Konstantinos Chrysafinos 1 and L. Steven Hou Introduction

Transcription

1 Mathematical Modelling and Numerical Analysis Modélisation Mathématique et Analyse Numérique Will be set by the publisher ANALYSIS AND APPROXIMATIONS OF THE EVOLUTIONARY STOKES EQUATIONS WITH INHOMOGENEOUS BOUNDARY AND DIVERGENCE DATA USING A PARABOLIC SADDLE POINT FORMULATION Konstantinos Chrysafinos 1 and L. Steven Hou 2 Abstract. This wor concerns the analysis and finite element approximations of the evolutionary Stoes equations, with inhomogeneous boundary and divergence data. The proposed wea formulation can be viewed as an attempt to develop the parabolic analog of the well nown saddle point theory for elliptic problems. Several results concerning the analysis and finite element approximations are presented. The ey feature of the wea formulation under consideration is the treatment of Dirichlet boundary conditions within the Lagrange multiplier framewor Mathematics Subject Classification. 65M12, 65M60, 76D05. The dates will be set by the publisher. 1. Introduction This wor concerns the analysis and finite element approximations of the evolutionary Stoes equations with inhomogeneous boundary and/or divergence data. In particular, we are interested in developing and analyzing an appropriate wea formulation for the following problem: Given data φ, ψ and initial velocity u 0 we see a pair ũ, p such that ũ t ν ũ + p = 0 in Ω 0, T ] div ũ = ψ in Ω 0, T ] 1.1 ũ = φ on Γ 0, T ] ũ0 = u 0 in Ω. Here Ω R d, d = 2, 3, denotes a bounded polygonal polyhedral when d = 3, and convex domain or a bounded domain with regular enough boundary Γ. Recall that the divergence Theorem implies the following compatibility condition, ψ., t = φ., t n. for a.e. t 0, T ]. Ω Γ It is worth noting that the analysis and finite element approximations of such problems are very important from the engineering view-point since they are closely related to boundary control problems with Dirichlet boundary control data, as well as to feedbac control problems see e.g. [19, 28]. In addition, another motive Keywords and phrases: Evolutionary Stoes Equations, Inhomogeneous Boundary and Divergence Data, Error Estimates, Finite Element Approximations, Lagrange Multipliers, Saddle Point Formulations 1 Department of Mathematics, School of Applied Mathematics and Physical Sciences, National Technical University of Athens, Zografou Campus, Athens 15780, Greece. 2 Department of Mathematics, Iowa State University, Ames, IA 50011, USA. c EDP Sciences, SMAI 1999

2 2 TITLE WILL BE SET BY THE PUBLISHER for this wor, is the analysis and finite element approximations of a wea formulation suitable for handling essential inhomogeneous Dirichlet boundary data for the evolutionary Stoes problem. Our main goal is to develop the parabolic analog of the well nown saddle point theory for elliptic problems and its finite element approximation within the context of mixed finite element methods. To our best nowledge there are no results regarding finite element approximations of such problems The parabolic saddle point framewor A wea formulation that resembles the classical saddle point formulation of the stationary Stoes equations will be developed. In particular, we examine wea problems of the following form: Given data g, u 0, find a solution pair u, p such that, for a.e. t 0, T ], u t t, v X,X + νaut, v + Bv, pt = 0 v X But, q = gt, q M,M q M u0, z = u 0, z z H, where X, M, H are suitable Banach spaces, X, M, H their duals, ν > 0 a positive constant, and A,, B, are continuous bilinear forms defined on X X and X M respectively. The precise functional analytic framewor is given in Section 2. A ey feature of our analysis is that the bilinear forms A, and B, are defined in way to handle evolutionary problems with essential inhomogeneous boundary data, in particular within the framewor of Lagrange multipliers. For instance, in the case of the evolutionary Stoes equations 1.1, we define the bilinear form Au, v = Ω u : v u, v X = H 1 Ω. All other terms, involving pressure and / or boundary terms resulting from integration by parts in space, are included into the bilinear form B,. The precise functional analytic formulation and its relation to Lagrange multipliers is presented in Section Related results and comments Evolutionary Navier-Stoes problems with inhomogeneous Dirichlet boundary data have been studied in the wors of [13, 14, 30]. Several results regarding the analysis of Dirichlet boundary value problems, as well as several applications to optimal boundary control problems were studied in [11, 12]. The evolutionary Stoes and Navier-Stoes equations with inhomogeneous divergence condition have also their own independent importance. To this end, we point out the wor of [30], where the Stoes and the Oseen s equations with inhomogeneous divergence condition were analyzed. The analysis of [30] is also applicable within the context of feedbac control. Saddle-point formulations suitable for space-time approximations, are studied in the recent wor of [18] for the Stoes and Navier-Stoes equations with Navier slip boundary conditions. The main target of the wor of [18] is the development of suitable wea formulations for space-time approximations with wavelet basis. A ey feature of the analysis presented here, is to impose regularity assumptions on the data to guarantee the existence of a suffieciently regular solution of 1.2 that allows the use of standard finite element approximations within the context of mixed finite elements. Our wor differs from the previously developed analysis of [30], since our main emphasis is to avoid the use of divergence-free spaces for the regularity of the time-derivative of the velocity u t, or very wea formulations resulting the validity of the pressure term in a distributional sense. Even though the various concepts of very wea solutions based on transposition techniques as presented in [30], guarantee existence and uniqueness under very low regularity assumptions on the data, they are not directly applicable within the framewor of finite element analysis. This is due to the fact that the finite element discretization of wea solutions based on 1.2

3 TITLE WILL BE SET BY THE PUBLISHER 3 transposition techniques typically require nonstandard finite element spaces. To the contrary, the parabolic saddle point formulation of 1.2 allows us to define finite element approximations in a more standard but not classical way and to obtain error estimates for the semi-discrete in space approximations for the velocity and the Lagrange multiplier. Special care is exercised in order to obtain estimates which resemble the symmetric structure of the ones of the classical saddle point theory of elliptic problems. In addition, we prove error estimates when essential inhomogeneous data are being used in the definition of the discrete analog of the wea formulation 1.2. These estimates can be used in many physical applications, including optimal boundary control problems. A particular choice of subspaces allowing the decoupling of the computation of the velocity and pressure from the computation of the and Lagrange multiplier is analyzed in [7]. For results related to the analysis and finite element approximations of parabolic problems with inhomogeneous Dirichlet boundary data, we also refer the reader to [3, 6]. The Lagrange multiplier framewor for the numerical treatment of essential inhomogeneous Dirichlet boundary data for elliptic problems, and for stationary Navier-Stoes equations has been considered in [1, 2, 20, 27, 34]. Saddle point problems are usually related to elliptic partial differential equations and result from certain minimization principles. The main concepts originate from solid and fluid mechanics since many problems in these areas can be viewed as saddle point problems. One of the main advantages of this approach, is the relation of saddle point problems to finite element methods of mixed type. Finite element spaces of mixed type were studied extensively in previous wors see e.g [4, 5]. For a comprehensive treatment of many important algorithms such as penalized, iterated penalized algorithms, augmented Lagrangian and Uzawa type, one may consult the classical wors of [5, 16, 33]. Even though parabolic problems of saddle point type are not related to an optimization principle, this particular type of formulation can be very useful for the analysis and finite element approximations of time dependent problems such as 1.1. This paper is organized as follows: In section 2 we present the notation and the main result concerning saddle point problems associated to elliptic partial differential equations. Furthermore, we state the main result concerning the existence and uniqueness of the solution of problem 1.2. In subsequent section 3 we establish the proof of the main theorem. In section 4, we present applications of the main theorem to the existence and uniqueness of wea solution for evolutionary problems with inhomogeneous boundary and divergence data. Finally, in section 5, we derive the main error estimates for the finite element approximations. Note that we also treat inhomogeneous essential boundary data Notation 2. Preliminaries and main results Let Ω is a bounded domain in R d, d = 2, 3 which can be either convex and polygonal convex and polyhedral, in d = 3 or with regular enough boundary Γ. We will denote all vector valued functions using the boldface notation u, v etc. We use the standard notation H m Ω, H s Γ for Hilbert spaces of order m, s R, defined on Ω and Γ repsectively, and their norms. Furthermore we denote by H 1 0 Ω {v H 1 Ω : v Γ = 0} and H 1 its dual. Abusing the notation we will not use different notation for their vector valued counterparts. For any Hilbert space U defined as above, the standard notation is being used for their corresponding time-space spaces L p 0, T ; U and their norms, i.e., T v L p 0,T ;U = 0 v p U dt 1 p, v L 0,T ;U = esssup t [0,T ] v U. We also employ the standard notation for the L 2 Ω inner product, L 2 Ω =,. In addition, we denote by X any vector valued version of the above spaces and by H a vector valued version of the above Hilbert spaces such that X H X form an evolution triple, i.e., X H with compact embedding for details see [35, Proposition 23.23], satisfying 1 d 2 dt ut 2 H = u tt, ut X,X. In practice, H is always the vector valued version of L 2 Ω and X a vector valued version of H 1 Ω, H0 1 Ω and/or their divergence free counterparts see

4 4 TITLE WILL BE SET BY THE PUBLISHER e.g. [15, 33]. Abusing the notation, we will denote by.,. the inner product of H, and by.,..,. X,X. Similarly, we denote by L 2 0, T ; X, L 0, T ; X the vector valued time dependent spaces with their norms defined as above. Finally, we will frequently use the space H 1 0, T ; X, endowed with norn u 2 H 1 0,T ;X = u 2 L 2 0,T ;X + u t 2 L 2 0,T ;X. For the pressure terms, we also use the space endowed with norm. L 2 Ω The elliptic saddle point problem L 2 0Ω = {p L 2 Ω : Ω pdx = 0}, The classical theory of elliptic saddle point problems can be described as follows: Find u, p X M such that, { νau, v + Bv, p = f, v v X 2.1 Bu, q = g, q M,M q M, where X, M are given spaces, f X, and g M are given data. We also assume that A, is a continuous bilinear form on X X, and B, is a continuous bilinear form on X M. Moreover, we define the auxiliary subspaces see e.g. [15] Zg := {u X : Bu, q = g, q M,M q M}, Z Z0. In addition we require that the bilinear forms satisfy the standard coercivity assumptions: inf Az, z α z 2 X z Z, 2.2 sup 0 q M 0 u X Bu, q u X q M β > The last inequality is usually called inf-sup condition see e.g., [1], [5], [15], [23], [26] and references within. The main result concerning the existence and uniqueness of a solution pair u, p X M is presented in the following theorem see e.g. [15]. Theorem 2.1. Let Au, v, Bv, q be bounded bilinear operators satisfying coercivity conditions Then, for any given f X, g M, there exists a unique pair u, p X M such that 2.1 holds The parabolic saddle point framewor and main results We close this section by stating the main result and some additional comments regarding the existence and uniqueness of parabolic saddle point problems. Theorem 2.2. Assume that the continuous bilinear forms A,, B, satisfy the coercivity properties Furthermore, suppose a semi-norm is defined by the bilinear form, u 2 X Au, u u X with Au, v 1 2 Au, u + 1 Av, v, u, v X If g H 1 0, T ; M, u 0 X, and Bu0, q = g0, q M,M, q M then there exists u L 0, T ; X H 1 0, T ; H, and p L 2 0, T ; M such that for a.e. t 0, T ], u t, v + νau, v + Bv, p = 0 v X Bu, q = g, q M,M q M u0, z = u 0, z z H. 2.5

5 TITLE WILL BE SET BY THE PUBLISHER 5 In addition, if we decompose u. = w. + z., where w. Z, z. Z, for a.e. w t L 2 0, T ; X. t 0, T ], then Remar 2.3. For the examples stated in the introduction, inequality 2.4, states that the bilinear form A, contains only gradient terms. In addition, note also that 2.4 implies the following inequality Au, u v 1 2 Au, u 1 2 Av, v 1 2 u 2 X 1 2 v 2 X. 2.6 The inhomogeneous evolutionary Stoes equations 1.1, can be included in the above setting provided that the data g are understood as a pair g ψ, φ H 1 0, T ; M with M M 1 M 2. Here M 1, M 2 denote appropriate spaces for the inhomogeneous divergence and boundary data respectively. Hence, we see velocity u ũ L 2 0, T ; X H 1 0, T ; X L 0, T ; H and a pair p p, λ L 2 0, T ; M 1 L 2 0, T ; M 2 consisting of the pressure p and the Lagrange multiplier λ terms respectively. Under our assumptions we prove the enhanced regularity u H 1 0, T ; H L 0, T ; X which is crucial in the development of error estimates. We emphasize that the Lagrange multiplier term λ contains all related boundary terms, including terms resulting from various applications of Green s Theorem. The presence of the Lagrange multiplier λ is an essential feature of our wor which distinguishes it from other approaches. Note also, that the classical evolutionary Stoes problem with inhomogeneous Dirichlet data can be fit into the above framewor for ψ 0. The above result can be extended for a nonzero forcing term f, when it is combined with an analogous result for the homogeneous case g 0. Remar 2.4. Let the assumptions of Theorem 2.2 hold, and let the forcing term f L 2 0, T ; H Then there exists u L 2 0, T ; X H 1 0, T ; H, and p L 2 0, T ; M such that for a.e. t 0, T ] u t, v + νau, v + Bv, p = f, v X,X v X Bu, q = g, q M,M q M u0, z = u 0, z z H. The regularity assumption for f and u 0 is due to the coupling between p and u t and our requirement for regularity p L 2 0, T ; M within the above wea formulation. Recall that even in case of the evolutionary Stoes equation with homogeneous data, if we restrict the regularity assumptions to f L 2 0, T ; Z, u 0 H, where Z = {v H 1 0 Ω : divv = 0}, and H = {v L 2 Ω : divv = 0, v n = 0}, then the existence of a pressure is only proved in a distributional sense see e.g. [33]. 3. Inhomogeneous parabolic saddle point problem In this section, we present the proof of the main Theorem 2.2. We will employ a semi-discretization in time approach, in order to fully utilize the inf-sup condition. In particular, we first obtain a-priori estimates for the semi-discrete solutions, and then we pass to the limit following the approach of [33, Chapter 3, Section 4] The semi-discrete in time approximation Let N be an integer, set = T N, and let {tm } N m=0, denote the partition points of [0, T ], where t m = m, with m = 0,..., N. We recursively define a family of elements of X, M, denoted by {u m } N, {p m } N respectively where u m, p m are in some sense approximations of functions u., p. respectively, on the interval t m 1 < t t m, with m = 1,..., N. Here, we denote by u 0 u0. Taing into account that g H 1 0, T ; M C[0, T ]; M we may also define elements g 0, g 2,..., g N of M as: g m = gt m, m = 0,..., N. 3.1

6 6 TITLE WILL BE SET BY THE PUBLISHER If {u i } m 1 i=1, {pi } m 1 i=1, are nown, we can define um, p m as elements of X, M respectively satisfying: { um u m 1, v + νau m, v + Bv, p m = 0 v X Bu m, q = g m, q M,M q M 3.2 We will also impose the compatibility condition Bu0, q = g0, q M,M, for all q M. The existence of the pair u m, p m X M can be easily justified by Theorem 2.1. Indeed, we can rewrite 3.2 as: { 1 um, v + νau m, v + Bv, p m = 1 um 1, v v X Bu m, q = g m, q M,M q M. Note that 1 um 1 + f m X are given data. Moreover, z m Z 1 zm, z m + νaz m, z m 1 zm 2 H + να z m 2 X να z m 2 X. Since the coercivity inequality and the inf-sup condition on B, hold, we may apply Theorem 2.1 to guarantee the existence and uniqueness of a pair u m, p m X M. Moreover, it easy to chec that the following inequality holds: There exists a positive constant C, depending only on α, β, ν, Ω such that u m X + p m M C 1 um 1 H + f m X + g m M For m = 1,..., N,we define the following auxiliary functions: u : [0, T ] X, u t = u m, t t m 1, t m ] p : [0, T ] M, p t = p m, t t m 1, t m ] ū : [0, T ] H,. ū is continuous, linear on each subinterval t m 1, t m ], and ū t m = u m. We also note that due to the inf-sup condition we may decompose u m X as u m = w m + z m, where w m Z and z m Z, and for all m = 1,.., N. We also define functions w : 0, T ] Z, in a similar fashion. The next lemma relates various quantities of the semi-discrete in time values g m in terms of regularity properties on data g. Lemma 3.1. Let g m be defined as in 3.1 and g H 1 0, T ; M. Then, g m M C <, gm g m 1 2 M C <. 3.3 Proof. The first estimate is obvious. For the second one, using standard calculations, Hölder s inequality and the fact that g t L 2 0, T ; M, we deduce that gm g m 1 M 1 t m g t t M dt 1 t m t m 1 1/2 t m 1 g t t 2 M dt Hence 3.4 implies, gm g m 1 t m 2 M g t 2 M dt. 3.5 t m 1

7 TITLE WILL BE SET BY THE PUBLISHER 7 Adding inequalities 3.5, we conclude gm g m 1 2 M N t m T g t 2 M dt g t t 2 M dt <. t m 1 0 We now derive estimates for the approximation pair u m, p m A priori estimates First we derive a priori estimates for w m Z using Lemma 3.1 and the inf-sup condition. Subsequently, we establish a priori estimates for the z m Z terms based on estimates on w m. Lemma 3.2. Assume that the bilinear forms are continuous and satisfy Suppose that g H 1 0, T ; M, u 0 u0 X are given data, g m, m = 0,...N are defined as in 3.1 with Bu 0, q = g0, q M.M, for all q M. Let u m, p m X M, m = 1,..., N satisfy 3.2. Then, wm w m 1 2 X C <, and w m 2 X C <, 3.6 where C > 0 depends only upon Ω, β. Proof. Note that Bu m u m 1, q = g m g m 1, q M,M condition, the fact that z m z m 1 Z, and 3.2 q M, for all m = 1,..., N, so using the inf-sup wm w m 1 1/2 X C sup q M C sup q M C sup q M B wm w m 1 1/2, q q M B um u m 1 1/2 gm g m 1, q 1/2. q M, q + B zm z m 1, q 1/2 q M Note that 3.4 implies gm g m 1 1/2 M, since gm g m 1 t m 1 1/2 M g t t 2 M dt 2 <. t m 1 Therefore, we deduce, or equivalently, squaring both sides, wm w m 1 1/2 X C sup q M gm g m 1, q 1/2 C gm g m 1 q M 1/2 M, 1 wm w m 1 2 X C gm g m 1 2 M.

8 8 TITLE WILL BE SET BY THE PUBLISHER Hence, the above inequality together with 3.3, wm w m 1 2 X C gm g m 1 2 M C g t 2 L 2 0,T ;M <. The other estimate is an immediate consequence of the inf-sup condition applied to Bu m, q = g m, q which states that w m X C g m M C < by 3.3, and hence, N wm 2 X C N gm 2 M C <. It remains to estimate several quantities related to {z m } N. Lemma 3.3. Suppose that the assumptions of Lemma 3.2 hold and the bilinear forms are continuous and satisfy Then, z m H C <, z m 2 X C <, z m z m 1 2 H C <, 3.7 where C denotes constants depending only upon Ω,α, β, and ν. Proof. We start from 3.2 and we substitute u m, u m 1 by their decomposition i.e., u m = w m + z m, u m 1 = w m 1 + z m 1, z m z m 1, v + νaz m, v + Bv, p m = w m w m 1, v νaw m, v. 3.8 Set v = 2z m Z and note that Bz m, p m = 0. Therefore, z m 2 H z m 1 2 H + z m z m 1 2 H + 2να z m 2 X C w m w m 1 H z m X + Cν w m X z m X 1 να z m 2 X + C ν wm w m 1 2 H + ν w m 2 X. Here, C is a constant depending on α and on the domain. Hence, z m 2 H z m 1 2 H + z m z m 1 2 H + να z m 2 X C ν wm w m 1 2 H + Cν w m 2 X. Using the above relation recursively, we obtain z N 2 H + z m z m 1 2 H + να C z 0 2 H + 1 ν z m 2 X w m w m 1 2 H + ν w m 2 X. Equations 3.6 of Lemma 3.2 guarantee that the last two sums are finite. Lemma 3.4. Under the assumptions of Lemma 3.3, the following estimates hold: 1 z m z m 1 2 H C <, and u m X C < for all m = 1,..., N, where C denotes constants depending only upon Ω,α, β, and ν.

9 TITLE WILL BE SET BY THE PUBLISHER 9 Proof. We start from equation 3.2, and we substitute u m, u m 1 by their decomposition, i.e., u m = w m + z m, u m 1 = w m 1 + z m 1, where z m, z m 1 Z and w m, w m 1 Z. { 1 zm z m 1, v + νau m, v + Bv, p m = 1 wm w m 1, v Bu m, q = g m 3.9, q M,M q M. Set v = z m z m 1 into 3.9. Therefore, after noting that Bz m z m 1, p m = 0, 1 zm z m 1 2 H + νau m, z m z m 1 wm w m 1 H z m z m 1 H 1 4 zm z m 1 2 H + wm w m 1 2 H. Using the decomposition once more together with 2.6, we rewrite the bilinear term as follows Combining the last two inequalities, Au m, z m z m 1 = Au m, u m u m 1 Au m, w m w m um 2 X 1 2 um 1 2 X Au m, w m w m zm z m 1 2 H + ν 2 um 2 X ν 2 um 1 2 X wm w m 1 2 H + νau m, w m w m 1 wm w m 1 2 H + ν u m X w m w m 1 X wm w m 1 2 H + ν 2 wm w m 1 2 X + ν 2 um 2 X Using the above relation recursively from m = 1 to m = N, we obtain 1 C z m z m 1 2 H + ν u N 2 X u 0 2 X + 1 w m w m 1 2 H + ν w m w m 1 2 X + ν u m 2 X. Lemmas guarantee that the above sums are finite. Returning to 3.10, and summing from 1 to m, we easily obtain u m 2 X C <. Collecting the estimates of Lemmas , we obtain the main stability estimates. Theorem 3.5. Assume that the bilinear forms A.,., B.,. are continuous, satisfy , and let g H 1 0, T ; M. Let u 0 = u0 X be given data, g m, m = 0,...N be defined as in 3.1 with Bu0, q = g0, q M.M g 0, q M,M, for all q M. Let u m, p m X M, m = 1,..., N satisfy 3.2. Then, the following quantities are bounded by constants C < depending only upon Ω, α, β, ν: u m 2 X, p m 2 M, u m u m 1 2 H, 1 1 u m u m 1 2 H, u m u m 1 2 X, um X for all m = 1,..., N.

10 10 TITLE WILL BE SET BY THE PUBLISHER Proof. Note that Lemmas and the triangle inequality imply the first three estimates for the sums of u m terms. From the inf-sup condition, it is clear that or equivalently, p m M C um u m 1 H + u m X + g m M N p m 2 M C um u m 1 2 H + u m 2 X + g m 2 M. It is now obvious that the desired estimate for the pressure holds, due to the first three estimates. Taing the supremum over v X into the first equation of 3.2, we obtain N um u m 1 2 X <. In order to estimate u m X we simply need to estimate u m H C <, since Lemma 3.4 states that u m X C <, for all m = 1,..., N. For this purpose, we return to 3.2 and set v = u m and q = p m. Thus, we deduce, u m u m 1, u m + νau m, u m = g m, p m M,M /2 g m 2 M + /2 pm 2 M. The proof now follows upon summing the above inequalities from 1 to m, and using the previous bounds on N gm 2 M and N i=1 pm 2 M. Now we are ready to prove the main Theorem 2.2 by using the above a-priori bounds of the auxiliary functions on the semi-discretized in time approach Proof of Theorem 2.2: The proof is similar to the one of [33, Chapter III, Section 4]. The functions u, ū, p defined as above, together with Theorem 3.5 remain bounded in L 2 0, T ; X L 0, T ; H, and L 2 0, T ; M respectively. Also note that ū t remains bounded in L 2 0, T ; H. Indeed, these are simply the interpretations of the stability estimates of Theorem 3.5. Moreover, [33, Lemma 4.8, pp 328] implies that u ū 0 in L 2 0, T ; H as. Therefore, we can extract subsequences, still denoted by u, ū, p, such that u u wealy in L 2 0, T ; X, u u wealy-* in L 0, T ; H p p wealy in L 2 0, T ; M, ū u wealy in L 2 0, T ; X ū u wealy in L dū 0, T ; H, du wealy in L 2 0, T ; X. dt dt But [33, Lemma 4.8, pp 328], also implies that u = u. Note also that the classical Aubin-Lions compactness Lemma see [33, Chapter 3, Section 3] implies that ū u strongly in L 2 0, T ; H, since X H X form an evolution triple and X H with compact embedding. It is evident that the limit u, p is the solution of 2.5. Indeed, using the definitions of the auxiliary functions, we can rewrite the equations 3.2 as: { dū t dt, v + νau, v + Bv, p = 0 v X Bu, q = g, q M,M q M, 3.11 where g is defined by: g t = g m, t t m 1, t m ]. Woring identically to [33, Lemma 4.9, pp 429] we obtain that g g wealy in L 2 0, T ; M.

11 TITLE WILL BE SET BY THE PUBLISHER 11 Hence, using the convergence results together with the continuity properties of the bilinear forms we pass the limit into 3.11 to obtain 2.5. The improved regularity on u L 0, T ; X is evident by the estimate of Theorem 3.5. The regularity on w t is due to the estimate of Lemma Applications to evolutionary problems with inhomogeneous data We apply the main Theorem 2.2 in order to prove the existence and uniqueness of a solution pair u, p of problems with inhomogeneous boundary and/or divergence data. First, we begin by treating the evolutionary Stoes problem with inhomogeneous divergence data, but with zero boundary condition. In particular, given u 0 and g, we see velocity u and pressure p such that, u t ν u + p = 0 in Ω 0, T ] div u = g in Ω 0, T ] u = 0 on Γ 0, T ] u0 = u 0 in Ω. First, we recast our problem into the parabolic saddle point framewor. Assume that X = H 1 0 Ω, H = L 2 Ω, M = L 2 0Ω and define the standard bilinear forms, Au, v = Ω u : v, Bv, q = Ω divv q for all u, v H 1 0 Ω, q L 2 0Ω. Here we denote u : v = d i,j=1 u i,jv i,j, with the second index denoting the derivative with respect to x j. Theorem 4.1. Suppose that u 0 H0 1 Ω, and g H 1 0, T ; L 2 0Ω with g0 = div u0. Then, there exists a unique wea solution pair u, p of 4.1 in the sense of 1.2, satisfying: 4.1 u L 0, T ; H 1 0 Ω H 1 0, T ; L 2 Ω, p L 2 0, T ; L 2 0Ω. For a.e. t 0, T ] let Z = {u H 1 0 Ω : Bu, q = 0, q L 2 0Ω}. Then, if u. is decomposed to u. = z. + w., for a.e t 0, T ] with z. Z, w. Z, we obtain, w t L 2 0, T ; H 1 0 Ω. Proof. It is an immediate consequence of the main Theorem 2.2. Indeed, note that Z {u H0 1 Ω : divu=0}, and hence the continuity and coercivity conditions can be easily proven see e.g. [15], as in the elliptic case. The second result, is an immediate consequence of the inf-sup condition see also regularity estimate of Lemma 3.2. The second application of Theorem 2.2 is the Lagrange multiplier method for a wea solution of the evolutionary Stoes, with inhomogeneous Dirichlet boundary data, i.e., the problem u t ν u + p = 0 in Ω 0, T div u = 0 in Ω 0, T u = φ on Γ 0, T u0 = u 0 in Ω, together with the compatibility condition Γ φ., t n. = 0 for a.e. t 0, T ]. In this problem, we enforce the boundary condition wealy which implies that we need to introduce an additional variable, the Lagrange multiplier λ corresponding to the boundary stress. Our preferred wea formulation, now can be defined as follows: We see u L 2 0, T ; H 1 Ω H 1 0, T ; H 1 Ω, p L 2 0, T ; L 2 0Ω and λ L 2 0, T ; H 1/2 Γ 4.2

12 12 TITLE WILL BE SET BY THE PUBLISHER such that for a.e. t 0, T ], and for all v H 1 Ω, q L 2 0Ω, s H 1/2 Γ, u t t, v + ν ut, v pt, divv λt, v H 1/2 Γ,H 1/2 Γ = 0 div ut, q = 0 ut, s H 1/2 Γ,H 1/2 Γ = φt, s H 1/2 Γ,H 1/2 Γ u0, v = u 0, v. 4.3 It is evident that if φ n = 0, and u, p, λ sufficiently smooth see e.g. [32], then the formulation 4.3 is Γ equivalent to 4.2. Next we put 4.3 into our parabolic saddle point framewor. For this purpose, we define X = H 1 Ω, M = L 2 0Ω H 1 2 Γ and we denote by au, v = u : v, bv, q = divv q, Ω Ω the standard bilinear forms associated to the evolutionary Stoes problem. Note that au, u u 2 X denotes a semi-norm, and satisfies 2.4. Now, it is clear that we can recast 4.3 as a parabolic saddle point problem by simply defining the bilinear forms, for a.e. t 0, T ], for all u., v H 1 Ω, q L 2 0Ω, s H 1/2 Γ, Au., v au., v, Bv., q, s bv., q v, s H 1/2 Γ,H 1/2 Γ. Then, problem 4.3 can be written as a parabolic saddle point problem 1.2, as follows: For a.e. t 0, T ], and for all v H 1 Ω and q, s L 2 0Ω H 1/2 Γ, u t, v + νau, v + Bv, p, λ = 0 Bu, q, s = φ, s H 1/2 Γ,H 1/2 Γ 4.4 u0, v = u 0, v. It remains to define the space Z = {u H 1 Ω : Bu, q, s = 0 q, s L 2 0Ω H 1/2 Γ}, upon which the coercivity condition on A.,. should be verified. We are ready to prove our main result. Theorem 4.2. Suppose that φ H 1 0, T ; H 1 2 Γ with φ0, s H 1/2 Γ,H 1/2 Γ = u0, s H 1/2 Γ,H 1/2 Γ, for all s H 1 Γ, and u 0 H 1 Ω, with divu 0 =0, then there exists a unique solution satisfying system 4.4. u L 0, T ; H 1 Ω H 1 0, T ; L 2 Ω, p L 2 0, T ; L 2 0Ω, λ L 2 0, T ; H 1 2 Γ Proof. It is easy to prove the continuity and coercivity assumption on Z for the bilinear form A,, since Z H0 1 Ω. The continuity of the bilinear form B, is also evident. Then the proof follows directly from Theorem 2.2, since the inf-sup condition is proved in [20, Proposition 3]. Remar 4.3. We note that more spacial regularity can be recovered, under additional assumptions. Indeed, the fact that u t L 2 0, T ; L 2 Ω may be used to improve the spacial regularity of u in a standard fashion and hence to recover a strong solution, by exploring techniques of parabolic regularity and classical boot-strap arguments provided that some additional compatibility conditions, and smoothness on the boundary are assumed for instance Γ C 1,1. For evolutionary Stoes equations, with inhomogeneous Dirichlet boundary data, if φ L 2 0, T ; H 3/2 Γ H 3/4 0, T ; L 2 Γ then we can recover L 2 0, T ; H 2 Ω regularity for the strong solution see for instance [32]. We note that the case of convex and polygonal domains requires further attention see for instance [17] since the polygonal structure of the domain acts as a barrier for higher regularity. However, for our analysis including the error estimates of the semi-discrete scheme, L 2 0, T ; H 2 Ω regularity for the velocity will not be necessary.

13 TITLE WILL BE SET BY THE PUBLISHER 13 Remar 4.4. Even though the regularity on g, u 0 is not optimal, compared to the notion of very wea solutions of [30], the above formulation clearly represents the parabolic analog of saddle point theory. Combining the above results, we may obtain the existence of a wea solution of the evolutionary Stoes equations, with inhomogeneous divergence and Dirichlet boundary data. We will treat the inhomogeneous Dirichlet boundary data for the evolutionary Stoes problem, as a parabolic saddle point problem by using a Lagrange multiplier principle similar to the elliptic case see e.g. [4]. As before, we denote by au, v = Ω u : v, bv, q = divv q. Ω Introducing the Lagrange multiplier, the wea formulation is given as follows: See u, p L 2 0, T ; H 1 Ω H 1 0, T ; H 1 Ω L 2 0, T ; L 2 Ω, and a Lagrange multiplier λ L 2 0, T ; H 1/2 Γ such that, for all v H 1 Ω, q L 2 Ω and s H 1/2 Γ, and for a.e. t 0, T ], u t, v + νau, v + bv, p λ, v H 1/2 Γ,H 1/2 Γ = 0 bu, q = ψ, q u, s H 1/2 Γ,H 1/2 Γ = φ, s H 1/2 Γ,H 1/2 Γ u0, v = u 0, v. In order to recast problem 4.5 as a parabolic saddle point problem, we define by and Au, v = au, v u, v H 1 Ω Bu, q, s = bu, q u, s H 1/2 Γ,H 1/2 Γ q, s L 2 0Ω H 1/2 Γ, Z = {u H 1 Ω : Bu, q, s = 0, q, s L 2 0Ω H 1/2 Ω}. Then, problem 4.5 can be rewritten as follows: For a.e. s H 1/2 Γ, 4.5 t 0, T ], for all v H 1 Ω, q L 2 0Ω and { ut, v + νau, v + Bv, λ, p = 0 Bu, s, q = ψ, q φ, s H 1/2 Γ,H Γ. 1/2 4.6 Theorem 4.5. Given initial and boundary data satisfying u 0 H 1 Ω, φ H 1 0, T ; H 1/2 Γ, ψ H 1 0, T ; L 2 Ω, and the compatibility conditions divu 0, q = ψ0, q for all q L 2 0Ω and u0, s H 1/2 Γ,H 1/2 Γ = φ0, s H 1/2 Γ,H 1/2 Γ for all s H 1/2 Γ, there exists a unique wea solution u, p, λ L 0, T ; H 1 Ω H 1 0, T ; L 2 Ω L 2 0, T ; L 2 0Ω L 2 0, T ; H 1/2 Γ of the wea problem 4.6. Let u be decomposed to u. = z. + w., with z. Z and w. Z for a.e. t 0, T ]. Then w t L 2 0, T ; H 1 Ω. Proof. Note that the continuity and coercivity assumption on bilinear form A.,. can be easily verified, since Z H0 1 Ω. It remains to prove the inf-sup condition, which can be verified identically to Theorem 4.2, since the bilinear form B, is defined as in Theorem 4.2.

14 14 TITLE WILL BE SET BY THE PUBLISHER 5. Finite element approximations of parabolic saddle point problems We now turn our attention to the error analysis of finite element approximations of such parabolic saddle point problems. The main goal is to derive best approximation type of estimates for semi-discrete in space approximations Preliminaries and assumptions Let V h X and M h M be standard finite element spaces, associated to the approximation of elliptic saddle point problems see e.g. [5, 15] satisfying the classical approximation theory properties: There exists an integer, and a constant C, independent of h such that, v H m+1 Ω X, 0 m and q H m Ω M, 0 m, the following inequalities hold: inf vh V h v vh X Ch m v H m+1 Ω X inf v h V h v vh H Ch m+1 v H m+1 Ω X inf qh M h q qh M Ch m q Hm Ω M. In addition, we assume that the discrete analog of the inf-sup condition holds for our choice of subspaces V h and M h : Bu h, q h inf sup 0 q h M h 0 u h V u h h X q h β, 5.2 M with β > 0 and independent of the discretization parameter h. First we note that it is possible to construct finite element spaces satisfying 5.1 and 5.2. Indeed, for the model problem 4.1, we may consider standard finite element spaces V0 h X H0 1 Ω, M h M L 2 0Ω, satisfying standard approximation properties 5.1 and the classical discrete inf-sup condition 5.2. For model problem 4.2, we use X = H 1 Ω, H = L 2 Ω for the velocity and M = M 1 M 2, with M 1 = L 2 0Ω for the pressure, and M 2 = H 1/2 Γ for the Lagrange multiplier term respectively. Therefore, we consider V h1 H 1 Ω, for the velocity and M h = M h1 1 M h2 2 L 2 0Ω H 1/2 Γ for the pressure and the boundary data respectively. We also assume that V h1 and M h1 1 satisfy the standard approximation properties 5.1. The approximation properties of M h2 2, in terms of the given regularity assumptions on data as well as on the boundary regularity are more complicated see for instance [20], since in 4.2 the computation of the velocity and pressure is coupled to that of the boundary stress terms. Hence, in order to satisfy the discrete inf-sup condition 5.2, the choice of M h2 2 should be related to that of V h1 and M h1 1. To this end, we first treat the case of convex and polygonal polyhedral in R 3 domains. We choose V h1 H 1 Ω and M h1 1 L 2 Ω such that the spaces V h1 0 = V h1 H0 1 Ω, and M1 h = M h1 1 L 2 0Ω satisfy 5.1 and the discrete inf-sup condition 5.2. Then, we choose M h2 2 H 1/2 Γ note that h 2 might be different from h 1 such that the following approximation and inverse estimates hold see e.g. [8] and [20]: There exists a constant C > 0 and an integer, 0 m, such that, 5.1 { infφ h M h φ φ h M2 Ch m 2 φ 2 H m 1, φ H m 1 2 Γ, 0 m 1, 2 Γ M 2 inf φ h M h φ φ h M2 Ch m 2 infû H m Ω,û Γ=φ û H m 2 Ω, φ H m Ω Γ, 1 m, 5.3 and φ h H s Γ Ch t s 2 φ h H t Γ, φ h M h2 2, 1/2 t s 1/2. Then, under the above assumptions we finally, set h = max{h 1, h 2 } and V h V h1, M h = M h1 1 M h2 2. Remar Despite the fact that the choice of M h2 2 is independent of the pair V h1, M h1 the dimension of M h2 2 can not exceed the one of V h1 Γ.

15 TITLE WILL BE SET BY THE PUBLISHER 15 2 The verification of the discrete inf-sup condition of the above pair V h and M h typically requires the existence of a suitably large constant C, such that h 2 Ch 1 see for example [20, Proposition 5]. 3 For the choice h = h 1 = h 2, and M h 2 = V h Γ, we note that M h 2 C Γ, and hence M h 2 H 1 Γ. It is clear that the first inequality of 5.3 is valid due to the approximation properties of V h, see [20, Lemma 13 and Proposition 14] at least when φ H m 1 2 Γ, 0 m 1. For the second inequality of 5.3, we note that for convex polyhedral domains, in general, it is not possible to define H s Γ when s > 1. Despite this fact, if v H s+ 1 2 Ω, with s > 1, it is still expected that its trace has approximation properties compatible with its regularity on Ω. We refer the reader to [20, Section 3] for a detailed discussion. The above approximation properties easily result to approximation properties in time-space spaces. For example see also [22, Section 2], there exists an integer and a constant C independent of h such that v L 2 0, T ; H m+1 Ω X, 0 m and q L 2 0, T ; H m Ω M 1, 0 m, the following inequalities hold: inf vh L 2 0,T ;V h v v h L 2 0,T ;X Ch m 1 v L 2 0,T ;H m+1 Ω X inf v h L 2 0,T ;V h v v h L2 0,T ;H Ch m+1 1 v L2 0,T ;H m+1 Ω X inf qh L 2 0,T ;M1 h q q h L2 0,T ;M 1 Ch m 1 q L2 0,T ;H m Ω M 1. Similarly, there exists a constant C > 0 and an integer, 0 m, such that, inf φ φ h L 2 0,T ;M2 h φh L 2 0,T ;M 2 Ch m 2 φ L 2 0,T ;H m 1, φ 2 Γ M L2 0, T ; H m 1 2 Γ, 0 m 1, 2 inf φ φ h L 2 0,T ;M2 h φh L2 0,T ;M 2 Ch m 2 infû L 2 0,T ;H m+1 Ω,û Γ=φ û L 2 0,T ;H m Ω X, φ L 2 0, T ; H m Ω Γ, 1 m. We will frequently combine the approximation properties of M 1 and M 2, using the space M = M 1 M 2, by denoting q q, φ M = M 1 M 2. In this case, recall that h = max{h 1, h 2 }, V h = V h1, M h = M h1 1 M h2 2. Then, the approximation property is stated as follows: q = q, φ such that q L 2 0, T ; H m Ω M 1, 0 m and φ L 2 0, T ; H m 1 2 Γ M 2, there exists a constant C > 0 such that, inf q q h L2 0,T ;M Ch m. q h L 2 0,T ;M h As before, we will abuse the notation to denote q h = q h, p h = p h, etc. To formulate the discrete analog of 2.5 we define the discretely divergence and/ or divergence-free analogs of the above finite element spaces by Z h g {x h. V h with Bx h., q h = g., q h q h M h for a.e t 0, T ]}. Note that Z h 0 Z h, where Z h = {v h V h : Bv h, q h = 0 q h M h }. Here, the bilinear form B.,. is defined in a similar spirit as in Section 4, i.e., it contains all boundary terms resulting from integration by parts, and related pressure terms. The semi-discrete in space finite element approximations of parabolic saddle point problem, can be defined as follows: Given u h 0 V h, and g H 1 0, T ; M we see a discrete solution pair u h, p h H 1 0, T ; V h L 2 0, T ; M h satisfying, for a.e. t 0, T ] u h t t, v h + νau h t, v h + Bv h, p h t = 0 v h V h Bu h t, q h = gt, q h q h M h u h 0 u h 0, v h = 0 v h V h. 5.4

16 16 TITLE WILL BE SET BY THE PUBLISHER Throughout the remaining of this wor, we assume that the discrete initial data u h 0 are chosen in a way to satisfy the standard approximation property, u 0 u h 0 H Ch m u 0 Hm Ω. Now, we turn our attention to the case of smooth domains for simplicity in R 2. In this case, it is assumed that the domain can be approximated appropriately by the corresponding finite element domain in the sense of [20, Section 3.4] or [34], and an approximation φ h of the boundary data φ is actually computed. As a consequence, we may construct our subspaces on the approximated polygonal domain, as above, while the second equation of the discrete formulation 5.4 is now modified to Bu h., q h = g h., q h q h M h, and for a.e t 0, T ]. This case is also very important within the context of optimal control problems, where the control is applied on the Dirichlet part of the boundary, and it is an actual unnown. We view this case as the essential data case. Typical choices for the approximation of g are the L 2 projections. For instance, recall that for the boundary data φ, one may choose the L 2 Γ projection operator from L 2 Γ to M h2 2. The rest of this Section is organized as follows: First, in Section 5.2, we consider 5.4, with g fixed which covers the case of convex and polygonal or polyhedral domains, while the case of smooth domains and the case of essential boundary data where g is approximated by an element g h will be treated subsequently in Section 5.3. In both cases, the role of the discrete inf-sup condition is carefully analyzed Preliminary best approximation estimates The ey difference between the inhomogeneous divergence and boundary data case, and the homogeneous one concerns the treatment of the inhomogeneous divergence data constraint equation. In addition, we note that the coupling between u t and p creates additional difficulties, within the context of numerical approximations. In order to obtain estimates for the differences u t u h t, and p p h, we will need to define various projections that satisfy the best approximation properties. We emphasize that we are interested in estimates at the natural energy norms, u t u h t L 2 0,T ;X and p p h L 2 0,T ;M respectively. We note that even for the homogeneous evolutionary Stoes equations the estimates on the pressure and the time-derivative are both suboptimal, due to the coupling between the time-derivative and the pressure through the incompressibility constraint. For this purpose, we will follow the techniques of [22]. We denote by P h the H projection P h : H V h such that P h v, w h = v, w h w h V h and by P h Z the discretely divergence-free analog, P h Z : H Zh which satisfies, P h Zv, z h = v, z h z h Z h. We also assume that P h satisfies stability properties in. X and. H norms, while PZ h stability property in. H. In particular, v X, satisfy the standard P h v X C v X, P h Zv H C v H. 5.5 In addition, the following inverse estimate P h v X C/h P h v H will be frequently used. We also note that P h Z v X C v X for all v X Z. Then, the following properties hold see e.g. [22, Section 2] for projections P h, P h Z in L2 0, T ; X: There exists a constant C > 0 independent of h, such that { v P h v L 2 0,T ;X 0 as h 0, v L 2 0, T ; X, v P h Z v L 2 0,T ;X C v L2 0,T ;X, v L 2 0, T ; X Z.

17 TITLE WILL BE SET BY THE PUBLISHER 17 Finally, there exists constant C and an integer such that for 0 m, the following error estimates for the projections P h, PZ h hold respectively: v P h v L 2 0,T ;X Ch m v L 2 0,T ;H m+1 Ω v P h v L2 0,T ;H Ch m+1 v L2 0,T ;H m+1 Ω v PZ hv L 2 0,T ;X Ch m v L2 0,T ;H m+1 Ω v L 2 0, T ; H m+1 Ω X, v L 2 0, T ; H m+1 Ω X, v L 2 0, T ; H m+1 Ω X Z. In the subsequent proposition, we obtain the basic estimate, which relates the error u u h to the best approximation error u x h, where x h L 2 0, T ; V h Z h g H 1 0, T ; V h and x h t L 2 0, T ; Z h g t. Note that if x h L 2 0, T ; V h Z h g and x h H 1 0, T ; V h then x h t L 2 0, T ; Z h g t, since we assume that the bilinear form B.,. does not contain time-dependent coefficients. We are now ready to obtain the preliminary best approximation estimate in Z h g for the velocity, while for the pressure the discrete inf-sup condition is needed, similar to the elliptic case see [15] for the stationary Stoes case. Theorem 5.2. Let g, u 0 satisfy the regularity assumptions of Theorem 2.2 and that the continuous bilinear forms A.,., B.,. satisfy the coercivity conditions Assume that u, u h are the solutions of the parabolic saddle point problem 1.2 and of the discrete parabolic saddle point problem 5.4 respectively. Moreover, let Az h, z h C z h 2 X, zh Z h. Suppose also that u h 0 V h. Then, for any arbitrary x h H 1 0, T ; V h L 2 0, T ; Z h g, q h L 2 0, T ; M h the following estimate holds: u u h 2 L 0,T ;H + u uh L2 0,T ;X C u 0 u h 0 H + inf p q h L 2 0,T ;M 5.6 q h L 2 0,T ;M h + inf u x h L 2 0,T ;X + u t x h t L 2 0,T ;X. x h H 1 0,T ;V h L 2 0,T ;Z h g Proof. The orthogonality condition states that for almost every t 0, T ] { u h t u t, v h + νau h u, v h + Bv h, p h p = 0 v h V h Bu h u, q h = 0 q h M h. 5.7 Let x h L 2 0, T ; V h Z h g, q h L 2 0, T ; M h be arbitrary elements. Then, adding and subtracting x h in 5.7, we obtain, u h t x h t, v h + νau h x h, v h + Bv h, p h p 5.8 = x h t u t, v h Ax h u, v h v h X h. Note that u h x h Z h, and hence Bu h x h, p h p = Bu h x h, q h p for any q h L 2 0, T ; M h. Setting v h = u h x h in 5.8 and using the coercivity inequality on Z h we obtain, 1 d 2 dt uh x h 2 H + C u h x h 2 X C x h t u t 2 X + xh u 2 X + q h p 2 M. The last inequality clearly implies estimate 5.6 by standard Grönwall Lemma. Proposition 5.3. Suppose that the assumptions of Theorem 5.2 and the discrete inf-sup condition 5.2 for the choice of V h, M h hold. Then, for any arbitrary x h H 1 0, T ; V h L 2 0, T ; Z h g, q h L 2 0, T ; M h the

18 18 TITLE WILL BE SET BY THE PUBLISHER following estimates hold: u t u h t L 2 0,T ;X + p p h L 2 0,T ;M C u 0 u h h 0 H + inf p q h L 2 0,T ;M q h L 2 0,T ;M h + inf u x h L 2 0,T ;X + u t x h x h H 1 0,T ;V h L 2 0,T ;Z h t L 2 0,T ;X, g u t u h t L 2 0,T ;Z C u 0 u h 0 H + inf p q h L 2 0,T ;M q h L 2 0,T ;M h + inf u x h L 2 0,T ;X + u t x h t L 2 0,T ;X. x h H 1 0,T ;V h L 2 0,T ;Z h g Here C > 0 denotes a constant depending only on Ω, ν, α, β. Proof. We begin by estimating the time-derivative. First, we note that if x h L 2 0, T ; Z h g then for a.e t 0, T ], we obtain u h. x h. Z h, and u h t. x h t. Z h. Recall, u h t. x h t. X = sup v X u h t. xh t.,v Adding and subtracting PZ h v, we obtain, u h t. x h t. X u h t. x h t., v PZ h = sup v + uh t. x h t., PZ hv. v X v X v X. Note that since u h t. x h t. Z h the definition of the projection PZ h implies that uh t. x h t., v PZ h v = 0. For the remaining term, from the orthogonality condition 5.8, we obtain ν Au u h t. x h h. x h., PZ h t. X sup v + BP Z hv, ph p v X v X v X + u t. x h t., P h Z v v X + Axh u, PZ hv. 5.9 v X Observe that BP h Z v, ph p = BP h Z v, qh p. Then, using the inverse estimate P h Z v X C h P h Z v H, squaring both sides, and integrating with respect to time we obtain the desired estimate. Once, we have shown an estimate on the time derivative on u t u h t L2 0,T ;X the estimate on the p p h term follows directly from the discrete inf-sup condition 5.2. Indeed, note that Bv h, p h q h = Bv h, p h p Bv h, p q h = u h t u t, v h νau h u, v h Bv h, p q h C u h t u t X v h X + u h u X v h X + v h X p q h M. Hence, dividing by v h X, taing the supremum over V h, using the discrete inf-sup 5.2 and standard algebra we derive the estimate on p p h term. At the second equality, we have used the orthogonality condition. For the last estimate, we note that u h t. x h u t. Z = sup h t. xh t.,v v Z v X and hence woring identically as above we obtain the analogue of 5.9, ν Au u h t. x h h. x h., PZ h t. Z sup v + BP Z hv, ph p v Z v X v X + u t. x h t., P h Z v v X + Axh u, P h Z v v X. 5.10

19 TITLE WILL BE SET BY THE PUBLISHER 19 The estimate now follows using similar arguments and the stability estimate P Z v X C v X for all v Z X. Remar 5.4. The structure of the estimate 5.10 on u h t x h t L 2 0,T ;Z is similar to the estimate of the velocity in L 2 0, T ; X and hence it leads to similar rates. In particular, we have shown the following best-approximation and almost symmetric error estimate: u u h L 2 0,T ;X + u t u h t L 2 0,T ;Z C u 0 u h 0 H + inf p q h L2 0,T ;M q h L 2 0,T ;M h + inf u x h L2 0,T ;X + u t x h t L2 0,T ;X. x h H 1 0,T ;V h L 2 0,T ;Z h g However, the above estimate is not useful since we cannot apply the inf-sup condition to recover an estimate for the pressure and the Lagrange multiplier term. Indeed, despite the fact that the estimate in L 2 0, T ; Z is of the same order to the velocity one, it seems unliely to obtain a better rate in L 2 0, T ; X norm simply because Bv, q = 0, v Z, q M. The reduced rate for the estimate on the time derivative and the pressure is even present for the homogeneous evolutionary Stoes equations, and it is due to the coupling between the time-derivative and the pressure. The rate reduction was caused because we have used a suboptimal bound for the time-derivative in L 2 0, T ; X norm by applying an inverse estimate. The inverse estimate was necessary since we cannot assume the stability property P Z v X C v X for any v X, but only if v X Z. On the other hand, the definition of the projection, implies the stability in the H norm. Next, we will relate the approximation properties on H 1 0, T ; V h L 2 0, T ; Z h g to standard best approximation properties on H 1 0, T ; V h. This is necessary in order to quantify the error estimate. The discrete inf-sup condition will be used similar to the elliptic case see [15, Theorem 1.1, pp 114]. We note that we will use the enhanced regularity w t H 1 0, T ; X, in order to obtain estimate on the time derivative via the inf-sup condition. Recall, that we have shown that u t L 2 0, T ; H and if the decomposition u. = w. + z., with w. Z, z. Z holds for a.e. t 0, T ] then w t L 2 0, T ; X. Lemma 5.5. Let the assumptions of Theorem 5.2 hold. In addition, suppose that the finite element subspaces V h, M h satisfy the discrete inf-sup condition 5.2. Then, for any v h H 1 0, T ; V h, q h L 2 0, T ; M h the following estimates hold: u u h L 0,T ;H + u u h L 2 0,T ;X + +h u t u h t L2 0,T ;X + p p h L2 0,T ;M C u 0 u h 0 H + inf v h H 1 0,T ;V h u v h L 2 0,T ;X + u t vt h L 2 0,T ;X + inf v h H 1 0,T ;V h w t vt h L 2 0,T ;X + p q h L 2 0,T ;M. Here we denote by u. = w. + z., where z. Z, w. Z for a.e. t 0, T ]. If in addition, u H 1 0, T ; H m+1 Ω X, and p L 2 0, T ; H m Ω M then there exists a constant C such that u u h L 0,T ;H + u u h L2 0,T ;X +h u t u h t L2 0,T ;X + p p h L2 0,T ;M Ch m. Proof. If the finite element subspaces V h, M h satisfy the discrete inf-sup condition 5.2 then using the Banach- Babuša-Nečas Lemma see e.g. [5, 15, 26] for a.e t 0, T ] there exists w h. Z h depending on q such that Bw h., q h = Bu. v h., q h, v h L 2 0, T ; V h and q h M h. 5.11

20 20 TITLE WILL BE SET BY THE PUBLISHER In addition, the discrete inf-sup implies w h. X C u. v h. X. Set x h. = w h. + v h. and note that x h. L 2 0, T ; V h, and x h. Z h g, since Bx h., q h = Bw h. + v h., q h = Bu., q h = g., q h, due to Therefore, u. x h. X u. v h. X + w h. X C u. v h. X. The last inequality implies that u x h L 2 0,T ;X C u v h L 2 0,T ;X, for any arbitrary v h L 2 0, T ; V h. Suppose now that v h H 1 0, T ; V h. The unique decomposition of u. = w. + z., with w. Z, z. Z implies that Bw h., q h = Bu. v h., q h = Bw. v h., q h. Hence, since w t, v h t L 2 0, T ; X, differentiating with respect to time, we deduce for a.e. t 0, T ], Bw h t., q h = Bw t. v h t., q h. The discrete inf-sup condition, and the Banach-Babuša-Nečas Lemma see e.g. [5, 15, 26] imply that there exists w h Z h such that B w h., q h = Bw t. v h t., q h. Therefore, we obtain B w h., q h = Bw t. v h t., q h = Bw h t., q h The discrete inf-sup implies w h. X C w t. vt h. X. Note that since w h L 2 0, T ; Z h we also deduce from 5.12 that wt h L 2 0, T ; Z h, and wt h L2 0,T ;X C w t. vt h. X. Hence using triangle inequality and the previous estimates we obtain the following estimate, u t x h t L2 0,T ;X C u t v h t L 2 0,T ;X + w h t L 2 0,T ;X C u t v h t L2 0,T ;X + w t v h t L2 0,T ;X, since wt h L2 0,T ;X is estimated by wt h L2 0,T ;X C w t vt h L2 0,T ;X. The desired estimate easily follows by substituting x h into the estimate of Theorem 5.2 and Proposition 5.3 see also Remar 5.4. The last estimate of the Theorem now follows by the approximation properties. We are ready to state the main best approximation type error estimates for the semi-discrete approximations of problems 4.1 and 4.2. First, we simply point out that we may recast problem 4.1 into the discrete parabolic saddle point framewor of 5.4, for V h H 1 0 Ω, M h L 2 0Ω and by defining the bilinear forms similar to Section 4. Then it is evident that the assumptions of Theorem 5.1, and Proposition 5.2 hold, and hence the estimates of Theorem 5.1, Proposition 5.2, and Lemma 5.3 hold. In particular, we have the following result: Corollary 5.6. Suppose that u 0 H0 1 Ω, and g H 1 0, T ; L 2 0Ω with g0 = div u0. Let V h, M h satisfy the approximation properties of Section 5.1, u h 0 V h an approximation of u 0. Then, there exists a constant C > 0 independent of h, such that, u u h L 0,T ;H + u u h L2 0,T ;X 0. If in addition, u H 1 0, T ; H m+1 Ω X, and p L 2 0, T ; H m Ω M then there exists a constant C such that u u h L 0,T ;H + u u h L 2 0,T ;X +h u t u h t L 2 0,T ;X + p p h L 2 0,T ;M Ch m. Now, for the model problem 4.2, we choose the spaces V h = V h1 for the velocity and M h = M h1 1 M h2 2 for the pressure and the boundary data term, satisfying the assumptions of Section 5.1. Then, denoting by