Convergence of a finite-volume scheme for a heat equation with a multiplicative Lipschitz noise

We study here the approximation by a finite-volume scheme of a heat equation forced by a Lipschitz continuous multiplicative noise in the sense of It\^o. More precisely, we consider a discretization which is semi-implicit in time and a two-point flux approximation scheme (TPFA) in space. We adapt the method based on the theorem of Prokhorov to obtain a convergence in distribution result, then Skorokhod's representation theorem yields the convergence of the scheme towards a martingale solution and the Gy\"{o}ngy-Krylov argument is used to prove convergence in probability of the scheme towards the unique variational solution of our parabolic problem.

For  > 0, we consider a nonlinear stochastic heat equation under Neumann boundary conditions: − ∆ d = ()  (), in Ω × (0,  ) × Λ; on Ω × (0,  ) × Λ; (1.1) where n denotes the unit normal vector to Λ outward to Λ.We assume the following hypotheses on the data: for all  ∈ R and a constant   ≥ 0 only depending on the Lipschitz constant  ≥ 0 of  and on (0).In particular, our scheme applies for square integrable, additive noise with appropriate measurability assumptions.

Concept of solution and main result
The theoretical framework associated with Problem (1.1) is well established in the literature.Indeed, we can find many existence and uniqueness results for various concepts of solutions associated with this problem such as mild solutions, variational solutions, pathwise solutions and weak solutions, see e.g.[16] and [32].In the present paper we will be interested in the concept of solution as defined below, which we will call a variational solution: Definition 1. Existence, uniqueness and regularity of this variational solution is well-known in the literature, see e.g.[31,32,35].The main result of this paper is to propose a finite-volume scheme for the approximation of such a variational solution and to show its stochastically strong convergence by passing to the limit with respect to the time and space discretization parameters.This is stated in the following convergence result: Theorem 1.3.Assume that hypotheses  1 and  2 hold.Let (  ) ∈N be a sequence of admissible finite-volume meshes of Λ in the sense of Definition 2.1 such that the mesh size ℎ  tends to 0 and (  ) ∈N ⊆ N * a sequence of positive numbers which tends to infinity.For a fixed  ∈ N, let   ℎ, and   ℎ, be respectively the right and left in time finite-volume approximations defined by (2.2), (2.5)-(2.6)with  =   and  =   .Then (  ℎ, ) ∈N and (  ℎ, ) ∈N converge in   (Ω;  2 (0,  ;  2 (Λ))) for any  ∈ [1,2) to the variational solution of Problem (1.1) in the sense of Definition 1.2.

State of the art
The study of numerical schemes for stochastic partial differential equations (SPDEs) has attracted a lot of attention in the last decades and there exists an extensive literature on this topic.A list of references for the numerical analysis of SPDEs and an overview of the state of the art is given in [2,17] and [34].
Concerning the theoretical and numerical study of stochastic heat equations, semigroup techniques may be used to construct mild solutions (see, e.g., [16]).However, from the point of view of applications and mathematical modeling, it is often interesting to consider first-order perturbations of the stochastic heat equation and more complicated, nonlinear second order operators, such as the -Laplacian or the porous medium operator.For these nonlinear SPDEs, the semigroup approach is not available and variational techniques have been developed in [31,35] and [32].
In the numerical analysis of variational solutions to parabolic SPDEs, spatial discretizations of finite-element type have been frequently used (see, e.g., [9,12] and the references therein).On the other hand, for stochastic scalar conservation laws, finite-volume schemes have been studied in [5-8, 21, 28, 33] and [22].To the best of our knowledge, there are only a few results on finite-volume schemes for parabolic SPDEs.Let us mention the work of [4] where the authors proposed a convergence result of a finite-volume scheme for the approximation of a stochastic heat equation with linear multiplicative noise.

Aim of the study
In this contribution, we want to extend the finite-volume approximation results in the hyperbolic case to the stochastic heat equation with Lipschitz continuous multiplicative noise.Having applications to nonlinear operators and also to degenerate parabolic-hyperbolic problems with stochastic force in mind for the future, we propose a method for the convergence of the scheme which does not rely on mild solutions and results from semigroup theory.Additionally, we may include a discrete gradient in the right-hand side of our scheme (2.6) in the future.Hence, further studies may be devoted to the convergence analysis of finite-volume schemes for equations with multiplicative noise involving first order spatial derivatives of the solution.The main technical challenge is the nonlinear multiplicative noise.Indeed, from the a priori estimates, we get up to subsequences weak convergence results in several functional spaces for our finite-volume approximations and this mode of convergence is not enough to identify the weak limit of the nonlinear term in the stochastic integral.Therefore, we first show the convergence towards a martingale solution by adapting the stochastic compactness method based on Skorokhod's representation theorem.Then, using a famous argument of pathwise uniqueness (see, e.g., [30]), we obtain the stochastically strong convergence result stated in Theorem 1. 3. In this contribution, we limit ourselfes to the convergence proof.The study of convergence rates is subject to current research activities and will be detailed in a forthcoming work.

Outline
The paper is organized as follows.The next section concerns the introduction of the finite-volume framework: the definition of an admissible finite-volume mesh on Λ will be stated and the associated notations of discrete unknowns will be given.Then the notions of discrete gradient and discrete  1 -seminorm will be introduced.In a last subsection, we will introduce our finite-volume scheme together with the associated finite-volume approximations.The remainder of the paper is then devoted to the proof of the convergence of this approximations towards the variational solution of (1.1).To do so, we will prove in Section 3 several stability estimates satisfied by these approximations, but also a boundedness result on the approximation of the stochastic integral.These estimates will allow us to pass the limit in the numerical scheme in Section 4.More precisely, we apply the classical stochastic compactness argument (see, e.g., [11]).By the theorem of Prokhorov, we will get convergence in law (up to subsequences) of our finite-volume approximations.At the cost of a change of probability space, the Skorokhod representation theorem will allow us to obtain almost sure convergence of the proposed finite-volume scheme.Then, a martingale identification argument will help us in order to recover at the limit the desired stochastic integral.In this way, we have shown that our finite-volume scheme converges to a martingale solution of (1.1), i.e., the stochastic basis is not fixed, but enters an unknown in the equation.Next, we show pathwise uniqueness of solutions to (1.1).This, together with a classical argument of Gyöngy and Krylov (see [30]) allows us to deduce convergence in probability of the scheme with respect to the initial stochastic basis.

Admissible finite-volume meshes and notations
In order to perform a finite-volume approximation of the variational solution of Problem (1.1) on [0,  ]×Λ we need first of all to set a choice for the temporal and spatial discretization.For the time discretization, let  ∈ N * be given.We define the fixed time step ∆ =   and divide the interval [0,  ] in 0 =  0 <  1 < . . .<   =  equidistantly with   = ∆ for all  ∈ {0, . . .,  − 1}.For the space discretization, we refer to [26] and consider finite-volume admissible meshes in the sense of the following definition.Once an admissible finite-volume mesh  of Λ is fixed, we will use the following notations. Notations.
-Let ,  ∈  be two neighbouring control volumes.For  = | ∈ ℰ int , let   be the length of  and  | the distance between   and   .-For neighbouring control volumes ,  ∈  , we denote the unit vector on the edge  = | pointing from  to  by n  .-For  = | ∈ ℰ int , the diamond   (see Fig. 2) is the open quadrangle whose diagonals are the edge  and the segment [  ,   ].For  ∈ ℰ ext ∩ ℰ  , we define   := .Then, Λ = ⋃︀ ∈ℰ   .-  :=  2 (  ) is the two-dimensional Lebesgue measure of the diamond   .Note that for  ∈ ℰ int , we Using these notations, we introduce a positive number reg( ) = max (where  is the maximum of edges incident to any vertex) that measures the regularity of a given mesh and is useful to perform the convergence analysis of finite-volume schemes.This number should be uniformly bounded when the mesh size tends to 0 for the convergence results to hold.

Discrete unknowns and piecewise constant functions
From now on and unless otherwise specified, we consider  ∈ N * , ∆ =   and  an admissible finite-volume mesh of Λ in the sense of Definition 2.1 with a mesh size ℎ.For  ∈ {0, . . .,  − 1} given, the idea of a finite-volume scheme for the approximation of Problem (1.1) is to associate to each control volume  ∈  and time   a discrete unknown value denoted    ∈ R, expected to be an approximation of (  ,   ), where  is the variational solution of (1.1).Before presenting the numerical scheme satisfied by the discrete unknowns {   ,  ∈  ,  ∈ {0, . . .,  − 1}}, let us introduce some general notations.
For any arbitrary vector (   ) ∈ ∈ R  ℎ we can define the piecewise constant function   ℎ : Λ → R by Note that since the mesh  is fixed, by the continuous mapping defined from R  ℎ to  2 (Λ) by the space R  ℎ can be considered as a finite-dimensional subspace of  2 (Λ) and we may naturally identify the function and the vector   ℎ ≡ (   ) ∈ ∈ R  ℎ .Then, knowing for all  ∈ {0, . . .,  } the function   ℎ , we can define the following piecewise constant functions in time and space   ℎ, ,   ℎ, : [0,  ] × Λ → R by (2.2) Remark 2.2.The superscripts  and  in (2.2) do not refer to the continuity properties of the associated functions (which may be chosen either càdlàg or càglàd).The difference is that   ℎ, is adapted whereas   ℎ, is not adapted.
As for the piecewise constant function in space, since  and  are fixed, by the continuous mapping defined from R  ℎ × to  2 (0,  ;  2 (Λ)) by the space R  ℎ × can be considered as a finite-dimensional subspace of  2 (0,  ;  2 (Λ)) and we may naturally identify We can also define the piecewise affine, continuous in time and piecewise constant in space reconstruction Remark 2.3.Note that in the rest of the paper, when we will consider a time and space function  : [0,  ]×Λ → R on all the space Λ (respectively the time interval [0,  ]) at a fixed time  ∈ [0,  ] (respectively at a fixed  ∈ Λ) we will omit the space (respectively time) variable in the notations and write () (respectively ()) instead of (, •) (respectively (•, )).

Discrete norms and discrete gradient
Fix  ∈ {0, . . .,  − 1} and consider for the remainder of this subsection an arbitrary vector (   ) ∈ ∈ R  ℎ and use its natural identification with the piecewise constant function in space   ℎ ≡ (   ) ∈ .We introduce in what follows the notions of discrete gradient and discrete norms for such a function   ℎ .Definition 2.4 (Discrete  2 -norm).We define the  2 -norm of   ℎ ∈ R  ℎ as follows Definition 2.5 (Discrete gradient).We define the gradient operator ∇ ℎ that maps scalar fields   ℎ ∈ R  ℎ into vector fields of (R 2 )  ℎ (where  ℎ is the number of edges in the mesh  ), we set We remark that ∇ ℎ  ℎ  is considered as a piecewise constant function, which is constant on the diamond   .Definition 2.6 (Discrete  1 -seminorm).We define the  1 -seminorm of   ℎ ∈ R  ℎ as follows Notation.If not marked otherwise, for an edge  ∈ ℰ int we denote the neighbouring control volumes by  and , i.e.,  = |.In particular we use this notation in sums.
Remark 2.7.Note that in particular, where the constant 2 corresponds to the space dimension  = 2.We have now all the necessary definitions and notations to present the finite-volume scheme studied in this paper.This is the aim of the next subsection.
Remark 2.9.The second term on the left-hand side of (2.6) is the classical two-point flux approximation of the Laplace operator obtained formally by integrating the Laplace operator on each control volume  ∈  , then applying the Gauss-Green theorem to the term ∫︀  ∆( +1 , )d and finally combining Taylor expansions of the function ( +1 , •) at the points   and   together with the orthogonality condition on the mesh (see [26], Sect.10) for more details on the two-point flux approximation of the Laplace operator with Neumann boundary conditions).The time-implicit discretization of the Laplace operator has several analytic advantages: First of all, calculations in the a-priori estimates are simplified.Secondly, we omit the use of a CFL-condition.Last but not least, for more general nonlinear operators such as the -Laplace operator, an implicit time discretization is more appropriate.However, an explicit time discretization of the noise is crucial and can not be omited due to the non-anticipative character of the Itô stochastic integral.
We can note that by multiplying equation (2.6) by   , summing over  ∈  and using equality (2.4), the numerical scheme can be rewritten as: For any  ∈ {0, . . .,  − 1}, find (2.7) The two formulations are equivalent but this "variational" formulation will be more useful in the analysis to follow.

Stability estimates
We will derive in this section several stability estimates satisfied by the discrete solution (  ℎ ) 1≤≤ ∈ (R  ℎ )  of the scheme (2.5)-(2.6)given by Proposition 2.10, and also by the associated right and left finite-volume approximations   ℎ, and   ℎ, defined by (2.2).

Bounds on the finite-volume approximations
We start by giving a bound on the discrete initial data.
Lemma 3.1.Let  0 be a given function satisfying assumption  1 .Then, the associated discrete initial data  0 ℎ ∈ R  ℎ defined by (2.5) satisfies P-a.s. in Ω, The proof is a direct consequence of the definition of  0 ℎ and the Cauchy-Schwarz inequality.
We can now give the bounds on the discrete solutions which is one of the key points of the proof of the convergence theorem.Proposition 3.2 (Bounds on the discrete solutions).There exists a constant  1 > 0, depending only on  0 ,   , |Λ| and  such that We consider the terms separately: For the first term on the left-hand side we find Taking expectation in (3.1), the first expression on the right-hand side of (3.1) vanishes, since    and ∆ +1  are independent and therefore In the second term we apply Young's inequality in order to keep all necessary terms.Then, taking expectation and using the Itô isometry we obtain for any  ∈  .Altogether we find Summing over  ∈ {0, . . .,  − 1} and multiplying with 2∆ we obtain Since the second and third term in (3.2) are nonnegative, from  2 and (1.2) it follows that Applying the discrete Gronwall lemma yields From (3.3) and Lemma 3.1 we may conclude that there exists a constant Υ > 0 such that sup for all  ∈ {1, . . . }.From (3.2), Lemma 3.1 and (3.4) it now follows that for all  ∈ {1, . . .,  }.
We are now interested in the bounds on the right and left finite-volume approximations defined by (2.2).As a direct consequence of Proposition 3.2 we get a  2 (Ω;  2 (0,  ;  2 (Λ)))-bound on these approximations.Thanks to Proposition 3.2 we can also obtain a  2 (Ω;  2 (0,  ;  2 (Λ)))-bound on the discrete gradients of the finite-volume approximations.
We end this section by a bound on the discrete solution which will be useful for obtaining the time translate estimate and the bounds on the Gagliardo seminorm.Note that the difficulty here is to have the maximum inside the expectation.Lemma 3.5.There exists a constant  3 ≥ 0 independent of the discretization parameters  ∈ N * and ℎ, such that Proof.For  ∈ N, we choose an arbitrary  ∈ {0, . . .,  − 1} and an arbitrary  ∈  .Testing the implicit scheme (2.7) with  +1  yields This provides by Cauchy-Schwarz and Young inequalities 1 2 .
We obtain .
We can estimate the second term by the Burkholder-Davis-Gundy inequality Now we apply Cauchy-Schwarz and Young inequalities (with  > 0),  2 and (1.2) to estimate Plugging the above estimate in (3.7) and again using  2 with (1.2), we arrive at )︂ .
Now, the assertion follows by Lemmas 3.1 and 3.3.
We start by giving an estimation of the space translate.We do not give the proof here as it is similar to the one given in Theorem 10.3 of [26].
Then there exists a constant  ≥ 0, only depending on Λ, such that for all  ∈ R 2 and almost every  ∈ [0,  ] and P-a.s in )︁ .

Convergence of the finite-volume scheme
We now have all the necessary material to pass to the limit in the numerical scheme.
In the sequel, for  ∈ N * , let (  )  be a sequence of admissible meshes of Λ in the sense of Definition 2.1 such that the mesh size ℎ  tends to 0 when  tends to +∞ and let (  )  ⊂ N be a sequence with lim →+∞   = +∞ and ∆  :=   .For the sake of simplicity we shall use the notations  =   , ℎ = size(  ), ∆ = ∆  and  =   when the -dependency is not useful for the understanding of the reader.
Our aim is to show that  is the unique solution to (1.1).But weak convergence is not enough to pass to the limit in the nonlinear diffusion term of our finite-volume scheme.Therefore we will apply the method of stochastic compactness.

The stochastic compactness argument
For better readability we define  :=  2 (0,  ;  2 (Λ)) and From Lemmas 3.9 and 3.11 we get immediately the following bound.Lemma 4.2.For any fixed  ∈ (0, 1  2 ), there exists a constant  6 depending on  0 and the mesh regularity reg( ) but not depending on the discretization parameter  ∈ N * , such that In the following, for a random variable  defined on a probability space (Ω, , P) the law of  will be denoted by P ∘  −1 .
In order to apply Skorokhod theorem and to prove the almost sure convergence, we begin by proving the convergence in law.
Proof.By Lemma A3 of [38] with  =  2 (0,  ;  2 (Λ)) and  = R  ℎ × it follows that there exists In the same manner with  =  2 (Λ) and  = R  ℎ it follows that there exists (ṽ 0  ) ∈ in R  ℎ such that We recall the notation of Subsection 2.2 and in particular that For any  ∈   we consider the non-negative, Borel measurable mapping and therefore, for all  ∈  and all  ∈   ,   (0, ) Thus, for all  ∈   ,  ∈ {0, . . .,   − 1} and P ′ -a.s. in

Identification of the stochastic integral
In this subsection, we adapt ideas taken from [11,18,34] and adjust the arguments to our specific situation.We show that each   is a Brownian motion with respect to the filtration given in Definition 4.8.With this result at hand, we may show that ( ∞ ()) ≥0 is a Brownian motion with respect to the filtration given in Definition 4.11.In Lemma 4.13 we prove that  ∞ is admissible for the stochastic Itô integral with respect to ( ∞ ()) ≥0 .Finally, in Lemma 4.14 we provide an approximation result for the stochastic Itô integrals.Definition 4.8.For  ∈ [0,  ] we define ℱ   to be the smallest sub--field of  ′ generated by  0  and   () for 0 ≤  ≤ .The right-continuous, P ′ -augmented filtration of (ℱ   ) ∈[0, ] denoted by (F   ) ∈[0, ] is defined by for any  ∈ [0,  ].
In the following, we want to show firstly that the stochastic process ( ∞ ()) ∈[0, ] :=  ∞ is a Brownian motion and secondly that a filtration may be chosen in order to have compatibility of  ∞ with stochastic integration in the sense of Itô with respect to  ∞ .Since  ∞ is a random variable taking values in  2 (0,  ;  2 (Λ)),  ∞ (, •) is only defined for a.e. ∈ [0,  ] and the construction of an appropriate filtration induced by  ∞ becomes delicate.
We now have all the necessary tools to pass to the limit in the scheme.Proposition 4.16.There exists a subsequence of (̂︀  ℎ, )  , still denoted by (̂︀  ℎ, )  , converging in   (Ω ′ ;  2 (0,  ;  2 (Λ))) (for any  ∈ [1, 2)) as  → +∞ towards a (F ∞  ) ∈[0, ] -adapted stochastic process  ∞ with values in  2 (Λ) and having P ′ -a.s.continuous paths.Moreover,  ∞ ∈  2 (Ω ′ ;  2 (0,  ;  1 (Λ))) and satisfies for all  ∈ [0,  ], First we sum (4.17) over each control volume  ∈  , then we integrate over each time interval [  ,  +1 ] for fixed  = 0, . . .,  − 1, then we sum over  = 0, . . .,  − 1 and finally we take the expectation to obtain In the following, we will pass to the limit with  → +∞ on the right-hand side of (4.18).Using partial integration we obtain Thanks to the convergence results of Lemma 4.15, passing to a not relabeled subsequence if necessary, we can pass to the limit and obtain Now our aim is to show the following convergence result: First, we note that by rearranging the sum in (4.18) the term  2, can be written as Then, since ∇ • n = 0 on Λ, thanks to the Stokes formula one has, Thus, we have Using Lemma 4.15 and passing to a not relabeled subsequence if necessary, one gets Concerning the second term in    P ′ -a.s. in Ω ′ for all  ∈ (R) with ( ) = 0 and all  ∈ (R 2 ) such that ∇ • n = 0 on Λ.By Theorem 1.1 of [24] the set { ∈ (R 2 ) | ∇ • n = 0 on Λ} is dense in  1 (Λ) and therefore (4.19) applies to all  ∈  1 (Λ).
To do so, we will proceed in several steps.First, pathwise uniqueness of the heat equation with multiplicative Lipschitz noise is a consequence of Proposition 4.18: Roughly speaking, martingale solutions of (1.1) on a joint stochastic basis and with respect to the same initial datum coincide.In the proof of Proposition 4.20, we construct two convergent finite-volume approximations with respect to a joint stochastic basis, namely (    ) and (    ), from the function (  ℎ, ) of our original finite-volume scheme using the theorems of Prokhorov and Skorokhod.Then, as a consequence of pathwise uniqueness, the limits coincide and we may apply Lemma 1.1 of [30] in order to obtain the convergence in probability of (  ℎ, ).Thanks to our previous result we can improve the convergence and pass to the limit in the originally given finite-volume scheme (see Lem. 4.21).Proposition 4.18.Let (Ω, , P, (ℱ  ) ≥0 , ( ()) ≥0 ) be a stochastic basis and  1 ,  2 be solutions to (1.1) with respect to the ℱ 0 -measurable initial values  1 0 and  2 0 in  2 (Ω;  2 (Λ)) respectively on (Ω, , P, (ℱ  ) ≥0 , ( ()) ≥0 ).Then, there exists a constant  ≥ 0 such that ]︁ for all  ∈ [0,  ].

Definition 2 . 1 .
(Admissible finite-volume mesh) An admissible finite-volume mesh  of Λ (see Fig.1) is given by a family of open polygonal and convex subsets , called control volumes of  , satisfying the following properties:-Λ = ⋃︀ ∈ .

Figure 1 .
Figure 1.Notations of the mesh  associated with Λ.

Figure 2 .
Figure 2. Notations on a diamond cell   for  ∈ ℰ int .
∈ ∈ R  ℎ , by summing over the edges we may rearrange the sum on the left-hand side and get the following rule of "discrete partial integration" Proposition 2.10 (Existence of a discrete solution).Assume that hypotheses  1 and  2 hold.Let  be an admissible finite-volume mesh of Λ in the sense of Definition 2.1 with a mesh size ℎ and  ∈ N * .Then, there exists a unique solution (  ℎ ) 1≤≤ ∈ (R  ℎ )  to Problem (2.6) associated with the initial vector  0 ℎ defined by (2.5).Additionally, for any  ∈ {0, . . .,  },   ℎ is a ℱ  -measurable random vector.  )(  −   ).