Error analysis for 2D stochastic Navier–Stokes equations in bounded domains

09/14/2021 ∙ by Dominic Breit, et al. ∙ Universität Tübingen Heriot-Watt University 0

We study a finite-element based space-time discretisation for the 2D stochastic Navier-Stokes equations in a bounded domain supplemented with no-slip boundary conditions. We prove optimal convergence rates in the energy norm with respect to convergence in probability, that is convergence of order (almost) 1/2 in time and 1 in space. This was previously only known in the space-periodic case, where higher order energy estimates for any given (deterministic) time are available. In contrast to this, in the Dirichlet-case estimates are only known for a (possibly large) stopping time. We overcome this problem by introducing an approach based on discrete stopping times. This replaces the localised estimates (with respect to the sample space) from earlier contributions.

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1. Introduction

We are concerned with the numerical approximation of the 2D stochastic Navier–Stokes equations in a smooth bounded domain . It describes the flow of a homogeneous incompressible fluid in terms of the velocity field and pressure function defined on a filtered probability space and reads as

(1.1)

-a.s. in , where , is the viscosity and is a given initial datum. The momentum equation is driven by a cylindrical Wiener process and the diffusion coefficient takes values in the space of Hilbert-Schmidt operators, see Section 2.1 for details.

Existence, regularity and long-time behaviour of solutions to (1.1) have been studied extensively over the three decades and we refer to [16] for a complete picture. Most of the available results consider (1.1) with respect to periodic boundary conditions. In some cases this is only for a simplification of the presentation. For instance, the existence of of stochastically strong solutions to (1.1) is not effected by the boundary condition. Looking at the spatial regularity of solutions the situation is completely different:

  • If — the two-dimensional torus — and (1.1) is supplemented with periodic boundary conditions one can obtain estimates in any Sobolev space provided the data (initial datum and diffusion coefficient) are sufficiently regular; cf. [16, Corollary 2.4.13].

  • If, on the other hand, is a bounded domain with smooth boundary and (1.1) is supplemented with the no-slip boundary condition

    (1.2)

    it is still an open problem if the solution satisfies

    (1.3)

    for any given , cf. [13, 15].

Moment estimates such as (1.3) are crucial for the numerical analysis. If they are not at disposal it is unclear how to obtain convergence rates for a discretisation of (1.1). Consequently, most if not all results are concerned with the space-periodic problem. In particular, it is shown in [5] and [7] for the space-periodic problem that for any

(1.4)

as (where and are arbitrary); see also [1, 2] for related results. Here is the solution to (1.1) and the approximation of with discretisation parameters and . The relation (1.4) tells us that the convergence with respect to convergence in probability is of order (almost) 1/2 in time and 1 in space. It seems to be an intrinsic feature of SPDEs with general non-Lipschitz nonlinearities such as (1.1) that the more common concept of a pathwise error (an error measured in ) is too strong (see [18] for first contributions). Hence (1.4) is the best result we can hope for. The proof of (1.4) is based on estimates in , which are localised with respect to the sample set. The size of the neglected sets shrinks asymptotically with respect to the discretisation parameters and is consequently not seen in (1.4). The localised -estimates in question rely on an iterative argument in the -th step of which one can only control the discrete solution up the step (to avoid problems with -adaptedness), while the continuous solution is estimated by means of the global regularity estimates being available in the periodic setting (recall the discussion above).
In contrast to the periodic situation, in the Dirichlet-case estimates are only known for a (possibly large) stopping time since the equality is no longer available. Incorporating the latter case into the framework of the localised estimates, the iterative argument just mentioned fails: controlling the continuous solution in the -th step only until the time is insufficient for the estimates, while “looking into” the interval in this set-up destroys the martingale character of certain stochastic integrals we have to estimate. We overcome this problem by using an approach based on discrete stopping times, which replaces the localised -estimates from earlier contributions. This allows to control all quantities even in the interval and, at the same time, preserves the martingale property of the stochastic integrals (see also the discussion in Remark 4). As a result we obtain ‘global-in-’ estimates up to the discrete stopping time, cf. Theorem 4. The discrete stopping times are constructed such that they converge to , where can be any given end-time. Consequently, the convergence in probability as in (1.4) follows for the Dirichlet-case, see our main result in Theorem 4. We believe that this strategy will be of use also for other SPDEs with non-Lipschitz nonlinearities.
It is worth to point out that our analysis does not require any additional structural conditions for the noise (we only need the diffusion coefficient to be regular enough). In particular, we do not need to assume that the diffusion coefficient takes values in the space of solenoidal functions. Proving the optimal convergence rate in this case (in the space-periodic setting) is the main improvement of [5] compared to [7]. It is based on a stochastic pressure decomposition and a careful analysis of the stochastic component of the pressure. We extend this idea to the case of bounded domains, which requires some additional care. As a matter of fact, we only obtain the correct decomposition if we have an analytically strong solution at hand, while weak solutions are sufficient for the periodic problem.

2. Mathematical framework

2.1. Probability setup

Let be a stochastic basis with a complete, right-continuous filtration. The process is a cylindrical -valued Wiener process, that is, with being mutually independent real-valued standard Wiener processes relative to , and a complete orthonormal system in a separable Hilbert space . Let us now give the precise definition of the diffusion coefficient taking values in the set of Hilbert-Schmidt operators , where can take the role of various Hilbert spaces. We assume that for , and for , together with

(2.1)
(2.2)
(2.3)

If we are interested in higher regularity some further assumptions are in place and we require additionally that for , together with

(2.4)
(2.5)

Assumption (2.1) allows us to define stochastic integrals. Given an -adapted process , the stochastic integral

is a well-defined process taking values in (see [10] for a detailed construction). Moreover, we can multiply by test functions to obtain

Similarly, we can define stochastic integrals with values in and respectively if belongs to the corresponding class.

2.2. The concept of solutions

In dimension two, pathwise uniqueness for analytically weak solutions is known under the assumption (2.1); we refer the reader for instance to Capiński–Cutland [9], Capiński [8]. Consequently, we may work with the definition of a weak pathwise solution.

Let be a given stochastic basis with a complete right-continuous filtration and an -cylindrical Wiener process . Let be an

-measurable random variable. Then

is called a weak pathwise solution to (1.1) with the initial condition provided

  1. the velocity field is -adapted and

  2. the momentum equation

    holds -a.s. for all and all .

Suppose that satisfies (2.2) and (2.3). Let be a given stochastic basis with a complete right-continuous filtration and an -cylindrical Wiener process . Let be an -measurable random variable such that for some . Then there exists a unique weak pathwise solution to (1.1) in the sense of Definition 2.2 with the initial condition .

We give the definition of a strong pathwise solution to (1.1) which exists up to a stopping time . The velocity field belongs -a.s. to .

Let be a given stochastic basis with a complete right-continuous filtration and an -cylindrical Wiener process . Let be an -measurable random variable with values in . The tuple is called a local strong pathwise solution to (1.1) with the initial condition provided

  1. is a -a.s. strictly positive -stopping time;

  2. the velocity field is -adapted and

  3. the momentum equation

    (2.6)

    holds -a.s. for all and all .

Note that (2.6) certainly implies the corresponding formulation in Definition 2.2. The reverse implication is only true for analytically strong solutions.
We finally define a maximal strong pathwise solution. [Maximal strong pathwise solution] Fix a stochastic basis with a cylindrical Wiener process and an initial condition as in Definition 2.2. A triplet

is a maximal strong pathwise solution to system (1.1) provided

  1. is a -a.s. strictly positive -stopping time;

  2. is an increasing sequence of -stopping times such that on the set , -a.s., and

    (2.7)

    with the convention that if the set above is empty;

  3. each triplet , , is a local strong pathwise solution in the sense of Definition 2.2.

We talk about a global solution if we have (in the framework of Definition 2.2) -a.s. The following result is shown in [17] (see also [13] for a similar statement). Suppose that (2.1)–(2.3) holds, and that . Then there is a unique maximal global strong pathwise solution to (1.1) in the sense of Definition 2.2.

2.3. Finite elements

We work with a standard finite element set-up for incompressible fluid mechanics, see e.g. [12]. We denote by a quasi-uniform subdivision of into triangles of maximal diameter . For and we denote by the polynomials on of degree less than or equal to . Let us characterize the finite element spaces and as

We will assume that to get (2.9) below. In order to guarantee stability of our approximations we relate and by the discrete inf-sup condition, that is we assume that

where does not depend on . This gives a relation between and (for instance the choice is excluded whereas is allowed). Finally, we define the space of discretely solenoidal finite element functions by

Let be the -orthogonal projection onto . The following results concerning the approximability of are well-known (see, for instance [14]). There is independent of such that we have

(2.8)

for all , and

(2.9)

for all . Similarly, if denotes the -orthogonal projection onto , we have

(2.10)

for all , and

(2.11)

for all . Note that (2.11) requires the assumption in the definition of , whereas (2.10) also holds for .

3. Regularity of solutions

3.1. Estimates for the continuous solution

In this section we derive the crucial estimates for the continuous solution, which hold up to the stopping time .

  1. Assume that for some and that satisfies (2.1). Then we have

    (3.1)

    where is the weak pathwise solution to (1.1), cf. Definition 2.2.

  2. Assume that for some and that satisfies (2.1)–(2.3). Then we have

    (3.2)

    where is the maximal strong pathwise solution to (1.1), cf. Definition 2.2.

  3. Assume that for some and that (2.1)–(2.5) holds. Then we have

    (3.3)

    where is the maximal strong pathwise solution to (1.1), cf. Definition 2.2.

Here is independent of .

Proof.

Part (a) is the standard a priori estimate, which is a consequence of applying Itô’s formula to .

For part (b) we follow [17], where the solution to a truncated problem is considered. For and with and in we set . Similar to Definition 2.2 we seek an -adapted stochastic process with

such that

(3.4)

which holds -a.s. for all and all . It is shown in [17, Lemma 3.7] that a unique strong solution to (3.6) exists in the class and that it satisfies

(3.5)

The proof is based on a Galerkin approximation which we mimick now in order to prove (3.2) and (3.3).

1) Galerkin approximation. Let

be a system of eigenfunctions to the Stokes operator

in

with eigenvalues

such that as . It is possible to choose the ’s such that the system is orthonormal in and orthogonal in . Moreover, we can assume that the ’s belong to . All of this will be needed to justify the following manipulations. For let and consider the unique solution to

(3.6)

for all . Problem (3.6) can be written as a system of SDEs with Lipschitz-continuous coefficients. Hence it is clear there is a unique solution, i.e., an -adapted process with values in . It is shown in [17, Prop. 3.2] that as

(3.7)

in probability. Applying Itô’s formula to and using the cancellation of the convective term one can prove for

(3.8)

where is independent of and .

2) Proof of (3.2). By construction we have -a.s. such that we can apply Itô’s formula to and use (3.6). This yields

where denotes the -orthogonal projection onto . We estimate now the terms , and . First of all, we have by definition of

where is arbitrary. Moreover, we obtain since

the expectation of which is bounded by (3.8). Finally, by Burkholder-Davis-Gundy inequality,

using (3.8). Choosing small enough we conclude that

(3.9)

uniformly in . By (3.7) we can pass to the limit and obtain a corresponding estimate for . Since we obtain (3.2).

3) Proof of (3.3). As far as (c) is concerned we argue similarly and apply Itô’s formula to the mapping which shows

It holds