Error estimates for finite element discretizations of the instationary Navier-Stokes equations

In this work we consider the two dimensional instationary Navier-Stokes equations with homogeneous Dirichlet/no-slip boundary conditions. We show error estimates for the fully discrete problem, where a discontinuous Galerkin method in time and inf-sup stable finite elements in space are used. Recently, best approximation type error estimates for the Stokes problem in the $L^\infty(I;L^2(\Omega))$, $L^2(I;H^1(\Omega))$ and $L^2(I;L^2(\Omega))$ norms have been shown. The main result of the present work extends the error estimate in the $L^\infty(I;L^2(\Omega))$ norm to the Navier-Stokes equations, by pursuing an error splitting approach and an appropriate duality argument. In order to discuss the stability of solutions to the discrete primal and dual equations, a specially tailored discrete Gronwall lemma is presented. The techniques developed towards showing the $L^\infty(I;L^2(\Omega))$ error estimate, also allow us to show best approximation type error estimates in the $L^2(I;H^1(\Omega))$ and $L^2(I;L^2(\Omega))$ norms, which complement this work.


INTRODUCTION
In this paper, we consider the instationary Navier-Stokes equations in two space dimensions with homogeneous boundary conditions, i.e.,          Here ν > 0 denotes the viscosity, I = (0, T ] ⊂ R a bounded, half-open interval for some fixed finite endtime T > 0, and Ω ⊂ R 2 a bounded convex polygonal domain.The equations are discretized in time by a a discontinous Galerkin (dG) method, i.e., the solution is approximated by piecewise polynomials in time, defined on subintervals of I, without any requirement of continuity at the time nodes, see, e.g., [24,47].The parameter indicating the time discretization will be denoted by k and corresponds to the length of the largest subinterval in the partition of I. To discretize in space, we use inf-sup stable pairs of finite element spaces for the velocity and pressure components.The parameter indicating the spatial discretization will be denoted by h and corresponds to the largest diameter of cells in the mesh.Due to the variational formulation of the dG time discretization, this discretization scheme is particularly suited for treating optimal control problems.See, e.g., [13,14] for optimal control problems governed by the Navier-Stokes equations, also [35,40] for optimal control of general parabolic problems.While in [13,14] the focus was put on low order schemes, recently the authors of [2] analyzed dG schemes of arbitrary order for the Navier-Stokes equations.Another advantage of the dG time discretization is the fact, that the maximal parabolic regularity, exhibited by parabolic problems, is preserved on the discrete level, and moreover can be extended to the limiting cases L 1 and L ∞ in time at the expense of a logarithmic factor, see [34,Theorems 11 & 12].The natural energy norm for the Navier-Stokes equations is the norm of the space L ∞ (I; L 2 (Ω) 2 ) ∩ L 2 (I; H 1 (Ω) 2 ).Indeed by formally testing (1) with the solution u, one obtains This bound is also preserved on the discrete level, i.e., holds for the fully discrete solution u kh , see Theorem 4.10.Our main goal in writing this paper, was the investigation of the discretization error in terms of the L ∞ (I; L 2 (Ω) 2 )-norm.In particular, such estimates are required for the analysis of optimal control problems, subject to state constraints pointwise in time, see [40], as the corresponding Lagrange multiplier in this case is a measure in time.There have been numerous approaches in the literature, deriving error estimates for the Navier-Stokes equations.In [13,Theorem 4.7], the same combination of dG-cG discretization schemes was used, and for f ∈ L 2 (I; L 2 (Ω) 2 ) an estimate was shown.Additional terms arise, when changes in the spatial mesh on different time intervals are permitted.Under the much stricter assumption f ∈ W 1,∞ (I; L 2 (Ω) 2 ) and for the implicit Euler time discretization, in [30] the estimate was shown.It was extended to the Crank-Nicholson scheme in [31] under the assumption of f ∈ W 2,∞ (I; L 2 (Ω) 2 ), yielding an error estimate at the time nodes of order O(k 2 + h 2 ).Beyond the references mentioned above, there are many works analyzing the stationary equations, e.g., [26,43], semidiscrete equations in space, e.g., [9,20], or semidiscrete equations in time, e.g., [4,23,44].Fully discrete error estimates for stabilized discretization schemes can be found e.g., in [2,5,18,33].The error estimates are most often derived assuming the necessary regularity of u, such that the discretization schemes can exhibit their full approximative power.The main drawback of the results in the literature is the fact, that usually the errors in L ∞ (I; L 2 (Ω)) and L 2 (I; H 1 (Ω)) are estimated in a combined fashion.Thus the estimate always has to account for the spatial error in the H 1 norm, yielding an order reduction for the estimate of the error in the spatial L 2 norm.The goal of this paper is to prove an error estimate which can be formulated as a best approximation type error estimate, which thus estimates the error in the L ∞ (I; L 2 (Ω)) norm in an isolated manner.Such an estimate for the instationary Stokes equations, has been derived recently in [7].More specifically, for velocity and pressure fields (w, r), solving the instationary Stokes equations on the continuous level, and their fully discrete approximations (w kh , r kh ), it holds where V kh is the used space of discretely divergence free space-time finite element functions.The operator R S h denotes a stationary Stokes projection, see [7] or Section 5 for a formal definition.Using this Stokes result, we will show that a best approximation type result also holds for the nonlinear Navier-Stokes equations.This main result is stated in Theorem 5.4.In Corollary 5.6, for f ∈ L ∞ (I; L 2 (Ω) 2 ), we then obtain an estimate in terms of O(l k (k + h 2 )), where l k denotes a logarithmic term depending on k.This result provides a better order of convergence compared to the estimate (2), shown in [13].The order of convergence in (3) from [30] is comparable, but requires a much stronger regularity assumption.The main tools used in this paper for proving the proposed L ∞ (I; L 2 (Ω)) error estimate are an error splitting approach, and a bootstrapping argument, to apply the corresponding error estimate for the Stokes equations to the first part of the error.In order to apply such an argument, understanding the precise regularity of the occuring nonlinear term (u • ∇)u is crucial.The second part of the error will be estimated by a duality argument.This is possible due to the variational nature of the dG time discretization.We will derive a stability result for a discrete dual equation.For this result, a specially adapted version of a discrete Gronwall lemma, Lemma 4.7, will be presented.For the analysis of the discrete dual problem, we require an estimate in the L 2 (I; H 1 (Ω)) norm, which is why we formulate this result first.We shall then show the error estimate in L ∞ (I; H 1 (Ω)) with the tools presented above.The proof of the error estimate in the L 2 (I; L 2 (Ω)) norm is then straight forward and concludes our work.Summarizing, our main results read and can be found in Theorems 5.3 to 5.5 and Corollaries 5.6 and 5.7.All three results are, up to logarithmic terms, optimal in terms of order of convergence and in terms of required regularity.The structure of this paper will be as follows.
First, we fix some notation and function spaces in Section 2. We proceed in Section 3 by stating the appropriate weak formulations of (1) with and without pressure and recall some known regularity results.We conclude the section with an analysis of the regularity properties of the nonlinear term (u • ∇)u.Section 4 will be devoted to the spatial and temporal discretizations.We present stability results for fully discrete primal and dual equations, and present a version of a discrete Gronwall lemma, which allows us to treat problems with right hand sides that are only L 1 in time.Lastly, in Section 5, we will recall the error estimates for the Stokes problem, and derive first the Navier-Stokes error estimate in L 2 (I; With this, we can apply improved stability results for the discrete Navier-Stokes equations and a discrete dual problem, which in the end allows us to show the error estimates in the L ∞ (I; L 2 (Ω)) and L 2 (I; L 2 (Ω)) norms.

PRELIMINARY
For a polygonal domain Ω ⊂ R 2 , 1 ≤ p ≤ ∞ and k ∈ N, we denote by L p (Ω), W k,p (Ω), H k (Ω) and H 1 0 (Ω) the usual Lebesgue and Sobolev spaces.The inner product on L 2 (Ω) will be denoted by (•, •) Ω .The space L 2 0 (Ω) is the subspace of L 2 (Ω), consisting of all functions, that have zero mean.For s ∈ R\N, s > 0 the fractional order Sobolev(-Slobodeckij) space W s,p (Ω) is defined, see e.g., [21], as In case p = 2 we again use the notation H s (Ω).Note that in this case H s (Ω) can equivalently be obtained via real or complex interpolation of the integer degree spaces H k (Ω).This is due to the fact, that in the Hilbert space setting, all resulting Bessel potential spaces H s 2 (Ω), Besov spaces B s 2,2 (Ω) and Sobolev-(Slobodeckij) spaces H s (Ω) coincide, see [48, pp. 12,39].For X being any function space over Ω, we denote by X * its topological dual space, and abbreviate the duality pairing by •, • Ω .We will also use the notation H 1 0 (Ω) * = H −1 (Ω).The structure of the Stokes and Navier-Stokes equations requires also the definition of some vector valued spaces, consisting of divergence free vector fields.We denote by ∇• the divergence operator and introduce the spaces and Note that instead of the definition via closures, in the case of Ω being bounded and Lipschitz, these spaces are alternatively characterized in the following way, see [46, Chapter 1, Theorems 1.4 & 1.6]: where by u • n we denote the normal trace of the vector field u.To improve readability, whenever vectorial spaces like H 1 (Ω) 2 would arise in the subscript of some norm, we shall drop the outer superscript (•) 2 .For a Banach space X and I = (0, T ] we denote by L p (I; X) the Bochner space of X valued functions, for which the following norm is finite , with the usual convention when p = +∞.It holds L p (I; L p (Ω)) ∼ = L p (I ×Ω), and for p = 2, we denote the inner product by (•, •) I×Ω .Whenever X is separable and 1 ≤ p < ∞, it holds (L p (I; X)) * ∼ = L p * (I; X * ), where 1/p + 1/p * = 1.
The duality pairing for such spaces will be denoted by •, • I×Ω .By W k,p (I; X) and H k (I; X) for k ∈ N we denote the spaces of functions v satisfying ∂ j t v ∈ L p (I; X), j = 0, ..., k.

NAVIER-STOKES EQUATIONS
We start by recalling some regularity results for the Navier-Stokes equations, and are going to prove some additional results, especially adapted to the situation considered in this paper.Throughout this paper, we shall always assume the convexity of Ω.We will state explicitly, whenever results also hold in a more general setting.It is a well known result, that for f ∈ L 2 (I; V * ) + L 1 (I; L 2 (Ω)) and u 0 ∈ H, there exists a unique weak solution, i.e., the following Proposition holds, see [46,Chapter 3 in the sense of distributions on I, and u(0) = u 0 .Moreover, there holds an estimate Note that the constant C above only depends on ν and Ω, but is independent of T , see [45, Theorems V.1.4.2, V.1.5.3, V.3.1.1].It is well known, that under the assumptions of Proposition 3.1, the nonlinearity satisfies for 1 ≤ s, q < 2: see [45,Lemma V.1.2.1].Equation ( 4) is the weak formulation of (1) in divergence free spaces.The proof of the above proposition relies heavily on the fact, that the trilinear form c(•, •, •) defined by posesses the properties summarized in the following lemma.
Lemma 3.2.Let Ω ⊂ R 2 be an open Lipschitz domain, then there holds the estimate due to which, the trilinear form c (•, •, •) satisfies for all u, v, w ∈ H 1 0 (Ω) 2 : Proof.The estimate for the L 4 (Ω) norm can be found in [25, Lemma II.In what follows, we often consider the trilinear form c integrated in time, which we denote by In analyzing the Navier-Stokes equations, the instationary Stokes equations frequently arise as an auxiliary problem.For initial data u 0 ∈ H and right hand side f ∈ L 1 (I; L 2 (Ω) 2 ), there exists a unique solution w ∈ L 2 (I; V )∩L ∞ (I; L 2 (Ω) 2 ) to the Stokes equations where the first line of ( 7) is understood in the sense of distributions on I.We introduce the Stokes operator A : 2 and the projection operator P : L 2 (Ω) 2 → H, defined by which is called the Helmholtz or Leray projection.Note, that with the vector-valued Laplacian 2 , the Stokes operator also satisfies the representation For convex Ω, the domains of the operators introduced above satisfy the representations see [17] for the H 2 regularity of the Stokes operator.The Stokes operator A generates an analytic semigroup in H, see [7,38], also [6] for a detailled general analysis.One important feature of the Stokes problem is the maximal parabolic regularity, which indicates, that both the time derivative ∂ t w and the Stokes operator Aw individually inherit certain regularity properties of f , see Proposition 3.4 below.For homogeneous initial data, this consequence of the analyticity of the semigroup has been shown in [19], see also [45, Chapter IV, Theorem 1.6.3].Since our analysis should also treat inhomogeneous initial data u 0 , we need to define the proper spaces for the initial data: see also [7] and [6, Chapter 1, Section 3.3], where Remark 3.3.Instead of using the spaces V 1−1/s explicitly as a requirement for the initial data, we can make use of the following imbedding results: For [7,Remarks 2.8,2.9].
The Stokes problem exhibits the following regularity properties, see [7,Proposition 2.6].
Then the solution w of the Stokes equations (7) satisfies For the nonlinear Navier-Stokes equations, the maximal parabolic regularity does not immediately transfer.Using H 2 regularity and bootstrapping arguments however, we can recover some of the regularity results.Note that from now on we explicitly require the convexity of Ω, whereas Proposition 3.1 also holds for general Lipschitz domains.Theorem 3.5 (H 2 regularity).Let u 0 ∈ V and f ∈ L 2 (I; L 2 (Ω) 2 ).Then the weak solution u to the Navier-Stokes equations (4) satisfies the improved regularity and there exist constants C 1 , C 2 > 0, such that there hold the bounds .
Proof.The proof of this result for C 2 domains can be found in [46,Chapter 3,Theorem 3.10].Instead of a C 2 boundary, we can also use the H 2 regularity for the Stokes operator on convex, polygonal domains, see, e.g., [17,Theorem 5.5] or [32,Theorem 2], to obtain the claimed regularity.The norm bounds are obtained by the Gronwall lemma.
We shall state a corresponding estimate for ∂ t u L 2 (I×Ω) after discussing the regularity of the nonlinearity.
Remark 3.6.By the Hölder inequality, with this H 2 regularity result, and the imbedding H 2 (Ω) ֒→ L ∞ (Ω), we immediately obtain The regularity u ∈ C(I; H 1 (Ω) 2 ) almost yields that u is uniformly bounded over the whole space time cylinder.However, since in two dimensions H 1 (Ω) ֒→ L ∞ (Ω), the boundedness in space has to be shown via an additional argument.
We first show, that the nonlinearity actually posesses more regularity, than what was claimed in Remark 3.6.
Theorem 3.7.Let the assertions of Theorem 3.5 be satisfied, i.e., u 0 ∈ V and f ∈ L 2 (I; L 2 (Ω) 2 ), and let u be the unique solution to the Navier-Stokes equations (4).Then the nonlinearity satisfies and for any 0 < δ ≤ min{2, s} there holds the estimate where the norms of u on the right hand side can be estimated by Theorem 3.5.
Proof.The proof follows the ideas of [45, Theorem V.1.8.2], however we will use interpolation spaces instead of fractional powers of the Stokes operator.With the Hölder inequality and the Sobolev imbedding H 1+ δ s (Ω) ֒→ C(Ω), it holds for any s < ∞ and δ > 0: Since δ ≤ s, we can express the space H 1+ δ s (Ω) as interpolation space [H 1 (Ω), H 2 (Ω)] δ/s , and obtain from [11, Theorem 1] the estimate u . All in all, we see that , by Theorem 3.5 and δ ≤ 2. With the Hölder inequality, we obtain the proposed estimate, which concludes the proof.

that the Navier-Stokes equations inherit the maximal parabolic regularity of the Stokes problem, in cases where
With Theorem 3.7, and slightly higher regularity of the data, we obtain the boundedness of u in the space-time cylinder: and for ε sufficiently small, there holds the estimate where the norms of u on the right hand side can be estimated by Theorem 3.5.
By a bootstrapping argument, since f ∈ L 2+ε (I; L 2 (Ω)) and u 0 ∈ V 1−1/ε , we can apply the maximal parabolic regularity of Proposition 3.4, and obtain together with H 2 regularity Note that here we have used, that for the L 2 (Ω) case, all corresponding Sobolev-Slobodeckij, Besov, Bessel-potential and interpolation spaces coincide, see, e.g., [39].For an overview over the topic of function spaces, see also [48].From the used embeddings, we moreover have the proposed estimate.This concludes the proof.
Corollary 3.10.Let the assertions of Theorem 3.9 be satisfied, i.e., f ∈ L 2+ε (I; L 2 (Ω) 2 ) and u 0 ∈ V 1−1/(2+ε) for some ε > 0. Then the nonlinear term in the Navier-Stokes equations satisfies and there holds the estimate where the norms of u on the right hand side can be estimated by Theorem 3.5.
Proof.This is a direct consequence of Theorems 3.5 and 3.9 and application of the Hölder inequality.
In the formulation of equation ( 4), we have used divergence free test functions.Whenever we want to test the equation with functions, that are not divergence free, we have to consider an alternative, equivalent formulation, that includes the pressure.The following theorem guarantees, that we can freely switch between the two formulations.It is based on the equivalence of the Stokes problem (7) in divergence free spaces, and the problem of finding (w, r) ∈ L 2 (I; for all (v, q) ∈ L 2 (I; H 1 0 (Ω) 2 ) × L 2 (I; L 2 0 (Ω)) and w(0) = u 0 .Theorem 3.11.Let the assertions of Theorem 3.5 be satisfied, i.e., u 0 ∈ V and f ∈ L 2 (I; L 2 (Ω) 2 ).Then there exists a unique solution (u, p) with such that u(0) = u 0 and for all (v, q) ∈ L 2 (I; where the norms of u on the right hand side can be estimated by Theorem 3.5.
Proof.This result can be shown using Theorem 3.7, with a bootstrapping argument.Moving to the right hand side, applying Remark 3.6 and using the result for Stokes, e.g., [7, Theorem 2.10, Corollary 2.11].

Spatial discretization
Let {T h } denote a family of quasi-uniform triangulations of Ω consisting of closed simplices.The index h denotes the maximum meshsize.We discretize the velocity u by a discrete function space U h ⊂ H 1 0 (Ω) 2 and the pressure p by the discrete space M h ⊂ L 2 0 (Ω), where (U h , M h ) satisfy the discrete, uniform LBB-condition with a constant β > 0 independent of h.Throughout this work, we will assume the following approximation properties of the spaces U h and M h .This assumption is valid, e.g., for Taylor-Hood and Mini finite elements, even on shape regular meshes, see [7, Assumption 7.2].

The (vector valued) discrete Laplacian
We introduce the space V h of discretely divergence free functions as The L 2 projection onto this space will be denoted by

and allows us to introduce the discrete Stokes operator
Having defined these discrete spaces and operators, we can now consider the Ritz projection for the stationary Stokes problem.For any (w, r) ∈ Note, that if w is discretely divergence free, i.e., (∇ • w, ψ h ) Ω = 0 for all ψ h ∈ M h , then it holds R S h (w, r) ∈ V h .Note that the space V h is in general not a subspace of the space V of pointwise divergence free functions.This means, that on the discrete space V h , the form c (•, •, •) does not posess the anti symmetry properties shown in Lemma 3.2.Hence we define, as in [16,30] an anti symmetric variant: By its definition, ĉ (•, •, •) now has the following antisymmetric properties on the space V h , which will later allow us to show the stability of the fully discrete solutions.
Proof.The last two identities are a direct consequence of the definition of ĉ (•, •, •).The first estimate follows from Lemma 3.2 and the imbedding H 1 0 (Ω) ֒→ L 4 (Ω).Note that due to the above lemma, formally we are still allowed to switch the second and third argument of ĉ (•, •, •).The original form c (•, •, •) however had a strict disctinction between the two arguments, as it contains the gradient of the second argument, but only the function values of the third argument.This is of importance when estimating the form in terms of its arguments.In ĉ (•, •, •) gradients occur in the second and third argument, thus switching the arguments does not allow us to obtain improved estimates.For this reason, we state the following lemma, which allows us to switch the second and third arguments of c (•, •, •) by introducing an additional term, even if the first argument is not (pointwise) divergence free.
Proof.The proof is simply an application of integration by parts, and can also be seen, e.g., from [2, Equation 2.9].
We conclude this subsection on the space discretization by recalling some important interpolation estimates.On the continuous level, applying (6) to the first order derivatives, and using H 2 regularity, yields Using the Stokes operator on the right hand side, instead of second order derivatives, allows us to translate this result to the discrete setting.This is facilitated by the following result, showing that for discretely divergence free functions, the discrete Laplacian ∆ h can be bounded in terms of the discrete Stokes operator A h : see [29,Corollary 4.4] or [27,Lemma 4.1].With this, we can translate (12) to the discrete setting, by considering for some fixed w h ∈ V h the solution w ∈ H 1 0 (Ω) 2 to the continuous problem By the stability of the Poisson Ritz projection in W 1,4 (Ω), ( 12) and ( 13), we then obtain the discrete version of (12).
Analogously, the Gagliardo-Nirenberg inequality, which is a consequence of [1, Theorem 3] together with H 2 regularity, has a discrete analogon.It can be shown using the standard discrete Gagliardo-Nirenberg inequality for all w h ∈ U h , which was proven in [28,Lemma 3.3].The proof stated there for smooth domains remains the same for convex domains.The discrete version of ( 15) is then again obtained by applying (13) and reads Straightforward calculations, using the definition of A h , also give With these considerations regarding the spatial discretization, we can now consider the fully discrete Navier-Stokes equations by also discretizing in time.

Temporal discretization
For discretization in time, we employ the discontinuous Galerkin method of order q (dG(q)), which is also used, e.g., in [7,16,24].The time interval We denote each timestep by k m = t m − t m−1 and for fixed M the maximal timestep by k := max 1≤m≤M k m , as well as the minimal one by k min := min 1≤m≤M k m .If we want to emphasize that I m belongs to a discretization level k, we denote it by I m,k .We make some standard assumptions on the properties on the time discretization: (1) There are constants C, β > 0 independent of k, such that (2) There is a constant κ > 0 independent of k, such that for all m = 1, 2, ..., M − 1 (3) It holds k ≤ T 4 .A dG(q) function with values in a given Banach space B is then given as a function in the space Where on each I m the space P q (I m ; B) is given as the spaces of polynomials in time up to degree q with values in B: Note that no continuity is required at the time nodes t m , which is why we use the following standard notations for one sided limits and jump terms: We introduce the compact notations Having defined these dG spaces, we introduce the following projection operator in time: for all m = 1, 2, ..., M .In case q = 0, the projection operator is defined solely by the second condition.
Remark 4.5.In this paper we will restrict ourselves to the two lowest cases q = 0 and q = 1.Since we work in a setting of low regularity of the right hand side f , the error estimates would not benefit from higher order schemes.Also, since the Navier-Stokes equations already pose a challenging large system to solve, higher order schemes in many applications are not feasible from the standpoint of computational cost.
Before we state the discretized version of the Navier-Stokes equations, let us recall the following version of a discrete Gronwall lemma, which is stated in [31, Lemma 5.1] for a constant timestep k, but its proof can easily be adapted to the setting of variable timesteps k m .Lemma 4.6.Let {k n }, {a n }, {b n }, {c n }, {γ n } be sequences of nonnegative numbers and B ≥ 0 a constant, such that for each n ∈ N 0 it holds and k m γ m < 1 for all m ∈ {0, ..., n}.Then with Continuous and discrete Gronwall lemmas are stated in many different forms in the literature, see, e.g., [36] and the references therein for an overview over different generalizations of the original lemma.Note, that, since the sum on the right hand side of the assumed bound goes up to n, the additional assumption k m γ m < 1 is needed.This is not the case in explicit forms of discrete Gronwall lemmas, where the sum on the right goes only up to n − 1, which is the form most often considered in the literature, e.g., [41].In the context of dG timestepping schemes, the sequences a n and b n often correspond to squared norms at time nodes or over subintervals.It therefore seems natural to also state the following version of a Gronwall lemma, where we also include a sum over non-squared contributions.To the best of the authors knowledge, a result like this has not been explicitly used in the literature.In later sections of this work, this lemma will facilitate the analysis of discrete problems with right hand sides that are L 1 in time.Here the norms of the solution occur in a non squared contribution.The following lemma shows, that the weights of the squared sum enter exponentially into the estimate, whereas the weights of the linear sum enter linearly in the estimate for x n .It can be understood as an adaptation of Bihari's inequality to the discrete setting, see [10,37].To improve readability, we drop the explicit mentioning of the timesteps k m , as we can always apply a transformation as done in the proof of Lemma 4.6.Proof.For n ∈ N 0 define δ n ≥ 0 such that and set X n := x 2 n + n m=0 b m + δ n .We will show from which the assertion follows by the definition of X n .Note that by assumption it holds and thus the sequences {X 2 n } n∈N0 and {X n } n∈N0 are monotonically increasing.By X n ≥ x n and the monotonicity of X n , from (19) we obtain Let us now recall, that for a, b > 0, an estimate To see this, we start by computing the roots of the quadratic polynomial, yielding x ≤ a 2 + a 2 4 + b.The root of the polynomial can then be estimated by We can thus apply Lemma 4.6 to X n and obtain after resubstituting the tilded quantities: Squaring and estimating the square of the sum yields the result.
We can now introduce the time-discretized formulation of the Navier-Stokes equations.We define the time-discrete bilinear form for the transient Stokes equations as in [7] by Since we frequently will test some discrete equations with their respective solutions, let us recall the following lemma.
Lemma 4.8.For any v k ∈ X q k (L 2 (Ω)) it holds Proof.For the first equality, we can express the integral over the time derivatives via and recombining terms gives the first identity.The proof of the second equality works completely anologous.
The fully discrete formulation of the transient Navier-Stokes equations, using the anti symmetrized trilinear form ĉ (•, •, •), introduced in (11), is then given as: Find u kh ∈ V kh , such that Before showing unique solvability of the discrete Navier-Stokes equations ( 21), we show stability for the discrete system in different norms under different assumptions on f .Solutions to the discrete problem also satisfy the same energy bounds as the weak solutions, i.e., there holds the following proposition, see [16, Lemma 5.1, Theorem A.1].
Proposition 4.9 (Stability of discrete Navier-Stokes).Let f ∈ L 2 (I; V * ), u 0 ∈ H and u kh ∈ V kh satisfy (21).Then there hold the bounds with a constant C depending on the domain Ω and the viscosity ν.
Note that contrary to the continuous setting, see Proposition 3.1, due to the exponent of 1 2 on the left hand side, the above proposition states a bound for u kh L ∞ (I;L 2 (Ω)) , which depends on the squared norms of the data u 0 and f .For the two low order cases q = 0 and q = 1, and f ∈ L 2 (I; L 2 (Ω) 2 ), [16, Lemma 5.1] also shows an estimate of the form With our discrete Gronwall estimate Lemma 4.7, we can generalize this result to f ∈ L 1 (I; L 2 (Ω) 2 ).Furthermore, by using the version of Gronwall's lemma presented here, contrary to [16], the bound does not grow exponentially in T .This is in agreement with the result of Proposition 3.1 in the continuous setting.We obtain the following result, which now yields an estimate for the norms of u kh , that is linear in terms of the data.
Theorem 4.10.Let f ∈ L 1 (I; L 2 (Ω) 2 ) + L 2 (I; V * ), u 0 ∈ H and u kh ∈ V kh satisfy equation (21) for either q = 0 or q = 1.Then there holds the bound Proof.We first prove the result for f ∈ L 1 (I; L 2 (Ω) 2 ) and remark at the end, which modifications are needed to also cover the L 2 (I; V * ) case.For notational simplicity, we use the convention u − kh,0 := u 0 and accordingly [u kh ] 0 = u + kh,0 − u 0 .By testing (21) with u kh | Im and using Lemma 4.3 we arrive on each time interval at: Applying Lemma 4.8 gives Multiplying by two and summing up the identity over the intervals 1, ..., n, we obtain From this point on, we have to treat the two cases q = 0 and q = 1 separately.Case 1: . Thus with the Hölder inequality, the terms in the last sum of ( 22) can be estimated as Hence, from ( 22) we obtain the following An application of Lemma 4.7 proves the assertion.Case 2: If q = 1, then the L ∞ (I m ) norm can be estimated by the evaluation at the two endpoints of the interval: With triangle inequality we can estimate the right sided limit in terms of a left sided limit and a jump: L 2 (Ω) for n = 1, ..., M yields from (22) the estimate or after an index shift In order to shorten the notation, we have added here a term f L 1 (In+1;L 2 (Ω)) x n , where in case n = M we use the convention I M+1 = ∅.Again we can apply Lemma 4.7, which yields the estimate Using Young's inequality, to estimate the term 2 f L 1 (I1;L 2 (Ω)) u 0 , concludes the proof.In case ), the contribution of f 1 can be treated as above.For the f 2 contribution, in (22) one applies the Hölder and Young inequalities, and absorbs the ∇u kh L 2 (Im;L 2 (Ω)) term to the left.Note that in both cases q = 0, 1, in the estimates before the application of Lemma 4.7, no x 2 m terms are summed on the right hand side, thus no exponential dependency is introduced in the final estimate.

The techniques presented above now allow us to show the unique solvability of the discrete equations (21).
Theorem 4.11.Let u 0 ∈ H and f ∈ L 1 (I; L 2 (Ω) 2 ) + L 2 (I; V * ).Let further either q = 0 or q = 1.Then there exists a unique solution u kh ∈ V kh of (21).
Proof.This result can be shown by applying a standard fixpoint argument, using the stability result of Theorem 4.10.
Since in general V h ⊂ V , the discrete solution u kh ∈ V kh is not divergence free, and thus, we are not allowed to use it as a test function for the divergence-free continuous formulation (4).Because of this, we introduce an equivalent formulation with pressure: Find (u kh , p kh ) ∈ Y kh , such that for all (v kh , q kh ) ∈ Y kh it holds where the mixed bilinear form B is defined by Theorem 4.12.The two formulations (21) and (23) are equivalent in the sense that, if u kh ∈ V kh satisfies (21), then there exists p kh ∈ M kh such that (u kh , p kh ) solves (23).Conversely, if (u kh , p kh ) satisfies (23), then u kh is an element of V kh and satisfies (21).
Proof.This can be shown by using the same arguments as [7,Proposition 4.3].
We next show the stability of the discrete solution u kh in stronger norms, comparable to the continuous result of Theorem 3.5.We will show this result, by applying the discrete Gronwall Lemma 4.7, to which end we need the two following technical lemmas, guaranteeing, that the coefficients in the Gronwall lemma become arbitrarily small, uniformly in m, as (k, h) → 0. To keep notation simple, we formally extend ζ to [0, T + ε] by values 0 such that the integration is well defined.
Proof.The proof relies on dominated convergence and Dini's theorem, see, e.g., [12, p. 125].We first define for fixed ε the function |ζ(s)| ds.Note that for each ε > 0 this function σ ε is continuous.To see this, let x ∈ I be fixed and let y → x.W.l.o.g.let y > x: where χ x,y (s) = 1 if s ∈ (x, y) ∪ (x + ε, y + ε), and 0 otherwise.For a fixed s, the integrand χ x,y (s)|ζ(s)| → 0 as y → x, thus the integrand converges pointwise to 0. By the dominated convergence theorem, this means that the integral converges to 0. For fixed ε > 0 this shows the continuity of σ ε .Moreover, by the same argument we can show, that for fixed x and ε → 0 it holds σ ε (x) → 0. Thus the sequence σ ε converges pointwise to 0. Note moreover, that for δ > ε, and fixed x, it holds σ δ (x) ≥ σ ε (x), as the integration covers a larger interval.We thus have shown, that when ε → 0 monotonically, also this pointwise convergence of σ ε (x) is monotone.Hence we can apply Dini's theorem, see [12, p. 125], to obtain σ ε L ∞ (I) → 0, as ε → 0.
Lemma 4.14.Let u ∈ L 2 (I; H 1 0 (Ω) 2 ), and u kh ∈ U kh such that u − u kh L 2 (I;H 1 (Ω)) → 0 as (k, h) → 0. Then it holds Proof.We first show the statement for u.To this end, note that for by Lemma 4.13.In order to show the result for the discrete solution u kh , we cannot directly apply the previous lemma, as u kh depends on k.We thus insert insert ±u and apply the triangle inequality to obtain With the claim for u and u kh → u in L 2 (I; H 1 (Ω)), we have shown the claim for u kh .
We now turn toward proving the stability of u kh in stronger norms.The proof follows the steps of the continuous result, shown in [46, Chapter 3, Theorem 3.10], using the discrete analogons of the inequalities presented in Section 4.1, and a discrete Gronwall lemma.For the application of the lemma, we need coefficients, that become small, as k → 0. These coefficients depend on u kh , and thus on k, h, hence we need to assume a convergence result in L 2 (I; H 1 (Ω)), in order to have coefficients that converge to 0 uniformly in m.We shall show later on in Theorem 5.3, that such a convergence result indeed holds.Making this assumption here, allows us to present the stability result jointly with similar results.There holds the following.Theorem 4.15.Let f ∈ L 2 (I; L 2 (Ω) 2 ) and u 0 ∈ V , and let the unique solution u kh ∈ V kh of (21) converge to the continuous solution u of (4) in L 2 (I; H 1 (Ω)) as (k, h) → 0. Then u kh satisfies with constants C 1 , C 2 independent of k, h.The L ∞ (I; L 2 (Ω)) and L 2 (I; H 1 (Ω)) norms of u kh can be estimated by the results of Theorem 4.10.
Proof.We test (21) with A h u kh | Im , which yields , using the Kronecker delta δ 0m to include the contribution of the initial data, in case m = 0.The definition of A h gives from the above identity Here the terms containing time derivatives and jumps can be combined according to Lemma 4.8.We first estimate the terms containing the data of the right hand side f and initial data u 0 .For the right hand side terms, it holds where we can absorb the A h u kh term.The contribution of the initial data is estimated via In case u 0 ∈ V , we can use the stability of P h in H 1 for continuously divergence free functions, see [49,Lemma 5.4], proving . We now turn towards estimating the trilinear form remaining in (24), which poses the main difficulty of this proof.Due to Lemma 4.4, the trilinear term satisfies the expression which we can estimate with Hölder's inequality by Using ( 16) and Young's inequality, we obtain the estimate The latter term can be absorbed, hence we proceed by discussing the first one.It holds

Let us introduce γ
. Using the L 2 (I; H 1 (Ω)) error estimates of Theorem 5.3 and Lemma 4.13, it holds γ m → 0 uniformly, as (k, h) → 0. This implies that by following the same steps as the proof of Theorem 4.10, we obtain the result as a consequence of the discrete Gronwall Lemma 4.7.
We conclude this section by showing two stability results for discrete dual equations.Their formulation is motivated by the following considerations.Due to the nonlinear structure of the Navier-Stokes equations, there does not hold a Galerkin orthogonality with respect to the bilinear form B, i.e., for solutions (u, p) of ( 9) and their discrete counterparts (u kh , p kh ) of ( 23), it holds for test functions (v kh , q kh ) ∈ Y kh : where the right hand side is nonzero in general.Since we want to use this orthogonality relation after testing the dual equation with u − u kh , we need to reformulate the trilinear terms, such that u − u kh occurs linearly.To this end, we use the identity where we linearize around the average of continuous and discrete solutions to the Navier-Stokes equations With these considerations, we have the following lemma: Lemma 4.16.Let (u, p) be a solution to the Navier-Stokes equations (9), and (u kh , p kh ) their discrete approximation solving (23).Let further uu kh denote the average of u and u kh as defined by (26).Then for any (v kh , q kh ) ∈ Y kh , it holds Proof.This result is an immediate consequence of the definitions of solutions to ( 9) and ( 23), together with the identity (25) This motivates the choice of uu kh as linearization point for setting up a dual equation: where the right hand side g will be chosen appropriately, see the proofs of Theorems 5.4 and 5.5.Note that the right hand side of ( 27) implicitly prescribes the final data z + kh,M = 0.This dual equation will help us in deriving the sought error estimates, which we will do in the following section.
Remark 4.17.To analyze this dual problem, it will be convenient, to have dual represenations of B and B at hand, which are obtained by partial integration on each I m and rearranging the terms.Note, that in this representation, the time derivative is applied to the second argument.It holds see also [7,34].
We first show unique solvability of the discrete dual problem, and the stability in L ∞ (I; L 2 (Ω))∩L 2 (I; H 1 (Ω)).Since both u and u kh occur in the formulation of the discrete problem, we need both results of Lemma 4.14 to hold true. 2 ) be solutions to the weak and fully discretized Navier-Stokes equations (4) and ( 21) respectively, such that u − u kh L 2 (I;H 1 (Ω)) → 0 as (k, h) → 0. Let further q = 0 or q = 1.Then for (k, h) small enough, problem (30) has a unique solution z kh ∈ V kh and for z kh we have the bound , where K : [0, +∞) → [0, +∞) is a strictly monotonically increasing, continuous nonlinear function, independent of k, h.
Proof.On the continuous level and for g ∈ L 2 (I; (H −1 (Ω)) 2 ), the proof of a corresponding estimate can be found in [15,Proposition 2.7].We adapt it to the discrete setting and to g ∈ L 1 (I; L 2 (Ω) 2 ), making use of the previously derived discrete Gronwall Lemma 4.7.We only have to prove the norm bound, since problem ( 27) is a quadratic system of linear equations, thus is solvable, if it is injective.The norm bound yields, that for right hand side g = 0, z kh = 0 is the only solution, thus the norm bound implies existence and uniqueness.For ease of notation, we use the convention Testing equation (30) with z kh χ Im , m = 1, ..., M , where by χ Im we denote the indicator function of the subinterval I m , yields: Applying Lemma 4.8 and writing the inner product as norm yields We proceed by estimating the trilinear forms.According to its definition (11), the first term vanishes, and for the second one, it holds After applying the Hölder inequality in space, we obtain After estimating the L 4 norms by Lemma 3.2 and applying the Hölder inequality in time, we arrive at ĉ ((z kh χ Im , uu kh , z kh )) ≤C ∇z kh L 2 (Im×Ω) z kh L ∞ (Im;L 2 (Ω)) ∇uu kh L 2 (Im×Ω) An application of Youngs inequality yields In order to abbreviate the notation, we introduce We insert the above estimate into (31), absorb terms, and multiply by 2, which yields After an application of Hölder's inequality, we obtain Since we have assumed q = 0 or q = 1, we can estimate the z kh L ∞ (Im;L 2 (Ω)) terms by evaluations at the right and left endpoints: With triangle inequality there hold the following estimates: . Note that by the terminal condition for z kh , it holds x M+1 = 0. Then after summing up (32) from m = n to m = M , we have: Shifting indices, we arrive at where for n = 1 we use the convention I 0 = ∅.Hence we are in the setting of Lemma 4.7, where formally we have to introduce an index transformation ñ = M − n.In order to apply the lemma, we need to verify 2(γ m + γ m−1 ) < 1 for all m.By Lemma 4.14, we obtain with triangle inequality, that sup 1≤m≤M ∇uu kh 2 L 2 (Im;L 2 (Ω)) → 0 as (k, h) → 0. Together with the bounds from Proposition 3.1 and Theorem 4.10 for u L ∞ (I;L 2 (Ω)) and u kh L ∞ (I;L 2 (Ω)) , we obtain that γ m → 0 uniformly in n for (k, h) → 0. Thus we can choose the discretization fine enough, such that γ m < 1/8, and thus we obtain from Lemma 4.7 We conclude this section by stating stability of the discrete dual solution in stronger norms, whenever the right hand side posesses more regularity.Theorem 4.20.Let the Assumptions of Theorem 4.19 hold true, i.e. let u,u kh ∈ L ∞ (I; L 2 (Ω) 2 )∩L 2 (I; H 1 (Ω) 2 ) be solutions to the weak and fully discretized Navier-Stokes equations (4) and ( 21) respectively, such that u−u kh L 2 (I;H 1 (Ω)) → 0 as (k, h) → 0. Let additionally f ∈ L 2 (I; L 2 (Ω) 2 ) and u 0 ∈ V , such that the results of Theorem 3.5 and Theorem 4.15 hold, and let g ∈ L 2 (I; L 2 (Ω) 2 ).Then for (k, h) small enough there holds the bound Proof.The proof follows the same steps as the proof the stability of u kh in stronger norms, presented in Theorem 4.15.We begin by testing the dual equation with A h z kh χ Im for arbitrary m = 1, ..., M , where χ Im denotes the characteristic function of the time interval I m .The nonlinear terms that occur for the discrete dual equation are of the form ĉ ((uu kh , A h z kh χ Im , z kh )) + ĉ ((A h z kh χ Im , uu kh , z kh )) .
By the Hölder inequality in space, (17) and the discrete Gagliardo-Nirenberg inequality (16), these terms can be estimated by Applying the continuous and discrete Gagliardo-Nirenberg inequalities (15) and (16), and (17) to uu kh , it remains An application of Young's inequality, absorbing terms and summing over the subintervals allows us to conclude the proof.By Theorem 4.19 it holds z kh ∈ L ∞ (I; L 2 (Ω) 2 ) with a bound independent on k, h and linear in g L 1 (I;L 2 (Ω)) .Further, the terms involving u, u kh are summable, since the L ∞ (I; L 2 (Ω)) norms of u, u kh remain bounded via Proposition 3.1 and Theorem 4.10, and the Au, A h u kh terms are summable by Theorem 3.5 and Theorem 4.15.

ERROR ESTIMATES
Before showing the fully discrete error estimates for the Navier-Stokes equations, we recapitulate the corresponding error estimates for the Stokes equations (8).The associated fully discrete formulation reads: Find (w kh , r kh ) ∈ Y kh satisfying B((w kh , r kh ), (φ kh , ψ kh )) = u 0 , φ In the recent contributions [7,49], best approximation type error estimates for the Stokes problem were shown in the norms of L ∞ (I; L 2 (Ω)), L 2 (I; L 2 (Ω)) and L 2 (I; H 1 (Ω)).There hold the following results, formulated in terms of best approximation error terms and the errors of the projection in time π τ , defined in (18), and the Stokes Ritz projection R S h , defined in (10).
Proposition 5.1 ( [49, Theorems 6.1& 6.3]).Let f ∈ L 2 (I; L 2 (Ω) 2 ) and u 0 ∈ V , let (w, r) and (w kh , r kh ) be the continuous and fully discrete solutions to the Stokes problems (8) and (34).Then for any χ kh ∈ V kh , there holds Proposition 5.2 ( [7, Corollary 6.4]).Let f ∈ L s (I; L 2 (Ω) 2 ) and u 0 ∈ V 1−1/s for some s > 1, and let (w, r) and (w kh , r kh ) be the continuous and fully discrete solutions to the Stokes equations ( 8) and (34).Then for any χ kh ∈ V kh , there holds Having shown these error estimates for the Stokes equations, the natural question arises, whether these results can be extended to the the Navier-Stokes equations.We first give a positive answer for the L 2 (I; H 1 (Ω)) error in Theorem 5.3.The main result of this work is the error estimate in the L ∞ (I; L 2 (Ω)) norm, presented in Theorem 5.4.With the same techniques, the proof of the L 2 (I; L 2 (Ω)) error estimate is then straightforward, and we state the result in Theorem 5.5.For the L 2 (I; H 1 (Ω)) error, [16,Theorem 5.2] shows a result, estimating the error for the Navier-Stokes equations in terms of the error of a Stokes problem.The discrete Stokes problem there is defined with right hand side ∂ t u − ν∆u, which corresponds to f − ∇p − (u • ∇)u, i.e. the pressure is included on the right.This means, that when applying the corresponding orthogonality relations, a pressure term remains.In the following result, we use a different right hand side for the discrete Stokes problem, yielding the following error estimate for the Navier-Stokes equations in L 2 (I; H 1 (Ω)).
Theorem 5.3.Let f ∈ L 2 (I; L 2 (Ω) 2 ), u 0 ∈ V and let (u, p), (u kh , p kh ) be the continuous and fully discrete solutions to the Navier-Stokes equations ( 9) and (23) respectively.Then for any χ kh ∈ V kh , there holds Proof.The proof uses the same arguments as the one of [16,Theorem 5.2].We repeat the main steps, in order to motivate, why due to the choice of our Stokes projection, no error term for the pressures arises on the right hand side.We denote by (ũ kh , pkh ) ∈ Y kh the solution to the discrete Stokes problem Subtracting the two and choosing (φ kh , ψ kh ) = (η kh , κ kh ) yields The two trilinear forms in (35) satisfy the relation Due to Lemma 4.3, the last term on the right hand side vanishes, and for the remaining ones, we obtain by the Hölder and Young inequalities Due to u kh and ũkh both being discretely divergence free, all pressure contributions in (35) vanish, and with the above estimates, after absorbing all ν 4 η kh L 2 (Im;H 1 (Ω)) terms to the left, we obtain To abbreviate notation we have used here [η kh ] 0 := η + kh,0 .Choosing k small enough, allows us to apply the discrete Gronwall lemma as in Theorem 4.10, which concludes the proof.
We now turn towards showing the main result of our work, i.e., the error estimate for the Navier-Stokes equations in the L ∞ (I; L 2 (Ω)) norm.We will split the error u − u kh into an error for a Stokes problem, and a remainder term, which we will estimate using the discrete dual equation (30).Note, that since we have shown the L 2 (I; H 1 (Ω)) error estimate in Theorem 5.3, we can apply the stability results of Theorem 4.19 for the discrete dual problem.Similar to the result for the Stokes equations of Proposition 5.2, the error estimate will consist of two terms with the first one being a best approximation error, and the second one being the error of the stationary Stokes Ritz projection introduced in (10).This result estimates the L ∞ (I; L 2 (Ω)) norm in an isolated fashion, and thus does not suffer from an order reduction, which is observed in results that estimate the error norm combined with the L 2 (I; H 1 (Ω)) norm.
Theorem 5.4.Let f ∈ L 2 (I; L 2 (Ω) 2 ) and u 0 ∈ V .Let (u, p) be the unique solution to the Navier-Stokes equations (9), and (u kh , p kh ) the corresponding solution to the discretized equations (23) for a discontinuous Galerkin method in time with order q = 0 or q = 1.Then there holds Proof.We denote by e := u − u kh the error, which we want to estimate, and introduce (ũ kh , pkh ) as the instationary Stokes-projection of (u, p), i.e., the solution to For the construction of such a function θ serving the purpose of a regularized Dirac measure, we refer to [42,Appendix A.5].We then define the dual solution z kh ∈ V kh such that for all φ kh ∈ V kh , it satisfies B(φ kh , z kh ) + ĉ ((uu kh , φ kh , z kh )) + ĉ ((φ kh , uu kh , z kh )) = η kh ( t)θ, φ kh I×Ω .
Thus we see directly η kh ( t) 2 L 2 (Ω) = −ĉ ((uu kh , ξ, z kh )) − ĉ ((ξ, uu kh , z kh )) .We want to make use of the L ∞ (I; L 2 (Ω)) estimate of ξ and thus have to move it to an argument of ĉ that has no spatial gradient applied to it.Since ĉ was obtained by anti-symmetrizing c, there are gradients in the second and third argument of ĉ, which is why we revert to the original trilinear form c. The first argument has no gradient, and thus η kh ( t) 2  L 2 (Ω) = − Canceling terms concludes the proof.
Using the same arguments from before, instead of the best-approximation type estimate, we can also directly use the error estimate for the Stokes projection, shown in [7,Theorem 7.4] and [49,Corollaries 6.2& 6.4] to obtain the following corollaries yielding explicit orders of convergence.

Proof.
By choosing k = 1, γ m = k m γ m and c m = k m c m , the quantities with • correspond to the notation of [31, Lemma 5.1].The result then directly follows from this redefinition.

a 2 + a 2 4 +
b ≤ a + √ b, which can be shown by subtracting a/2 and squaring both sides.Applied to equation (20), we have thus for every n ∈ N 0 the estimate where we have defined B := 0 as well as c0 := d 0 + c 0 + B , and cm :=