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Uniform in time error estimates for a finite element method applied to a downscaling data assimilation algorithm for the Navier-Stokes equations

In this paper we analyze a finite element method applied to a continuous downscaling data assimilation algorithm for the numerical approximation of the two and three dimensional Navier-Stokes equations corresponding to given measurements on a coarse spatial scale. For representing the coarse mesh measurements we consider different types of interpolation operators including a Lagrange interpolant. We obtain uniform-in-time estimates for the error between a finite element approximation and the reference solution corresponding to the coarse mesh measurements. We consider both the case of a plain Galerkin method and a Galerkin method with grad-div stabilization. For the stabilized method we prove error bounds in which the constants do not depend on inverse powers of the viscosity. Some numerical experiments illustrate the theoretical results.


Introduction
Data assimilation refers to a class of techniques that combine experimental data and simulation in order to obtain better predictions in a physical system. There is a vast literature on data assimilation methods, specially in the recent years (see e.g., [4], [12], [29], [31], [34], and the references therein). One of these techniques is nudging, where a penalty term is added in order to drive the approximate solution towards coarse mesh or large scale spatial observations of the data. In a recent work [6], a new approach, known as continuous data assimilation, is introduced for a large class of dissipative partial differential equations, including Rayleigh-Bénard convection [14], the planetary geostrophic ocean dynamics model [15], etc. (see also references therein). Continuous data assimilation has also been used in numerical studies, for example, with the Chafee-Infante reaction-diffusion equation the Kuramoto-Sivashinsky equation (in the context of feedback control) [32], Rayleigh-Bénard convection equations [3], [13], and the Navier-Stokes equations [22], [25]. However, there is much less numerical analysis of this technique. The present work concerns with the numerical analysis of continuous data assimilation for the Navier-Stokes equations when discretized with mixed finite element methods (MFE).
To be more precise, we consider the Navier-Stokes equations ∂tu − ν∆u + (u · ∇)u + ∇p = f in (0, T ] × Ω, in a bounded domain Ω ⊂ R d , d ∈ {2, 3}. In (1), u is the velocity field, p the kinematic pressure, ν > 0 the kinematic viscosity coefficient, and f represents the accelerations due to external body forces acting on the fluid. The Navier-Stokes equations (1) must be complemented with boundary conditions. For simplicity, homogeneous Dirichlet boundary conditions u = 0 on ∂Ω will be considered in the present paper.
Following [33] we consider given measurements, corresponding to a solution u of (1), observed at a coarse spatial mesh. The measurements are assumed to be continuous in time and error-free. We denote by IH(u) the operator used for interpolating these measurements, where H denotes the resolution of the coarse spatial mesh of the observed measurements. Since the initial condition for u is missing one cannot compute u by simulating equation (1) directly. To overcome this difficulty it was suggested in [6] to consider instead a solution v of the following approximating system where β is the relaxation (nudging) parameter.
In the case of the Navier-Stokes equations (and indeed, of many other nonlinear systems), it is well-known that for relatively not so small Reynolds numbers, solutions are unstable and even chaotic. For this reason, it is expected that any small error in the initial data could lead to exponentially growing error in the solutions. Notably, the instabilities in the NSE occur at the large spatial scales, while the fine scales are stabilized by the viscosity. For this reason once the large spatial scales are stabilized, as it is done in the proposed downscaling data assimilation approximation, equation (2), the corresponding solution are stable and converge to the same solution u that is corresponding to IH(u). This is the very reason that small errors are not magnified in time and allows to obtain uniform in time error bounds.
In this paper we consider a semidiscrete method with inf-sup stable mixed finite elements for equation (2). In particular, we establish uniform-in-time error estimates for approximating the unknown reference solution, u, that corresponds to the coarse-mesh measurement IH(u). The spatial error bounds are optimal, in the sense that the rate of convergence is that of the best interpolant.
We now comment on the analysis of numerical methods for (2). In [33], a semidiscrete postprocessed Galerkin spectral method for the two-dimensional Navier-Stokes equations is studied. Under suitable conditions on the nudging parameter β, the coarse mesh resolution H and the number of modes in the spectral method, uniform-in-time error estimates are obtained for the difference between the numerical approximation to v and u. Furthermore, the use of a postprocessing technique introduced in [20] [21], allows for higher convergence rates than a standard spectral Galerkin method. A fully-discrete method for the spatial discretization in [33] is analyzed in [28], where the backward Euler method is used for time discretization. Fully implicity and semi-implicit methods are considered, and optimal uniform-in-time error estimates are obtained with the same convergence rate in space as in [33]. It must be noted that in [33] and [28], as opposed to the present paper and [30], they deal only with the two-dimensional case. This, and a more elaborate analysis allows the authos of [33] and [28] to obtain error bounds with smaller constants than in the present paper or in [30]. On the other hand, if constants similar to K(u, p) in Theorem 3.1 below or to those in Theorem 3.8 in [30] are allowed to feature in the error bounds, the analysis in the present paper is easy to translate to the spectral methods in [33] for the three-dimensional case as well.
More closely related to the present work is [30], where the authors consider fully discrete approximations to equation (2) where the spatial discretization is performed with the same MFE as in the present paper, although a grad-div stabilization term is added in [30]. First and second order in time schemes are analyzed in [30], and, as in [33], [28] and the present paper, uniform-in-time error bounds are obtained. However, compared with the results in the present paper, error bounds in [30] are suboptimal in space, since one order of convergence in terms of the spatial mesh h is lost with respect to the error bounds we prove here. Also, the analysis we present here is much simpler than in [30] as it is the case of the method (plain Galerkin method here instead of Galerkin method plus grad-div stabilization in [30]). We want to remark that the analysis we present here for the semidiscrete method is easily extended to fully discrete methods following techniques in [17], [18], while keeping the optimal order in space we have obtained. Furthermore, combining the analysis in the present paper with the techniques in [17], [18], it may be possible to obtain the same suboptimal (in space) error bounds in [30] but with a significant difference, which is that, as in [17], [18], error constants do not depend on inverse powers of the viscosity parameter ν. This difference is of importance in may applications where viscosity is orders of magnitude smaller than the velocity. However, obtaining such error bounds for equation (2) is outside the scope of the present paper and will be subject of future work.
The rest of the paper is as follows. Section 2 is devoted to preliminary material, and in section 3 we introduce and analyze the finite element method for equation (2)

Preliminaries and Notation
Throughout the paper, W s,p (D) will denote the Sobolev space of real-valued functions defined on the domain D ⊂ R d with distributional derivatives of order up to s in L p (D). We denote by | · |s,p,D standard seminorm, and, following [11], for W s,p (D) we will use the norm · s,p,D defined by where |D| stands for the Lebesgue meassure of D so that f m,p,D |D| m d − 1 p is scale invariant.If s is not a positive integer, W s,p (D) is defined by interpolation [1]. In the case s = 0 one has W 0,p (D) = L p (D). As it is standard, W s,p (D) d will be endowed with the product norm and, since no confusion can arise, it will be denoted again by · W s,p (D) . The case p = 2 will be distinguished by using H s (D) to denote the space W s,2 (D). The space H 1 0 (D) is the closure in H 1 (D) of the set of infinitely differentiable functions with compact support in D. For simplicity, · s (resp. | · |s) is used to denote the norm (resp. semi norm) both in H s (Ω) or H s (Ω) d . The exact meaning will be clear by the context. The inner product of L 2 (Ω) or L 2 (Ω) d will be denoted by (·, ·) and the corresponding norm by · 0 in general D is skipped in the notation for the norm when D = Ω. For vector-valued functions, the same conventions will be used as before. The norm of the dual space H −1 (Ω) of H 1 0 (Ω) is denoted by · −1. As usual, L 2 (Ω) is always identified with its dual, so one has The following Sobolev's embedding [1] will be used in the analysis: For s > 0, let 1 ≤ p < d/s and q be such that 1 If p > d/s the above relation is valid for q ′ = ∞. A similar embedding inequality holds for vector-valued functions. We will also use the following interpolation inequality which is a consequence of Sobolev's inequality and the convexity inequality (see, e.g., [19, § II.1]), and Agmon's inequality The case d = 2 is a direct consequence of [2, Theorem 3.9]. For d = 3, a proof for domains of class C 2 can be found in [11,Lemma 4.10]. By means of the Calderón extension theorem (see e.g., [1,Theorem 4.32] the proof is also valid for bounded Lipschitz domains. In all previous inequalities, the constants cs, c1 and cA are scale-invariant, as it will be the case of all constants in the present paper unless explicitly stated to the contrary. Let H and V be the Hilbert spaces H = {u ∈ L 2 (Ω)) d , | div(u) = 0, u · n | ∂Ω = 0}, V = {u ∈ H 1 0 (Ω)) d , | div(u) = 0}, endowed with the inner product of L 2 (Ω) d and H 1 0 (Ω) d , respectively. Let where h is the maximum diameter of the elements τ h i ∈ T h , and φ h i are the mappings from the reference simplex τ0 onto τ h i . We shall assume that the partitions are shape-regular and quasi-uniform. Let r ≥ 2, we consider the finite element spaces where P r−1 (τ0) denotes the space of polynomials of degree at most r − 1 on τ0.
When Ω has polygonal or polyhedral boundary Ω h = Ω and mappings φ h i form the reference simplex are affine. When Ω has a smooth boundary, for the purpose of analysis we will assume that Ω h exactly matches Ω, as it is done for example in [10], [35], although at a price of a more complex analysis discrepancies between Ω h and Ω can also be taken into account (see, e.g., [5], [36]).
We shall denote by (X h,r , Q h,r−1 ) the MFE pair known as Hood-Taylor elements [8,38], when r ≥ 3, where and, when r = 2, the MFE pair known as the mini-element [9] where Q h,1 = S h,2 ∩L 2 (Ω h )/R, and X h,2 = (S 0 h,2 ) d ⊕ B h . Here, B h is spanned by the bubble functions bτ , τ ∈ T h , defined by bτ (x) = (d + 1) d+1 λ1(x) · · · λ d+1 (x), if x ∈ τ and 0 elsewhere, where λ1(x), . . . , λ d+1 (x) denote the barycentric coordinates of x. For these elements a uniform inf-sup condition is satisfied (see [8]), that is, there exists a constant βis > 0 independent of the mesh grid size h such that The velocity vector field will be approximated by elements of the discrete divergence-free space For each fixed time t ∈ [0, T ] the solution (u, p) of (1) is also the solution of a Stokes problem with right-hand side f − ut − (u · ∇)u. We will denote by (s h , q h ) ∈ (X h,r , Q h,r−1 ), its MFE approximation satisfying (∇ · s h , ψ h ) = 0 ∀ψ h ∈ Q h,r−1 .
We observe that s h = S h (u) : V → V h,r is the discrete Stokes projection of the solution (u, p) of (1) (see [26]) and satisfies The following bound holds: where here and in the sequel we denote The proof of (8) for Ω = Ω h can be found in [27]. Under the same conditions, the bound for the pressure is (cf. [24]) where the constant C β is depends on the constant βis in the inf-sup condition (6). Assuming that Ω is of class C m , with m ≥ 3, and using standard duality arguments and (8), one obtains u − s h −s ≤ CNr(u, p)h r+s , 0 ≤ s ≤ min(r − 2, 1). (11) Finally, for the interpolation operator IH we will assume that it satisfies the following approximation identity property We will also assume the L 2 -stability of the interpolation operator: The Bernardi-Girault [7], Girault-Lions [23], or the Scott-Zhang [37] interpolation operators satisfy the above properties in (12) and (13). We would like to stress here that we are not assuming for the error analysis condition (3.105) in [33], i.e., we do not assume that

The finite element method
We consider the following method to approximate (2).
where b h (·, ·, ·) is defined in the following way Hereafter, we denote by (·, ·) both the inner product in L 2 and the duality action between H −1 and H 1 0 , depending on the context. It is straightforward to verify that b h enjoys the skew-symmetry property Let us observe that taking ϕ h ∈ V h,r from (14) we get We have the follwing result.
where K(u, p) is the constant in (29) below. If condition (13) does not hold, then the factor h r in the estimate above should be replaced by Hh r−1 .
Proof: Following [5] we compare u h with s h , where s h satisfies (7). Let us denote by e h = u h − s h . Then, we get the error equation where the truncation error satisfies Taking ϕ h = e h in (17) we get We will bound the terms on the right-hand side of (18). For the nonlinear term, using the skew-symmetry property (15) and (3), and assuming d = 3, we get where c1 is the constant in (3) for s = 1. In the case d = 2, the first term on the right-hand side above, using (4) and Young's inequality is bounded as follows, For the next term on the right-hand side of (18), using (12), we get To bound the next term on the right-hand side of (18) applying (13) we get We remark that in case property (13) does not hold, adding and subtracting u − s h and using (12) we get and then one can continue arguing as we do above. In this case the rate of convergence of the method would be O(Hh s−1 ) instead of O(h s ). Finally, for the last term on the right-hand side of (18) we get Denoting by and inserting (19), (20), (21) and (22) into (18) we get Now, we will impose conditions so that β is large enough that β/2 − L1 − L2 ≥ β/4. To this end we will impose L1 ≤ β/8 and L2 ≤ β/8. The first condition is (See Remak 3.4 below for an equivalent condition in terms of norms of u and p). While the second condition is Let us observe that condition (26) is analogous to condition (3.110) in [33].
Assuming from now on that conditions (25) and (26)  Therefore, From (27) we get Consequently, To bound the last term on the right-hand side of (28) we apply (8) Finally, for the truncation error, applying (11) we can bound or, in case we use the mini-element or the boundary is not of class C 3 , applying (8) so that we conclude the proof by applying again (8).
Then, the rate of convergence of the method is optimal e h 0 = O(h s ) and, as in [33] we have obtained uniform in time error estimates.

Lemma 3.2 The following bound holds
and Nj (u, p) is the quantity in (9).

Conclusions
We have analyzed a semidiscretization, in space by inf-sup stable mixed finite elements, of a continuous downscaling data assimilation method for the two-and three-dimensional Navier-Stokes equations. The data assimilation method, introduced in [6], combines observational data (measurements), on large spatial scales or coarse mesh, with simulations in order to improve predictions of the physical phenomenon being studied. Although this data assimilation method has already been used in a number of computational studies, fewer work has been devoted to its numerical analysis. In the present paper, uniform error bounds in time have been obtained for the approximation to velocity field, under standard assumptions in finite element analysis. The order of convergence proved is optimal, in the sense that it is the best that can be obtained with the finite element space being used (i.e., errors of the same order as interpolation). The techniques of analysis used in the present paper allow to improve the only available error bounds for a closely-related finite element method in [30].