝔼⁢[∥Xj∥𝕎1,22]≤C⁢h2+∑j=0Jk⁢𝔼⁢[∥𝒫𝕃2⁢Ej∥𝕃22].

We use (4.13), (5.1) and (5.4), together with Young’s inequality to get

∑j=0J𝐁2,j2≤C⁢h2⁢k⁢∑j=0J𝔼⁢[(∥Xj∥𝕎1,24+∥Yj∥𝕎1,24)⁢∥Xj∥𝕎1,22]≤C⁢h2⁢k⁢∑j=0J𝔼⁢[∥Xj∥𝕎1,28+∥Yj∥𝕎1,28+∥Xj∥𝕎1,24]≤C⁢h2.

It remains to bound 𝐁3,j. We decompose 𝐁3,j as follows:

𝐁3,j=k2𝔼[((|Xj|2-|Yj|2)(Xj-Xj-1),𝒫𝕃2Ej)𝕃2]+k2𝔼[((|Yj|2-1)(Ej-Ej-1),𝒫𝕃2Ej)𝕃2]=:𝐁3,j1+𝐁3,j2.

Thanks to generalized Hölder’s inequality, the Lq-stability (1≤q≤∞) of 𝒫𝕃2, the embedding 𝕎1,2↪𝕃6 for d≤3, estimates (5.4), (5.1) and (4.13), we have

𝐁3,j2≤C⁢k⁢𝔼⁢[∥Ej-Ej-1∥𝕃2⁢∥Ej∥𝕎1,2⁢∥|Yj|2-1∥𝕃3]≤C⁢k⁢𝔼⁢[(∥𝒫𝕃2⁢(Ej-Ej-1)∥𝕃2+∥Xj-𝒫𝕃2⁢Xj-(Xj-1-𝒫𝕃2⁢Xj-1)∥𝕃2)⁢∥Ej∥𝕎1,2⁢∥|Yj|2-1∥𝕃3]≤k16⁢𝔼⁢[∥Ej∥𝕎1,22]+C⁢k⁢𝔼⁢[∥𝒫𝕃2⁢(Ej-Ej-1)∥𝕃21+1⁢∥|Yj|2-1∥𝕃32]+C⁢k⁢𝔼⁢[∥Xj-𝒫𝕃2⁢Xj-(Xj-1-𝒫𝕃2⁢Xj-1)∥𝕃22⁢∥|Yj|2-1∥𝕃32]≤k16⁢𝔼⁢[∥∇⁡Ej∥𝕃22]+C⁢k⁢𝔼⁢[∥𝒫𝕃2⁢Ej∥𝕃22]+18⁢𝔼⁢[∥𝒫𝕃2⁢(Ej-Ej-1)∥𝕃22]+C⁢k2⁢𝔼⁢[∥𝒫𝕃2⁢(Ej-Ej-1)∥𝕃22⁢∥|Yj|2-1∥𝕃34]+C⁢k⁢h2⁢𝔼⁢[∥Xj∥𝕎1,22]+C⁢k⁢h2⁢𝔼⁢[(∥Xj∥𝕎1,22+∥Xj-1∥𝕎1,22)⁢∥|Yj|2-1∥𝕃32]≤k16⁢𝔼⁢[∥∇⁡Ej∥𝕃22]+18⁢𝔼⁢[∥𝒫𝕃2⁢(Ej-Ej-1)∥𝕃22]+C⁢k⁢𝔼⁢[∥𝒫𝕃2⁢Ej∥𝕃22]+C⁢k⁢h2+C⁢k2⁢𝔼⁢[∥Ej-Ej-1∥𝕃22⁢(1+∥Yj∥𝕎1,28)]+C⁢k⁢h2⁢𝔼⁢[1+∥Xj∥𝕎1,24+∥Xj-1∥𝕎1,24+∥Yj∥𝕎1,28]≤k16⁢𝔼⁢[∥∇⁡Ej∥𝕃22]+18⁢𝔼⁢[∥𝒫𝕃2⁢(Ej-Ej-1)∥𝕃22]+C⁢k⁢𝔼⁢[∥𝒫𝕃2⁢Ej∥𝕃22]+C⁢k⁢(h2+k).

Next we estimate 𝐁3,j1. We use the generalized Hölder’s inequality, the Lq-stability (1≤q≤∞) of 𝒫𝕃2, the embedding 𝕎1,2↪𝕃6 for d≤3, Young’s inequality, estimates (5.1), (5.4) and (4.6), (4.7) and (4.13), along with Lemma 4.1 to get

𝐁3,j1≤C⁢k⁢𝔼⁢[∥𝒫𝕃2⁢Ej∥𝕃6⁢∥Xj-Xj-1∥𝕃2⁢∥|Xj|2-|Yj|2∥𝕃3]≤C⁢k⁢𝔼⁢[∥Ej∥𝕎1,2⁢∥Xj-Xj-1∥𝕃2⁢(∥Xj∥𝕃62+∥Yj∥𝕃62)]≤k16⁢𝔼⁢[∥∇⁡Ej∥𝕃22]+k16⁢𝔼⁢[∥Ej∥𝕃22]+C⁢k⁢𝔼⁢[∥Xj-Xj-1∥𝕃22⁢(∥Xj∥𝕃62+∥Yj∥𝕃62)2]≤k16⁢𝔼⁢[∥∇⁡Ej∥𝕃22]+C⁢k⁢𝔼⁢[∥𝒫𝕃2⁢Ej∥𝕃22]+C⁢k⁢h2⁢𝔼⁢[∥Xj∥𝕎1,22]+C⁢k2⁢𝔼⁢[∥Xj∥𝕎1,28+∥Yj∥𝕎1,28]+C⁢𝔼⁢[∥Xj-Xj-1∥𝕃24]≤k16⁢𝔼⁢[∥∇⁡Ej∥𝕃22]+C⁢k⁢𝔼⁢[∥𝒫𝕃2⁢Ej∥𝕃22]+C⁢k⁢(h2+k)+C⁢𝔼⁢[k2⁢𝒥2⁢(Xj-1)+|Δj⁢W|4]≤k16⁢𝔼⁢[∥∇⁡Ej∥𝕃22]+C⁢k⁢𝔼⁢[∥𝒫𝕃2⁢Ej∥𝕃22]+C⁢k⁢(h2+k)+C⁢k2⁢(1+𝔼⁢[𝒥2⁢(Xj-1)])≤k16⁢𝔼⁢[∥∇⁡Ej∥𝕃22]+C⁢k⁢𝔼⁢[∥𝒫𝕃2⁢Ej∥𝕃22]+C⁢k⁢(h2+k).

Putting things together in (5.2) and using the discrete Gronwall’s lemma (implicit form) then yields

sup0≤j≤J⁡𝔼⁢[∥𝒫𝕃2⁢Ej∥𝕃22]+k⁢∑j=0J𝔼⁢[∥∇⁡(Xj-Yj)∥𝕃22]≤C⁢(k+h2).(5.5)

Thus, thanks to (4.13), (5.4) and (5.5), we conclude that

sup0≤j≤J⁡𝔼⁢[∥Xj-Yj∥𝕃22]+k⁢∑j=0J𝔼⁢[∥∇⁡(Xj-Yj)∥𝕃22]≤sup0≤j≤J⁡𝔼⁢[∥𝒫𝕃2⁢Ej∥𝕃22]+k⁢∑j=0J𝔼⁢[∥∇⁡(Xj-Yj)∥𝕃22]+C⁢h2⁢sup0≤j≤J⁡𝔼⁢[∥Xj∥𝕎1,22]≤C⁢(k+h2).

This finishes the proof. ∎

5.1 Proof of Main Theorem

Let Assumption A.1 hold and x∈𝕎per2,2. Then thanks to Theorem 4.2, for every δ>0, there exist constants 0≤Cδ<∞ and k1≡k1⁢(T,x)>0 such that for all k≤k1 sufficiently small,

sup0≤j≤J⁡𝔼⁢[∥Xtj-Xj∥𝕃22]+k⁢∑j=0J𝔼⁢[∥∇⁡(Xtj-Xj)∥𝕃22]≤Cδ⁢k1-δ,(5.6)

where {Xt:t∈[0,T]} solves (2.2) while {Xj:0≤j≤J} solves (4.1). Again, Theorem 5.2 asserts that there exist constants C>0, independent of the discretization parameters h,k>0 and k2≡k2⁢(T,x)>0 such that for all k≤k2 sufficiently small,

sup0≤j≤J⁡𝔼⁢[∥Xj-Yj∥𝕃22]+k⁢∑j=0J𝔼⁢[∥∇⁡(Xj-Yj)∥𝕃22]≤C⁢(k+h2).(5.7)

Let k0=min⁡{k1,k2}. Then (5.6) and (5.7) hold true for all k≤k0 sufficiently small. We combine (5.6) and (5.7) to conclude the proof of the main theorem.

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About the article

Received: 2017-05-28

Accepted: 2017-06-21

Published Online: 2017-07-18

Published in Print: 2018-04-01


Citation Information: Computational Methods in Applied Mathematics, Volume 18, Issue 2, Pages 297–311, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2017-0023.

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