Error estimates of finite element methods for fractional stochastic Navier–Stokes equations

Based on the Itô’s isometry and the properties of the solution operator defined by the Mittag-Leffler function, this paper gives a detailed numerical analysis of the finite element method for fractional stochastic Navier–Stokes equations driven by white noise. The discretization in space is derived by the finite element method and the time discretization is obtained by the backward Euler scheme. The noise is approximated by using the generalized \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L_{2}$\end{document}L2-projection operator. Optimal strong convergence error estimates in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L_{2}$\end{document}L2-norm are obtained.

Fractional Navier-Stokes equations (FNSEs) are widely regarded as some of the most fascinating problems in fluid mechanics, in particular, they could even lead to a better understanding of the physical phenomena and mechanisms of turbulence in fluids [25]. Furthermore, the presence of noises could give rise to some statistical features and important phenomena, for example, a unique invariant measure and ergodic behavior driven by degenerate noise have been established. At the same time, the stochastic perturbations cannot be avoided in a physical system, sometimes they even cannot be ignored. Hence fractional stochastic Navier-Stokes equations have been proposed, which display the behavior of a viscous velocity field of an incompressible liquid and have wide application value in the fields of physics, chemistry, population dynamics, and so on [26][27][28].
This article is devoted to the study of the error estimates of the finite element method for the incompressible fractional stochastic Navier-Stokes equations where ⊂ R 2 is a bounded and connected polygonal domain, u represents the velocity field, p is the associated pressure, u 0 is the initial velocity and the right-hand side terṁ W denotes the white noise, Lu =u; B α := R D 1-α t is the Riemann-Liouville fractional derivative in time defined for 0 < α < 1 by where I α is the temporal Riemann-Liouville fractional integral operator of order α.
The above-mentioned problem has many physical applications in various areas. Particularly, when α = 1, problem (1.1) reduces to the classical stochastic Navier-Stokes equations, numerical approximations of which have been carried out by the authors [29,30]. For the fractional stochastic Navier-Stokes equations, the well-posedness has been studied in [26,27]. So far, for most fractional stochastic differential equations, it is very difficult to get exact solutions, so it is necessary to propose numerical methods. However, to the best of our knowledge, numerical analysis of such a problem for fractional stochastic Navier-Stokes equations is missing in the literature. Therefore, this article aims to fill the gap, by studying and obtaining the strong convergence approximations of fractional stochastic Navier-Stokes equations like (1.1).
In this article, our goal is to give some detailed numerical analysis of the finite element method for problem (1.1). Because the mild solution of fractional stochastic Navier-Stokes equations is provided by the solution operator E(t) defined through the Mittag-Leffler function, it is different from the classic stochastic Navier-Stokes equations related to the analytic semigroup e t . The properties of the semigroup and the semigroup theory have been studied in detail in [31,32]. However, for the solution operator E(t), as far as we are know, similar properties are less studied. The novelty of this paper is to derive the properties of the solution operator E(t) which is defined through the Mittag-Leffler function and establish the Hölder regularity of the weak solutions for fractional stochastic Navier-Stokes equations. Firstly, we deduce some regularity results and stability properties of E(t) which play a key role in the error analysis. The discretization in space is derived by the finite element method and the time discretization is obtained by the backward Euler scheme. Based on the error estimates for the corresponding deterministic problem and Itô isometry, finally the strong convergence error estimates for the fully discrete schemes of fractional stochastic Navier-Stokes equations are obtained.
The structure of this paper is as follows: In Sect. 2, we introduce some preliminaries and notations, as well as give the definition of the Mittag-Leffler function. In Sect. 3, we give the semidiscrete Galerkin approximations in space and then obtain the fully discrete scheme. In Sect. 4, we present several lemmas about the operator E(t) which play a crucial role in the proof of the error estimate. Finally, in Sect. 5, we will give the fully discrete error estimates for the fractional stochastic Navier-Stokes equations.

Preliminaries
Throughout the paper, we denote by C a constant that may not be of the same form from one occurrence to another, even in the same line. In this section, we introduce some notations and some important preliminaries.
Let · U and · H be the norms of separable Hilbert spaces U and H, respectively. Let L(U, H) denote the space of bounded linear operators from U to H, and let L 2 (U, H) be the space of Hilbert-Schmidt operators with norm The norms in the Sobolev spaces H s ( ), s ≥ 0, are denoted by · s . And we assume that a family {V h } of finite-dimensional subspaces of H 1 0 is such that, for some integer r ≥ 2 and small h (cf. [31]), where H s denotes the Sobolev space of order s. Let ( , F, P) be a probability space and let E be the expectation. For any Hilbert space, we define is a U-valued Gaussian random variable with zero mean and covariance operator (ts)Q, It is known (see, e.g., Sect. 2.1 in [33]) that for a given Q-Wiener process satisfying (i)-(iv) one can always find a normal filtration {F t } t≥0 so that (v)-(vi) hold. Suppose that are the eigenpairs of Q with orthonormal eigenvectors and {β j (t)} ∞ j=1 are realvalued mutually independent standard Brownian motions. Then W (t) has the orthogonal expansion It is then possible to define the stochastic integral t 0 ψ(s) dW (s) together with Itô's isometry, The operator P h : L 2 ( ) → V h denotes the projection operator defined by For the reader's convenience, the definition of Mittag-Leffler function will be provided. We shall use the extended Mittag-Leffler function E α,β (z) [25] defined by where (·) is the standard Gamma function defined as

Discretization of fractional stochastic problem
Let be the divergence-free projection operator of the Helmholtz decomposition (cf., [34,35]). In order to consider a velocity u satisfying P-a.s. (almost surely) ∇ · u = 0, we project the fractional stochastic Navier-Stokes equation onto the space of divergence-free vector fields, thereby removing the pressure p(x, t). Then, applying the Helmholtz projection on both sides of Eq. (1.1), we obtain where A = -, B(u, u) := ((u · ∇)u). The bilinear operator B(·, ·) satisfies the following inequality (cf., [36,37]): which has important applications when establishing strong convergence error estimates for the fully discrete schemes of fractional stochastic Navier-Stokes equations. We shall assume that Also we assume that the operator A is self-adjoint and there exist eigenvectors ϕ j corresponding to eigenvalues λ j such that (cf., [28,29]) In a standard way, the fractional powers A s , s ∈ R, of A are introduced by Now we introduce the operator E(t) by where α ∈ (0, 1) denotes the Caputo fractional derivative of order α and E α,1 is the Mittag-Leffler function. By making use of time fractional Duhamel's priciple [38][39][40], the solution u(t) of (3.1) at time t = t n can be written as  Then the semidiscrete problem corresponding to (3.1) is to find the process u h (t) ∈ V h such that The operator E h (t) is introduced by where {λ h j } N j=1 and {ϕ h j } N j=1 are respectively the eigenvalues and eigenfunctions of the discrete Laplace operator A h . Then the semidiscrete problem (3.6) has the abstract integral equation given by For a fixed time step size t > 0, we put t n = n t and define a piecewise-constant approximation U n h ≈ u(t n ) by applying the DG method [41][42][43], namely where U n h = U h (tn ) = lim t→tn U h (t) denotes the one-sided limit from below at the nth time level. Thus, U h (t) = U n h for t n-1 < t ≤ t n . A short calculation shows that and, for j ≥ 1, Then, the fully discrete mild formulation for (3.1) can be obtained as where the detailed definition of B n,h can be found in [44].

Some important lemmas for operator E(t)
In order to give the error estimates for the stochastic fractional problem, we will derive some lemmas for operator E(t).
The following lemma presents the stability and smoothing estimate for operator E(t), which play a key role in the error analysis of FEM approximations.
Next, several important properties of the Mittag-Leffler function E α,β (·) will be given.
In particular, if m = 1, we obtain The following estimates are crucial for the error analysis in the sequel.

Lemma 4.3
Let Then, for all t > 0, we have Besides, we get Proof For the proof of (4.2), we refer to [3] and omit it here. Subsequently, we will give the detailed proof of equality (4.3). By virtue of Lemma 4.2, we have which completes the proof.
Next we will derive the properties of operator E(t) which will be used throughout this paper.

Lemma 4.4 Let
Proof Firstly, we prove (i). By Lemma 4.1, with = q = 0, p = 2μ, one has which gives The proof of (i) is completed.
For (ii), by making use of (4.3), we obtain the second to last inequality of which is derived from (4.2) with p = 2 -2μ, q = 0 in Lemma 4.3. This completes the proof of the lemma.
In the following, the regularity of the mild solution in time will be given.

Theorem 4.1 (Temporal regularity)
Let u be the solution of (3.1). Then for t 1 Proof Let 0 ≤ t 1 < t 2 ≤ T be arbitrary. By making use of the mild solution formulation (3.5), it can be obtained that In the sequel, each term will be estimated separately. For the first term L 1 , by virtue of Lemmas 4.3 and 4.4, one has The second term L 2 can be split into two terms:  Similarly, the third term L 3 can be written as By making use of Itô's isometry and Lemma 4.4, it can be deduced that The term L 32 is estimated analogously by using Lemma 4.4, namely Combining (4.4)-(4.8) yields the result.

Error estimates for the stochastic fractional N-S equations
In this section, we will give the fully discrete error estimates for the stochastic fractional Navier-Stokes equations. Let e n = U n hu(t n ). Then, by (3.9) and (3.5), it can be obtained that  Next, each term will be estimated in turn. In order to prove the main error estimates, we need the following useful conclusions for the corresponding deterministic problem, see [44] for more details.
The following lemma is the time discrete version with smooth initial data.
For the term II 2 , by making use of Lemma 4.4 and property (3.2) of B(·, ·), one can arrive at The estimate for II 3 is a straightforward application of Lemma 5.1 and property (3.2) of B(·, ·) yielding  Hence, by (5.6) and (5.7), the proof is completed.
Based on the above conclusions, the main theorem of the paper can now be obtained.