Finite Element Approximations for a linear Cahn-Hilliard-Cook equation driven by the space derivative of a space-time white noise

We consider an initial- and Dirichlet boundary- value problem for a linear Cahn-Hilliard-Cook equation, in one space dimension, forced by the space derivative of a space-time white noise. First, we propose an approximate regularized stochastic parabolic problem discretizing the noise using linear splines. Then fully-discrete approximations to the solution of the regularized problem are constructed using, for the discretization in space, a Galerkin finite element method based on H2-piecewise polynomials, and, for time-stepping, the Backward Euler method. Finally, we derive strong a priori estimates for the modeling error and for the numerical approximation error to the solution of the regularized problem.


Introduction
Let T > 0, D = (0, 1) and (Ω, F , P ) be a complete probability space. Then we consider the following model initial-and Dirichlet boundary-value problem for a linear Cahn-Hilliard-Cook equation: find a stochastic function u : [0, T ] × D → R such that (1.1) x u(t, ·) ∂D = 0 ∀ t ∈ (0, T ], m = 0, 1, a.s. in Ω, whereẆ denotes a space-time white noise on [0, T ] × D (see, e.g., [23], [11]) and µ is a real constant for which there exists κ ∈ N such that where N is the set of all positive integers. The above stochastic partial differential equation combines two independent characteristics. On the one hand it corresponds to the linearization of the Cahn-Hilliard-Cook equation around a homogeneous initial state, in the spinodal region, that governs the dynamics of spinodal decomposition in metal alloys; see e.g. [4], and references therein. On the other hand the forcing noise is a derivative of a space-time white noise that physically arises in generalized Cahn-Hilliard equations, which are equations of conservative type describing the evolution of an order parameter in phase transitions (see [10]; cf. [12], [2], [19]). The mild solution of the problem above (cf. [6]) is given by the formula  with λ k := k π for k ∈ N, and ε k (z) := √ 2 sin(λ k z) for z ∈ D and k ∈ N. Observe that Ψ(t; x, y) = −∂ y G(t; x, y), where G(t; x, y) = ∞ k=1 e −λ 2 k (λ 2 k −µ)t ε k (x) ε k (y) for all (t, x, y) ∈ (0, T ] × D × D, is the space-time Green kernel of the corresponding deterministic parabolic problem: find a deterministic function w : [0, T ] × D → R such that (1.5) x w(t, ·) ∂D = 0 ∀ t ∈ (0, T ], m = 0, 1, The goal of the paper at hand is to propose and analyze a methodology of constructing finite element approximations to u. 1.1. The regularized problem. Our first step is to construct below an approximate to (1.1) regularized problem getting inspiration from the work [1] for the stochastic heat equation with additive space-time white noise (cf. [14], [15]).
First, we let S ⋆ be the space of functions which are continuous on D and piecewise linear over the above specified partition of D, i.e., S ⋆ := s ∈ C(D; R) : s D j ∈ P 1 (D j ) for j = 1, . . . , J ⋆ ⊂ H 1 (D).
3. An overview of the paper and related references. Our analysis first focus on the estimation of the modeling error, i.e. the difference u − u, in terms of the discretization parameters ∆t and ∆x. Indeed, working with the integral representation of u and u, we obtain (see Theorem 3.1) where C me is a positive constant that is independent of ∆x, ∆t and ǫ. Next target in our analysis, is to provide the fully discrete approximations of u defined in Section 1.2 with a convergence result, which is achieved by proving the following strong error estimate (see Theorem 5.3) for all ǫ 1 ∈ (0, 1 8 ] and ǫ 2 ∈ (0, ν(r)] with ν(2) = 1 3 and ν(3) = 1 2 , where C ne is a positive constant independent of ǫ 1 , ǫ 2 , ∆τ , h, ∆x and ∆t. To get the error estimate (1.11) we use as an auxilliary tool the Backward-Euler time-discrete approximations of u which are defined in Section 4. Thus, we can see the numerical approximation error as a sum of two types of error: the time-discretization error and the space-discretization error. The time-discretization error is the approximation error of the Backward Euler time-discrete approximations which is estimated in Theorem 4.2, while the space-discretization error is the error of approximating the Backward Euler time-discrete approximations by the Backward Euler finite element approximations, which is estimated in Proposition 5.2.
Let us expose some related bibliography. The work [18] contains a general convergence analysis for a class of time-discrete approximations to the solution of stochastic parabolic problems, the assumptions of which may cover problem (1.1). However, the approach we adopt here is different since first we introduce a space-time discretization of the noise and then we analyze time-discrete approximations to the solution. We would like to note that we are not aware of another work providing a rigorous convergence analysis for fully discrete finite element approximations to a stochastic parabolic equation forced by the space derivative of a space-time white noise. We refer the reader to our previous work [14], [15] and to [16] for the construction and the convergence analysis of Backward Euler finite element approximations of the solution to the problem (1.1) when µ = 0 and an additive space-time white noiseẆ is forced instead of ∂ xẆ . Finally, we refer the reader to [8], [1], [13], [3], [22] and [24] for the analysis of the finite element method for second order stochastic parabolic problems forced by an additive space-time white noise.
We close the section by an overview of the paper. Section 2 introduces notation, and recalls or proves several results often used in the paper. Section 3 is dedicated to the estimation of the modeling error. Section 4 defines the Backward Euler time-discrete approximations of u and analyzes its convergence. Section 5 contains the error analysis for the Backward Euler fully-discrete approximations of u.

Notation and Preliminaries
2.1. Function spaces and operators. Let I ⊂ R be a bounded interval. We denote by L 2 (I) the space of the Lebesgue measurable functions which are square integrable on I with respect to Lebesgue's measure dx, provided with the standard norm g 0,I := I |g(x)| 2 dx 1 2 for g ∈ L 2 (I). The standard inner product in L 2 (I) that produces the norm · 0,I is written as (·, ·) 0,I , i.e., (g 1 , g 2 ) 0,I := I g 1 (x)g 2 (x) dx for g 1 , g 2 ∈ L 2 (I). Let N 0 be the set of the nonnegative integers. For s ∈ N 0 , H s (I) will be the Sobolev space of functions having generalized derivatives up to order s in the space L 2 (I), and by · s,I its usual norm, i.e. g s,I := s ℓ=0 ∂ ℓ g 2 0,I 1 2 for g ∈ H s (I). Also, by H 1 0 (I) we denote the subspace of H 1 (I) consisting of functions which vanish at the endpoints of I in the sense of trace. We note that in H 1 0 (I) the, well-known, Poincaré-Friedrich inequality holds, i.e., there exists a nonegative constant C PF such that is a solution to the eigenvalue/eigenfunction problem: find . For s ≥ 0, the pair (V s (D), · V s ) is a complete subspace of L 2 (D) and we set (Ḣ s (D), · Ḣs ) := (V s (D), · V s ). For s < 0, we define (Ḣ s (D), · Ḣs ) as the completion of (V s (D), · V s ), or, equivalently, as the dual of (Ḣ −s (D), · Ḣ−s ). Let m ∈ N 0 . It is well-known (see [21]) that and there exist positive constants C m,A and C m,B such that Also, we define on L 2 (D) the negative norm · −m,D by v −m,D := sup (v,ϕ)0,D ϕ m,D : ϕ ∈Ḣ m (D) and ϕ = 0 , ∀ v ∈ L 2 (D), for which, using (2.3), it is easy to conclude that there exists a constant C −m > 0 such that Let L 2 = (L 2 (D), (·, ·) 0,D ) and L(L 2 ) be the space of linear, bounded operators from L 2 to L 2 . We say that, an operator Γ ∈ L(L 2 ) is Hilbert-Schmidt, when Γ HS := ∞ k=1 Γ(ε k ) 2 0,D 1 2 < +∞, where Γ HS is the so called Hilbert-Schmidt norm of Γ. We note that the quantity Γ HS does not change when we replace (ε k ) ∞ k=1 by another complete orthonormal system of L 2 , as it is the sequence (ϕ k ) ∞ k=0 with ϕ 0 (z) := 1 and ϕ k (x) := √ 2 cos(λ k z) for k ∈ N and z ∈ D. It is well known (see, e.g., [7]) that an operator Γ ∈ L(L 2 ) is Hilbert-Schmidt iff there exists a measurable function g : D × D → R such that (Γ(v))(·) = D g(·, y) v(y) dy for v ∈ L 2 (D), and then, it holds that Let L HS (L 2 ) be the set of Hilbert Schmidt operators of L(L 2 ) and Φ : [0, T ] → L HS (L 2 ). Also, for a random variable X, let E[X] be its expected value, i.e., E[X] := Ω X dP . Then, the Itô isometry property for stochastic integrals, which we will use often in the paper, reads for n = 1, . . . , N ⋆ and for g ∈ L 2 ((0, T ) × D), for which holds that Now, in the lemma below, we relate the stochastic integral of the projection Π of a deterministic function to its space-time L 2 −inner product with the discrete space-time white noise kernel W defined in Section 1.1 (cf. Lemma 2.1 in [14]).
Proof. To obtain (2.9) we work, using (2.7) and the properties of the stochastic integral, as follows: We close this section by observing that: if c ⋆ > 0, then and if (H, (·, ·) H ) is a real inner product space, then
where C R,m is a positive constant which is independent of f but depends on the D and m. Observing and in view (2.14), the map γ B : is an inner product on L 2 (D). Let (S(t)w 0 ) t∈[0,T ] be the standard semigroup notation for the solution w of (1.5). Then, the following a priori bounds hold (see Appendix A): for ℓ ∈ N 0 , β ≥ 0 and p ≥ 0, there exists a constant C β,ℓ,µ,µT > 0 such that: 2.3. Discrete spaces and operators. For r ∈ {2, 3}, let M r h ⊂ H 1 0 (D) ∩ H 2 (D) be a finite element space consisting of functions which are piecewise polynomials of degree at most r over a partition of D in intervals with maximum mesh-length h. It is well-known (cf., e.g., [5]) that the following approximation property holds: where C FM ,r is a positive constant that depends on r and is independent of h and v. Then, we define the discrete elliptic operators where the operator Λ B,h is invertible since where C is a positive constant independent of h and f .
To simplify the notation we define B : . It is easily seen that Later in the proof we shall use the symbol C for a generic constant that is independent of h and f , and may changes value from one line to the other. First, we observe that e 2 0,D = B(e, v). Then, we use the Galerkin orthogonality to get Using again (2.23) and the Galerkin orthogonality, we obtain Combining (2.24), (2.25) and (2.17), we arrive at (2.26) Let r = 2. We use (2.26) and (2.15) to get from which we conclude (2.22) for r = 2. Let r = 3. We use (2.26) with s ′ = 3 and (2.15) to obtain Then, as a simple consequence of (2.21), the following inequality holds Thus, observing that , and using (2.27), we easily conclude that γ B,h is an inner product in L 2 (D). We close this section with the following useful lemma.

An Estimate for the Modeling Error
In this section, we estimate the modeling error in terms of ∆t and ∆x (cf. Theorem 3.1 in [14]).
Theorem 3.1. Let u be the solution of (1.1) and u be the solution of (1.6). Then, there exists a real constant C > 0, independent of ∆t and ∆x, such that where ω 0 (∆t) := 1 + ∆t  . Now, we introduce the splitting x; s, y) Also, to simplify the notation in the rest of the proof, we set µ k := λ 2 k (λ 2 k − µ) for k ∈ N, and use the symbol C to denote a generic constant that is independent of ∆t and ∆x and may changes value from one line to the other.
Let w 0 ∈Ḣ 2 (D). According to the discussion in the begining of this section, when κ = 1 or κ ≥ 2 and ∆τ µ 2 < 1 4 , the existence and uniqueness of the time-discrete approximations (W m ) M m=0 is secured. We omit the case κ = 1 since then the operator Λ B is invertible and the proof of (4.5) follows moving along the lines of the proof of Proposition 4.1 in [14], or alternatively moving along the lines of the proof below using the operator T B instead of T B . Here, we will proceed with the proof of (4.5) under the assumption ∆τ µ 2 < 1 4 , without using somewhere a possible invertibilty of Λ B . In the sequel, we will use the symbol C to denote a generic constant that is independent of ∆t and may changes value from one line to the other. Let Now, take the L 2 (D)−inner product with E m of both sides of (4.6), to obtain Using (2.11), (4.7) and (2.15), we arrive at   Next, we use the Cauchy-Schwarz inequality to bound σ m as follows:  Combining (4.8), (4.11) and (4.10), we have Finally, use (4.13) and (2.16) (with β = 0, ℓ = 1, p = 0) to obtain (4.14) which establishes (4.5) for θ = 1. First, we observe that (4.4) is written equivalently as from which, after taking the L 2 (D)−inner product with W m , we obtain . . , M. Then, we combine (2.11) and (4.15) to have from which, applying a simple induction argument, we conclude that   In addition we have Thus, the estimate (4.5) for θ = 0 follows easily combining (4.19) and (4.20).

The Stochastic
Case. Next theorem combines the convergence result of Proposition 4.1 with a discrete Duhamel's principle in order to prove a discrete in time L ∞ t (L 2 P (L 2 x )) convergence estimate for the time discrete approximations of u (cf. [14], [22]).  1)-(4.2). Also, we assume that κ = 1, or κ ≥ 2 and ∆τ µ 2 < 1 4 . Then, there exists a constant C > 0, independent of ∆t, ∆x and ∆τ , such that where ω 1 (∆τ, ǫ) := ǫ − 1 2 + (∆τ ) ǫ (1 + (∆τ ) . Also, for m ∈ N, we denote by G ΛΦ,m the Green function of the operator Λ m−1 Φ. In the sequel, we will use the symbol C to denote a generic constant that is independent of ∆t, ∆τ and ∆x, and may changes value from one line to the other.
Using (4.2) and a simple induction argument, we conclude that which is written, equivalently, as follows: Now, we introduce the splitting By the definition of the Hilbert-Schmidt norm, we have Let θ ∈ [0, 1 8 ). Using the deterministic error estimate (4.5) and (2.10), we obtain

Convergence of the Fully-Discrete Approximations
To get an error estimate for the fully-discrete approximations of u defined by (1.8)-(1.9), we proceed by comparing them with their time-discrete approximations defined by (4.1)-(4.2) and using a discrete Duhamel principle (cf. [14], [22]).
Next, we derive a discrete in time L 2 t (L 2 x ) estimate for the error approximating the Backward Euler time-discrete approximations of the solution to (1.5) defined in (4.3)-(4.4), by the Backward Euler finite element approximations defined in (5.1)-(5.2). The main difference with the case µ = 0 which has been considered in [14], is that, our assumption (1.2) on µ, can not ensure the coerciveness of the discrete elliptic operator Λ B,h . Proof. The error estimate (5.3) follows by interpolation, after showing that holds for θ = 0 and θ = 1.
In the sequel, we will use the symbol C to denote a generic constant that is independent of ∆τ and h, and may changes value from one line to the other. Let for m = 1, . . . , M . Then, combine (5.5) and (5.6), to get the following error equation Taking the L 2 (D)−inner product with E m of both sides of (5.7), it follows that from which, after using (2.11), we conclude that Since 2∆τ µ 2 < 1, (5.8) yields for m = 1, . . . , M . Applying a simple induction argument based on (5.8) and then using that 4∆τ µ 2 < 1, we get Summing with respect to m from 1 up to M , using (5.10) and observing that T B,h E 0 = 0, (5.8) gives Let r = 3. Then, by (2.22), (5.11) and the Poincaré-Friedrich inequality, we obtain Taking the L 2 (D)−inner product of (4.4) with ∂ 4 W m and then integrating by parts, we obtain . . , M. Using (2.11), (5.13) and the Cauchy-Schwarz inequality we obtain . . , M, which, after using the geometric mean inequality, yields (5.14) Since 2 µ 2 ∆τ < 1, from (5.14) follows that . . , M, from which, applying a simple induction argument, we conclude that Next, sum both side of (5.14) with respect to m, from 1 up to M , and use (5.15) to conclude that Taking the L 2 (D)−inner product of (4.4) with ∂ 2 W m , and then integrating by parts, it follows that . . , M. Using (2.11), (5.17), the Cauchy-Schwarz inequality and the geometric mean inequality, we obtain . . , M. Since 2 µ 2 ∆τ < 1, proceeding as in obtaining (5.15) and (5.16) from (5.14), we arrive at Summing with respect to m from 1 up to M , and using (5.24), (5.23) gives Finally, using (5.25), (2.28) and (2.4) we obtain Proof. Let δ ǫ,h : M r h × M r h → R be the inner product on M r h given by We can construct a basis (χ j ) n h j=1 of M r h which is L 2 (D)−orthonormal, i.e., (χ i , χ j ) 0,D = δ ij for i, j = 1, . . . , n h , and δ ǫ,h −orthogonal, i.e., there exist (λ ǫ,h,ℓ ) n h ℓ=1 ⊂ (0, +∞) such that δ ǫ,h (χ i , χ j ) = λ ǫ,h,i δ ij for i, j = 1, . . . , n h (see Section 8.7 in [9]). Thus, there are (µ j ) n h j=1 ⊂ R such that ψ h = n h j=1 µ j χ j , and (5.27) is equivalent to µ i = 1 λ ǫ,h,i (f, χ i ) 0,D for i = 1, . . . , n h . Finally, we obtain (5.28) with A h,ǫ (x, y) = n h j=1 χj (x)χj (y) λ ǫ,h,j . Our second step is to compare, in a discrete in time L ∞ t (L 2 P (L 2 x )) norm, the Backward Euler timediscrete approximations of u with the Backward Euler finite element approximations of u. Proof. Let I : L 2 (D) → L 2 (D) be the identity operator and Λ h : L 2 (D) → M r h be the inverse discrete elliptic operator given by Λ h := (I + ∆τ Λ B,h ) −1 P h , having a Green function G Λ h = A h,∆τ according to Lemma 5.1 and taking into account that µ 2 ∆τ < 4. Also, we define an operator Φ h : L 2 (D) → M r h by (Φ h f )(x) := D G Φ h (x, y) f (y) dy for f ∈ L 2 (D) and x ∈ D, where G Φ h (x, y) = −∂ y G Λ h (x, y). Then, we have that Λ h f ′ = Φ h f for all f ∈ H 1 (D). Also, for ℓ ∈ N, we denote by G Λ h ,Φ h ,ℓ the Green function of Λ ℓ h Φ h . In the sequel, we will use the symbol C to denote a generic constant that is independent of ∆t, ∆x, h and ∆τ , and may changes value from one line to the other.
The available error estimates allow us to conclude a discrete in time L ∞ t (L 2 P (L 2 x )) convergence of the Backward Euler fully-discrete approximations of u. Proof. The estimate is a simple consequence of the error bounds (5.29) and (4.21).