Optimal error bounds for two-grid schemes applied to the Navier–Stokes equations

Applied Mathematics and Computation 218 (2012) 7034–7051

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Optimal error bounds for two-grid schemes appliedto the Navier–Stokes equations

Javier de Frutos a,1, Bosco García-Archilla b,2, Julia Novo c,⇑,3

a IMUVA, Instituto de Matemáticas, Universidad de Valladolid, Spainb Departamento de Matemática Aplicada II, Universidad de Sevilla, Spainc Departamento de Matemáticas, Universidad Autónoma de Madrid, Spain

a r t i c l e i n f o

Keywords:Two-grid methodsMixed finite elementsNavier–Stokes equationsOptimal error estimates

0096-3003/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.amc.2011.12.051

⇑ Corresponding author.E-mail addresses: [email protected] (J. de Fruto

1 Research supported by Spanish MICINN under gr2 Research supported by Spanish MICINN under gr3 Research supported by Spanish MICINN under gr

a b s t r a c t

We consider two-grid mixed-finite element schemes for the spatial discretization of theincompressible Navier–Stokes equations. A standard mixed-finite element method isapplied over the coarse grid to approximate the nonlinear Navier–Stokes equations whilea linear evolutionary problem is solved over the fine grid. The previously computed Galer-kin approximation to the velocity is used to linearize the convective term. For the analysiswe take into account the lack of regularity of the solutions of the Navier–Stokes equationsat the initial time in the absence of nonlocal compatibility conditions of the data. Optimalerror bounds are obtained.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

In this paper we study two-grid mixed finite-element (MFE) methods for the spatial discretization of the incompressibleNavier–Stokes equations

ut � Duþ ðu � rÞuþrp ¼ f ;

divðuÞ ¼ 0;ð1Þ

in a bounded domain X � Rd ðd ¼ 2;3Þ with a smooth boundary subject to homogeneous Dirichlet boundary conditionsu ¼ 0 on @X. In (1), u is the velocity field, p the pressure, and f a given force field. As in [25–27] we assume that the fluiddensity and viscosity have been normalized by an adequate change of scale in space and time. We approximate the solutionu and p corresponding to a given initial condition

uð�;0Þ ¼ u0: ð2Þ

Two-grid methods are a well established technique for nonlinear steady problems, see [37,38]. The main idea in a two-levelmethod involves the discretization of the equations over two meshes of different size. A nonlinear system over the coarsemesh is solved in the first step of the method. In a second step, a linearized equation based on the approximation overthe coarse mesh is solved on the fine mesh. In [30–32] several two-level methods are considered to approximate the steadyNavier–Stokes equations. In these papers, depending on the algorithm, the second step is based on the solution of a discrete

. All rights reserved.

s), [email protected] (B. García-Archilla), [email protected] (J. Novo).ant MTM2010-14919, and by JCyL grant VA001A10-1.ant MTM2009-07849.ant MTM2010-14919.

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Stokes problem, a linear Oseen problem or one step of the Newton method over the fine mesh with the coarse mesh approx-imation as initial guess.

Several two-level or two-grid schemes have also been considered in the literature to approximate the evolutionary non-linear Navier–Stokes equations (1) and (2). Again, two approximations to the velocity (and correspondingly two approxima-tions to the pressure), are computed. One of them is defined by a discretization of the nonlinear equations over a coarsemesh, uH , and another one, ~uh, is defined by an appropriate linearization over a fine mesh. Different classes of algorithmscan be seen as two level methods. In particular, although they were originally developed from different ideas, the so callednonlinear Galerkin methods, postprocessed and dynamical postprocessed methods, fall into this category.

Postprocessed Galerkin methods were first introduced for spectral methods in [14,20,21] (see also [34]) and later ex-tended to finite element methods in [8,7,15,16,18]. In all these works the main idea is the following: one first computethe standard Galerkin approximation to the velocity and pressure over a coarse mesh ðuH; pHÞ of size H and then computethe postprocessed approximation in a finer mesh at selected times in which one wants to obtain an improved approximation.More precisely, the postprocessed approximation ð~uh; ~phÞ computed at a given time t� is an approximation in a mesh of sizeh� H to the following (steady) Stokes problem:

�D~uþr~p ¼ f � uH;tðt�Þ � ðuHðt�Þ � rÞuHðt�Þdivð~uÞ ¼ 0

�in X;

~u ¼ 0; on @X:ð3Þ

Here, uHðtÞ; t 2 ð0; T�, is the standard MFE approximation computed in the coarse mesh in a time interval ð0; T� and t� 2 ð0; T�.Note that the computation of ðuHðtÞ; pHðtÞÞ; t 2 ð0; T�, is completely independent of the computation of ð~uhðt�Þ; ~phðt�ÞÞ in thefine mesh. The postprocessed approximation improves the rate of convergence of the standard Galerkin approximation overthe coarse mesh in the following sense. If the rate of convergence of the Galerkin approximation to the velocity in theL2ðXÞd ðj ¼ 0Þ or H1ðXÞd ðj ¼ 1Þ norm is OðHr�jÞ then the rate of convergence of the postprocessed approximation to the veloc-ity is OðHrþ1�j logðHÞj jÞ þ Oðhr�jÞ. Analogous results are obtained for the pressure. For first order mixed finite element methodsthe improvement in the rate of convergence of the velocity is only achieved in the H1ðXÞd norm, [7]. Then, if one wants toachieve the optimal accuracy of the fine level in the H1ðXÞd norm, one can first compute the Galerkin approximation on acoarse mesh of size H ¼ hðr�1Þ=r and then compute the postprocessed approximation over the fine mesh of size h at the de-sired time levels. For example, one should take H ¼ h1=2 and H ¼ h2=3 for linear and quadratic mixed finite elements, respec-tively. It can be expected that the computational cost of the postprocessed approximation is smaller than that of the Galerkinapproximation on the same fine mesh, since in the former method the time evolution is done on the coarse mesh, and only atselected time levels are computations done on the fine mesh. This has been confirmed by the numerical experiments in [8](see also [17,20]).

In [34] a related algorithm, the so-called dynamical postprocessing, is introduced for the Fourier case. In this algorithm,the standard Galerkin approximation, ðuH; pHÞ, is computed over a coarse mesh in the first level, as before. For the secondlevel an approximation to a linear evolutionary problem, instead of the steady problem (3), is computed. More precisely,the dynamical postprocessing involves the approximation, at each time step, over a mesh of size h� H of the problem:

~ut � D~uþr~p ¼ f � ðuH � rÞuH

divð~uÞ ¼ 0

�in X;

~u ¼ 0; on @X:

ð4Þ

Note that in the dynamical postprocessing, the computation of ðuHðtÞ; pHðtÞÞ and ð~uhðtÞ; ~phðtÞÞ; t 2 ½0; T�, is coupled. The rateof convergence of the dynamical postprocessing scheme is proved in [34] to be the same as the rate of convergence of thestandard postprocessing. In the case of highly oscillatory solutions the dynamical algorithm is shown to be more efficientthan the standard postprocessing in some one dimensional examples. The dynamical postprocessing method is also consid-ered in [36], named now as two-level method, in the case of mixed finite elements. In [36], the author treats the fully discretecase integrating in time with the backward Euler method. A similar two-level scheme is also considered and analyzed in [23]where the author uses first order mixed finite elements in space, Crank-Nicolson extrapolation for the time integration overthe coarse mesh and the backward Euler method for the time integration over the fine mesh.

The so-called nonlinear Galerkin methods are also two-level methods that have been used to compute approximations to(1) and (2). They were first introduced for Fourier spectral methods [13,35], and later extended to mixed finite elementmethods in [5]. In this work the authors obtain the rate of convergence of the nonlinear Galerkin method in the case of firstorder elements. The rate of convergence is the same one of the postprocessed method. The main difference between the non-linear Galerkin methods and the postprocessed or two-grid methods is that in the former the approximation on the coarsemesh takes into account the influence of the fine mesh, whereas in the latter it is just the standard Galerkin method (i.e.,computed independently of the fine mesh).

In this paper we analyze two two-grid algorithms in the context of spatial mixed finite element discretizations to approx-imate the solutions of (1) and (2). The two algorithms we consider are very similar to the dynamical postprocessing method.The difference is the treatment of the nonlinearity in the second level. In the dynamical postprocessing method the nonlinearconvective term of the fine level is approximated by the data ðuH � rÞuH (see the right-hand-side of (4)). In the two algo-rithms we consider in the present paper, the approximation to the velocity of the coarse mesh uH is used to linearize the

7036 J. de Frutos et al. / Applied Mathematics and Computation 218 (2012) 7034–7051

nonlinear convective term of the fine level. In the first algorithm, the linearized convective term of the fine level is ðuH � rÞ~uh.In the second algorithm uH is regarded as an initial guess to perform one Newton step in the fine level. For the spatial dis-cretization we consider mixed finite elements of first, second and third order. More precisely, we consider the mini-elementand the quadratic and cubic Hood-Taylor elements. The analysis for other mixed finite elements of the same order is similar.As in [26], [15] due to the lack of regularity at t ¼ 0 of the solution of (1) and (2) no better than OðH5Þ error bounds can beexpected. For this reason we do not analyze higher than cubic finite element discretizations. For the temporal discretizationwe use the backward Euler method or the two-step backward differentiation formula. The analysis of the fully discrete meth-ods is similar to the one appeared in [16] and it is only briefly indicated in this paper.

This is not the first time these two algorithms have been considered. The first algorithm was introduced in [22], where theauthors analyze the semi-discrete in space case for first order finite elements. In [2] the authors extend this analysis to thefully discrete case and in [1] the second order Hood-Taylor finite element is used for the spatial discretization and the two-step backward differentiation formula for the time integration. In [29] the second algorithm is analyzed for the Fourier spec-tral case while in [33] the analysis is extended to the case of first order mixed finite elements considering the fully discretecase coupled with the Crank–Nicolson scheme for the time integration. As opposed to the above mentioned works on thesame methods, in the present paper we take into account the lack of regularity suffered by the solutions of the Navier–Stokesequations at the initial time. Then, for the analysis in the present paper we do not assume more than second-order spatialderivatives bounded in L2 up to initial time t ¼ 0, since demanding further regularity requires the data to satisfy nonlocalcompatibility conditions unlikely to be fulfilled in practical situations [25,26]. This is the first time these methods are ana-lyzed under realistic regularity assumptions. Also, this is the first time the cubic case is considered and the first time thequadratic case is considered for the second method.

There are some other improvements with respect to previous works. In [1] the authors get an error bound of orderOðH3 þ h2 þ ðDtÞ2Þ for the fine approximation to the velocity ~uh in the H1ðXÞd norm whenever the following inequality is sat-isfied a1H3

6 ðDtÞ2 6 a2H3;a1 and a2 being constants independent of H and Dt. With the technique of this paper an errorbound of order Oðj logðhÞjj logðHÞjH4 þ h2 þ ðDtÞ2Þ for the same fully discrete method in the H1ðXÞd norm can be obtainedfor H and Dt independently chosen. With the new error bound obtained in this paper one can achieve the rate of convergenceof the fine mesh in the H1ðXÞd norm by taking H ¼ h1=2 instead of H ¼ h2=3. This fact improves the efficiency of the methodcompared with the (same order) standard Galerkin method over the fine mesh. Also, the authors of [1] remark that they haveobserved the same rate of convergence for the two-grid method with H ¼ h1=2 and H ¼ h2=3 in the numerical tests they havecarried out, which supports the improved rate of convergence we obtain in this paper. We want to remark that in all thenumerical experiments of [29,33,1] the two-grid algorithms improve the efficiency of the standard Galerkin method inthe sense that a given error can be achieved with less computational cost with the new algorithms than with the standardGalerkin method. In [29] a comparison in the Fourier case between the standard postprocessing, the dynamical postprocess-ing and the second two-grid algorithm is also included. Although the computational cost of the two-grid approximation overthe fine mesh is bigger than that of the postprocessed approximations, the two-grid algorithm produces smaller errors in thecase of moderate to high Reynolds numbers. Finally, comparing the two algorithms we analyze in this paper we remark thatwith the second algorithm better error bounds are obtained in terms of H. Although this fact could make the choice of thesecond algorithm preferable for computations, it turns out in practice to be rather inefficient to solve the linear systemsaccurately. For this reason, some authors suggest solving instead an Oseen problem leading then to the first algorithm,see [30].

The rest of the paper is as follows. In Section 2 we introduce some preliminaries and notations. In Section 3 we carry outthe error analysis of the first two-grid algorithm in the semi-discrete in space case. In Section 4 we consider the analysis ofthe second two-grid algorithm in the semi-discrete in space case. Finally, in Section 5 we consider the fully discrete case inte-grating in time with the backward Euler method or the two-step backward differentiation formula.

2. Preliminaries and notations

We will assume that X is a bounded domain in Rd; d ¼ 2;3, not necessarily convex and smooth enough. When dealingwith linear elements (r ¼ 2 below) X may also be a convex polygonal or polyhedral domain. We will consider the Hilbertspaces

H ¼ fu 2 ðL2ðXÞÞdjdivðuÞ ¼ 0; u � nj@X ¼ 0g;V ¼ fu 2 ðH1

0ðXÞÞdjdivðuÞ ¼ 0g

endowed with the inner product of L2ðXÞd and H10ðXÞ

d, respectively. For l P 0 integer and 1 6 q 61, we consider the stan-dard Sobolev spaces Wl;qðXÞd of functions with derivatives up to order l in LqðXÞ, and HlðXÞd ¼Wl;2ðXÞd. We will denote byk � kl the norm in HlðXÞd, and k � k�l will represent the norm of its dual space. We consider also the quotient spaces HlðXÞ=Rwith norm kpkHl=R ¼ inffkpþ ckljc 2 Rg.

Let us recall the following Sobolev’s imbedding inequalities [4]: for q 2 ½1;1Þ, there exists a constant C ¼ CðX; qÞ such that

kvkLq0 ðXÞd 6 CkvkWs;qðXÞd ;1q0

P1q� s

d> 0; q <1; v 2Ws;qðXÞd: ð5Þ


For q0 ¼ 1, (5) holds with 1q <

sd.

Let P : L2ðXÞd _ H be the L2ðXÞd projection onto H. We denote by A the Stokes operator on X:

A : DðAÞ � H! H; A ¼ �PD; DðAÞ ¼ H2ðXÞd \ V :

Applying Leray’s projector to (1), the equations can be written in the form

ut þ Auþ Bðu;uÞ ¼ Pf in X;

where Bðu;vÞ ¼ Pðu � rÞv for u, v in H10ðXÞ

d.We shall use the trilinear form bð�; �; �Þ defined by

bðu; v;wÞ ¼ ðFðu;vÞ;wÞ 8u; v;w 2 H10ðXÞ

d;

where

Fðu;vÞ ¼ ðu � rÞv þ 12ðr � uÞv 8u; v 2 H1

0ðXÞd:

It is straightforward to verify that b enjoys the skew-symmetry property

bðu; v;wÞ ¼ �bðu;w;vÞ 8u;v ;w 2 H10ðXÞ

d: ð6Þ

Let us observe that Bðu;vÞ ¼ PFðu;vÞ for u 2 V ;v 2 H10ðXÞ

d.We shall assume that

kuðtÞk1 6 M1; kuðtÞk2 6 M2; 0 6 t 6 T

and, for k P 2 integer,

sup06t6T

@bk=2ct f

�� k�1�2bk=2c

þXbðk�2Þ=2c

j¼0

sup06t6T

@ jtf

�� k�2j�2

< þ1;

so that, according to Theorems 2.4 and 2.5 in [25], there exist positive constants Mk and Kk such that for k P 2

kuðtÞkk þ kutðtÞkk�2 þ kpðtÞkHk�1=R6 MksðtÞ1�k=2 ð7Þ

and for k P 3, if X is of class Ck,
Z t
0rk�3ðsÞðkuðsÞk2

k þ kusðsÞk2k�2 þ kpðsÞk

2Hk�1=R

þ kpsðsÞk2Hk�3=R

Þds 6 K2k ; ð8Þ

where sðtÞ ¼minðt;1Þ and rn ¼ e�aðt�sÞsnðsÞ for some a > 0. Observe that, for t 6 T <1, we can take sðtÞ ¼ t and rnðsÞ ¼ sn.For simplicity, we will take these values of s and rn. We note that no further than k 6 6 will be needed in the present paper.

Let T h ¼ ðshi ;/

hi Þi2Ih

; h > 0, be a family of partitions of suitable domains Xh, where h is the maximum diameter of the ele-ments sh

i 2 T h and /hi are the mappings of the reference simplex s0 onto sh

i (see comments following (12) below on the dis-crepancies between X and Xh). We restrict ourselves to quasi-uniform and regular meshes T h.

Let r P 2, we consider the finite-element spaces

Sh;r ¼ vh 2 CðXhÞjvhjshi� /h

i 2 Pr�1ðs0Þn o

;

S0h;r ¼ vh 2 CðXhÞjvhjsh

i� /h

i 2 Pr�1ðs0Þ; vhðxÞ ¼ 08x 2 @Xh

n o;

where Pr�1ðs0Þ denotes the space of polynomials of degree at most r � 1 on s0. Since we are assuming that the meshes arequasi-uniform, the following inverse inequality holds for each vh 2 ðS0

h;rÞd (see, e.g., [11, Theorem 3.2.6]).

kvhkWm;qðsÞd 6 Chl�m�dð1q0�

1qÞkvhkWl;q0 ðsÞd ; ð9Þ

where 0 6 l 6 m 6 1; 1 6 q0 6 q 61, and s is an element in the partition T h.We shall denote by ðXh;r ;Q h;r�1Þ the so-called Hood–Taylor element [9,28], when r P 3, where

Xh;r ¼ S0h;r

� �d; Q h;r�1 ¼ Sh;r�1 \ L2ðXhÞ=R; r P 3

and the so-called mini-element [10] when r ¼ 2, where Qh;1 ¼ Sh;2 \ L2ðXhÞ=R, and Xh;2 ¼ ðS0h;2Þ

d Bh. Here, Bh is spanned bythe bubble functions bs, s 2 T h, defined by bsðxÞ ¼ ðdþ 1Þdþ1k1ðxÞ � � � kdþ1ðxÞ, if x 2 s and 0 elsewhere, where k1ðxÞ; . . . ; kdþ1ðxÞdenote the barycentric coordinates of x. For these mixed elements a uniform inf-sup condition is satisfied (see [9]); that is,there exists a constant b > 0 independent of the mesh grid size h such that

infqh2Qh;r�1

supvh2Xh;r

ðqh;r � vhÞkvhk1kqhkL2=R

P b: ð10Þ


The approximate velocity belongs to the discretely divergence-free space

Vh;r ¼ Xh;r \ vh 2 H10ðXhÞjðqh;r � vhÞ ¼ 0 8qh 2 Q h;r�1

n o:

We observe that, for the Hood–Taylor element, Vh;r is not a subspace of V. Let Ph : L2ðXÞd _ Vh;r be the discrete Leray’s pro-jection defined by

ðPhu;vhÞ ¼ ðu;vhÞ 8vh 2 Vh;r :

We will use the following well-known bounds

kðI �PhÞukj 6 Chl�jkukl; 1 6 l 6 r; j ¼ 0;1: ð11Þ

Let us remark that, as discussed in [6,8] for the above estimate to hold when Xh–X, the computational domain Xh must besuch that the value dðhÞ ¼maxx2@Xh

disðx; @XÞ satisfies

dðhÞ ¼ Oðh2ðr�1ÞÞ: ð12Þ

This can be achieved if, for example, @X is piecewise of class C2ðr�1Þ, and superparametric approximation at the boundary isused. In this case, as argued in [6,8] (see also [15,16]) one can ignore the discrepancies between X and Xh in the argumentsthat follow. Consequently in what follows we will assume that @X is of class Cr and piecewise of class C2ðr�1Þ. Assuming that Xhas such a smooth enough boundary, we also have

A�m=2PðI �PhÞu��

06 Chlþminðm;r�2Þkukl; 1 6 l 6 r; m ¼ 1;2: ð13Þ

We will denote by Ah : Vh;r ! Vh;r the discrete Stokes operator defined by

ðrvh;r/hÞ ¼ ðAhvh;/hÞ ¼ A1=2h vh;A

1=2h /h

� �8vh;/h 2 Vh;r :

Since ðA�1=2h Phf ;vhÞ ¼ ðf ;A�1=2

h vhÞ, for all vh 2 Vh;r , it follows that

kA�1=2h Phfk0 6 Ckfk�1: ð14Þ

Moreover it holds for f 2 L2ðXÞd, see [15]:

kA�s=2h Phfk0 6 Chskfk0 þ kA

�s=2Pfk0 s ¼ 1;2: ð15Þ

Let A denote either A ¼ A or A ¼ Ah. Notice that both are positive self-adjoint operators with compact resolvent in H and Vh,respectively. Let us consider then for a 2 R and t > 0 the operators Aa and e�tA, which are defined by means of the spectralproperties of A (see, e.g., [12, p. 33], [19]). An easy calculation shows that

kAae�tAk0 6 ðae�1Þat�a; a P 0; t > 0; ð16Þ

where, here and in what follows, k � k0 when applied to an operator denotes the associated operator norm. Also, using thechange of variables s ¼ s=t, it is easy to show that
Z t
0s�1=2 A1=2

h e�ðt�sÞAh

�� 0

ds 61ffiffiffiffiffiffi2ep B

12;12

� �; ð17Þ

where B is the Beta function (see, e.g., [3]).

3. Semi-discretization in space. The first two-grid algorithm

In this section we carry out the error analysis of the first two-grid algorithm for the Hood–Taylor mixed finite elementwith r ¼ 3 or 4. At the end of the section we include the results that can be obtained for the mini-element with a similarbut simpler analysis than the one showed along the section.

The first algorithm we consider is the following. Let us choose h < H so that VH;r � Vh;r . Then, in the first level we computethe standard mixed finite-element approximation to (1) and (2). That is, given uHð0Þ ¼ PHðu0Þ, we compute uHðtÞ 2 XH;r andpHðtÞ 2 QH;r�1; t 2 ð0; T�, satisfying, for all /H 2 XH;r and wH 2 QH;r�1

ðuH;t;/HÞ þ ðruH;r/HÞ þ bðuH;uH;/HÞ þ ðrpH;/HÞ ¼ ðf ;/HÞ; ð18Þðr � uH;wHÞ ¼ 0: ð19Þ

In the second level we solve a linearized problem on a finer grid and given ~uhð0Þ ¼ Phu0 we compute ~uhðtÞ 2 Xh;r and~phðtÞ 2 Q h;r�1; t 2 ð0; T�, satisfying, for all /h 2 Xh;r and wh 2 Q h;r�1

ð~uh;t ;/hÞ þ ðr~uh;r/hÞ þ ðuH � r~uh;/hÞ þ ðr~ph;/hÞ ¼ ðf ;/hÞ; ð20Þðr � ~uh;whÞ ¼ 0: ð21Þ


To obtain the error bounds for ð~uh; ~phÞwe will follow the error analysis of [15] and introduce an auxiliary approximation (see[15, Section 4.1]). For a u and p solution of (1) and (2) let us consider the approximations vh : ½0; T� ! Xh;r andgh : ½0; T� ! Qh;r�1, respectively, solutions of

ðvh;t;/hÞ þ ðrvh;r/hÞ þ ðrgh;/hÞ ¼ ðf ;/hÞ � bðu;u;/hÞ; ð22Þðr � vh;whÞ ¼ 0; ð23Þ

for all /h 2 Xh;r and wh 2 Q h;r�1, with initial condition vhð0Þ ¼ Phu0. We will also use the following notation:

zh ¼ Phu� vh: ð24Þ

Next, we state some lemmas that are needed in the proof of the main theorems. The first one summarizes previous results.

Lemma 1. Let ðu; pÞ be the solution of (1) and (2). There exists a positive constant C such that the discrete velocity vh defined by(22) and (23) and the Hood–Taylor element approximation to u;uH, satisfy the following bounds for j ¼ 0;1, and t 2 ð0; T�:

kvHðtÞ � uHðtÞkj 6C

tðr�2Þ=2 j logðHÞjHrþ1�j; 3 6 r 6 4; ð25Þ

kuHðtÞ � uðtÞkj 6C

tðr�2Þ=2 Hr�j; 2 6 r 6 5; ð26ÞZ t

0kuHðsÞ � uðsÞk2

j ds 6 CH2ð3�jÞ; 3 6 r 6 4: ð27Þ

Proof. The bound (25) is proved in Theorems 4.7 and 4.15 in [15]. The case j ¼ 0 in (27) is proved in [25, Theorem 3.1] and[26, Theorem 3.1]. The case j ¼ 1 follows from the case j ¼ 0 by applying (9) and (11), see also Corollaries 4.18 and 4.16 in[15]. Finally, (27) is proved in Lemmas 5.1 and 5.2 in [26]. h

For the convenience of the reader, we will reproduce here the following two Lemmas, the first one from [8] and the sec-ond one from [15].

Lemma 2. For any f 2 Cð½0; T�; L2ðXÞdÞ, the following estimate holds for all t 2 ½0; T�:

Z t

0kAhe�ðt�sÞAh Phf ðsÞk0ds 6 Cj logðhÞjmax

06t6Tkf ðtÞk0:

Lemma 3. Let ðu; pÞ be the solution of (1) and (2). Then, there exists a positive constant C such that the error zh ¼ Phu� vh in (24)satisfies the following bound:

kAð�1þjÞ=2h zhk0 6

Ctðr�2Þ=2 hrþ1�j

; j ¼ 0;1;2; r P 3: ð28Þ

Lemma 4. For each a > 0 there exist positive constants K > 0 and h0 depending on a and M2 such that, for h < H < h0,h1 ¼ h2 ¼ h or fh1;h2g ¼ fh;Hg, and every w1

h1ð�Þ 2 Vh1 ;r and ;w2

h2ð�Þ 2 Vh2 ;r satisfying the threshold condition

uðtÞ �w1h1ðtÞ

�� j6 ah2�j

1 ; uðtÞ �w2h2ðtÞ

�� j6 ah2�j

2 ; j ¼ 0;1; t 2 ½0; T� ð29Þ

for whðtÞ 2 H10ðXÞ; t 2 ½0; T�, satisfying

kwhðtÞkj 6 2a maxðh1; h2Þ2�j;

the following bounds hold:

Fðwh;w2h2Þ

�� 0þ Fðw1

h1;whÞ

�� 06 Kkwhk1; ð30Þ

Fðwh;w2h2Þ

�� 1þ Fðw1

h1;whÞ

�� 16 Kkwhk0; ð31Þ

kBhðwh;w2h2Þk0 þ Bhðw1

h1;whÞ

�� 06 K whk k1; ð32Þ

A�1=2h ðBhðwh;w2

h2ÞÞ

�� 0þ A�1=2

h ðBhðw1h1;whÞÞ

�� 06 Kkwhk0; ð33Þ

where Fðu;vÞ can be either ðu � rÞv þ 12 ðr � uÞv or ðu � rÞv , and Bh ¼ PhF .


Proof. The proof of the present lemma can be found in that of Lemma 4.4 in [15] for Fðu;vÞ ¼ ðu � rÞv þ 12 ðr � uÞv and

wh 2 Vh;r . With obvious changes, the proof is also valid when Fðu;vÞ ¼ ðu � rÞv , as well as when wh R Vh;r . h

Remark 1. We will apply the above inequalities for wh ¼ w1h1�w2

h2; wh ¼ w1

h1� u and wh ¼ w2

h2� u. Let us also remark that

the Lemma 4 also holds if either w1h1

or w2h2

is replaced by u. In what follows we will apply Lemma 4 to uh and vh both sat-isfying the threshold condition (29) for an appropriate value of a (see [15, Remark 4.1]).

Lemma 5. For v 2 ðH2ðXÞÞd \ V there exists a positive constant K ¼ Kðkvk2Þ such that w 2 H10ðXÞ

d the following bound holds fore ¼ v �w:

kA�1P½Fðv ; eÞ þ Fðe; vÞ�k0 6 Kkv �wk�1; ð34Þ
where Fðu; vÞ can be either ðu � rÞv þ 1
2 ðr � uÞv or ðu � rÞv .

Proof. The proof of this result when Fðu;vÞ ¼ ðu � rÞv þ 12 ðr � uÞv can be found as part of the proof of [8, Lemma 3.4]. With

obvious changes, the proof is also valid when Fðu;vÞ ¼ ðu � rÞv . h

Let us observe that the approximation over the finer grid ~uh and the recently defined vh satisfy

~uh;t þ Ah~uh þPhðuH � r~uhÞ ¼ Phf ; uhð0Þ ¼ Phu0; ð35Þvh;t þ Ahvh þPhðu � ruÞ ¼ Phf ; vhð0Þ ¼ Phu0; ð36Þ

respectively. Then eh ¼ vh � ~uh satisfies

eh;t þ Aheh þPhðuH � rehÞ ¼ Phqh;H; ehð0Þ ¼ 0; ð37Þ

where

qh;H ¼ uH � rvh � u � ru:

In the proof of Theorem 1 below we will use the following lemmas.

Lemma 6. Let ðu; pÞ be the solution of (1) and (2). There exists a positive constant C such that the following inequality holdsfor r ¼ 3;4:

kA�1h Phqh;Hk0 6

Ctðr�2Þ=2 j logðHÞjHrþ1; t 2 ð0; T�:

Proof. Let us write qh;H ¼ q1h;H þ q2

h;H , where

q1h;H ¼ ððuH � uÞ � rvhÞ; q2

h;H ¼ ðu � rðvh � uÞÞ: ð38Þ

By applying (15) we have

kA�1h Phqj

h;Hk0 6 Ch2kqjh;Hk0 þ kA

�1Pqjh;Hk0; j ¼ 1;2:

To bound kq1h;Hk0 let us recall Remark 1 and apply (30) to get

kq1h;Hk0 6 CkuH � uk1 6 C

Hr�1

tðr�2Þ=2 ; ð39Þ

where we have applied (26) from Lemma 1 in the last inequality. Applying (30) we also get

kq2h;Hk0 6 Ckvh � uk1 6 C

hr�1

tðr�2Þ=2 ; ð40Þ

where in the last inequality we have applied standard bounds for Ph (see (11)) together with the estimates (28) for zh inLemma 3. Let us next bound kA�1Pq1

h;Hk0. We will use the decomposition

q1h;H ¼ ððu� uHÞ � rÞðvh � uÞ þ ððu� uHÞ � rÞu: ð41Þ

Then, we obtain

kA�1Pq1h;Hk0 ¼ kA

�1PðððuH � uÞ � rÞðvh � uÞÞk0 þ kA�1PðððuH � uÞ � rÞuÞk0: ð42Þ

To bound the second term in (42) we apply (34) from Lemma 5 to get

kA�1PðððuH � uÞ � rÞuÞk0 6 CkuH � uk�1:


Applying then (25) together with (13) and (28) we get

kuH � uk�1 6 kuH � vHk0 þ kvH � uk�1 6C

tðr�2Þ=2 j logðHÞjHrþ1 þ Ctðr�2Þ=2 Hrþ1: ð43Þ

To bound the first term in (42) we argue by duality, using (5), we get

kA�1PðððuH � uÞ � rÞðvh � uÞÞk0 ¼ supk/k0¼1

ðððuH � uÞ � rÞðvh � uÞ;A�1P/Þ 6 supk/k0¼1

kuH � uk0kvh � uk1kA�1P/k1

6 supk/k0¼1

CkuH � uk0kvh � uk1kA�1P/k2 6 CkuH � uk0kvh � uk1:

Now, in view of the case r ¼ 2 in (26) and using again (11) and (28) we conclude

kA�1PðððuH � uÞ � rÞðvh � uÞÞk0 6C

tðr�2Þ=2 H2hr�1:

Finally, to bound kA�1Pq2h;Hk0 we apply again (34) to bound this norm in terms of kvh � uk�1 which, as we shown in (43), is

bounded by Ctð2�rÞ=2hrþ1. h

Lemma 7. Let ðu; pÞ be the solution of (1) and (2). There exists a positive constant C such that the following inequalities hold forr ¼ 3;4:

kqh;Hk�1 6C

tðr�1Þ=2 j logðHÞjHrþ1 þ Ctðr�2Þ=2 hr

; t 2 ð0; T�; ð44Þ

kqh;Hk�1 6C

t1=2 H3; t 2 ð0; T�: ð45Þ

Proof. The proof is very similar to the one of the previous lemma. We will prove (44) since the proof of (45) is completelyanalogous and yet easier. We use the decomposition (38).

For q2h;H we apply (31) to get

kq2h;Hk�1 6 Ckvh � uk0 6 C

hr

tðr�2Þ=2 ;

where we have applied (11) and (28) in the last inequality. For q1h;H we will use the decomposition (41). For the first term in

(41) using (5) we have

kððuH � uÞ � rÞðvh � uÞk�1 ¼ supk/k1¼1

ðððuH � uÞ � rÞðvh � uÞ;/Þ 6 supk/k1¼1

kuH � ukL2dkvh � uk1k/kL2d=ðd�1Þ

6 CkuH � uk1kvh � uk1 6 CHr�1

tðr�2Þ=2

h2

t1=2 ;

where we have applied (26) and (11) and (28) in the last inequality. Finally, for the second term using (5), (7) and (43) weobtain

kððuH � uÞ � rÞuk�1 ¼ supk/k1¼1

ðððuH � uÞ � rÞu;/Þ 6 supk/k1¼1

kuH � uk�1k/ruk1 6 supk/k1¼1

kuH � uk�1ðkrukW1;2d=ðd�1Þ k/kL2d

6 supk/k0¼1

kuH � uk�1 kDukL2d=ðd�1Þ þ kruk1k/k1

6

Ctðr�2Þ=2 j logðHÞjHrþ1kuk3

6C

tðr�1Þ=2 j logðHÞjHrþ1: �

Lemma 8. Let ðu; pÞ be the solution of (1) and (2). Then there exists a positive constant C such that the discrete velocity vh definedby (36) and the approximation to u over the finer grid, ~uh satisfy the following bound:

kAl=2h ðvhðtÞ � ~uhðtÞÞk0 6 CH3�l; r P 3; l ¼ 0;1; t 2 ð0; T�:

Proof. Let us consider yhðtÞ ¼ Al=2h ehðtÞ. From (37) it follows that

yhðtÞ ¼Z t

0e�ðt�sÞAh Al=2

h Ph ðuH � rÞehð ÞdsþZ t


h Phqh;HðsÞds:


Applying (16), and taking into account that as a consequence of (30) and (33) we have kAð�1þlÞ=2h PhððuH � rÞehÞk0 6 CkAl=2

h ehk0,it follows that

kyhðtÞk0 6

Z t

0

Cffiffiffiffiffiffiffiffiffiffit � sp kyhk0 dsþ

Z t


h Phqh;HðsÞds�� ;

so that a generalized Gronwall lemma [24, pp. 188–189] allow us to write

max06t6T

kyhðtÞk0 6 C max06t6T

Z t


h Phqh;HðsÞds��

0: ð46Þ

Using (17) we obtain

max06t6T

kyhðtÞk0 6 CB12;12

� �max06s6T

s1=2 Að�1þlÞ=2h Phqh;H

�� 0: ð47Þ

To conclude we apply (14) and (45) from Lemma 7 in the case l ¼ 0, and (39) and (40) in the case l ¼ 1. h

Lemma 9. Let ðu; pÞ be the solution of (1) and (2). Then, there exists a positive constant C such that the discrete velocity vh definedby (36) and the approximation to u over the finer grid, ~uh satisfy the following bound:

kA�1=2h ðvhðtÞ � ~uhðtÞÞk0 6 Cj logðHÞjH4; r P 3; t 2 ð0; T�:

Proof. The proof follows the steps of the proof of Lemma 4.6 in [15]. Let us consider yhðtÞ ¼ A�1=2h ehðtÞ. From (37) it follows

that

yhðtÞ ¼Z t

0e�ðt�sÞAh A�1=2



h Phqh;HðsÞds:

Applying (16) we have that

kyhðtÞk0 6

Z t

0

Cffiffiffiffiffiffiffiffiffiffit � sp A�1

h PhððuH � rÞehÞ��

0dsþ

Z t



0: ð48Þ

Let us now bound kA�1h PhððuH � rÞehÞk0. We will argue as in the proof of [15, (4.23) Lemma 4.4]. Let us first observe that

h2kehk1 6 CkA�1=2h ehk0. Using (15) we get

A�1h PhððuH � rÞehÞ

�� 06 Ch2kðuH � rÞehk0 þ kA

�1PððuH � rÞehÞk0 6 Ch2kehk1 þ kA�1PððuH � rÞehÞk0

6 CkA�1=2h ehk0 þ kA

�1PððuH � rÞehÞk0:

Let us now bound the second term on the right hand side above. We write

kA�1PððuH � rÞehÞk0 6 kA�1PðððuH � uÞ � rÞehÞk0 þ kA

�1Pððu � rÞehÞk0:

For the first term arguing by duality we get

kA�1PðððuH � uÞ � rÞehÞk0 6 CkuH � uk0kehk1 6 CH2kehk1:

For the second one, arguing again by duality and integrating by parts, we get

kA�1Pððu � rÞehÞk0 6 Ckehk�1kuk2 6 Ckyhk0:

We finally obtain

A�1h PhððuH � rÞehÞ

�� 06 Ckyhk0 þ CH2kehk1: ð49Þ

Going back to (48) we obtain

kyhðtÞk0 6

Z t

0


Z t



0þ Ct1=2H2 max

06s6tkehðsÞk1; ð50Þ


max06t6T


Z t


h Phqh;H ds��

0þ H2 max

06t6Tkehk1

� �:



max06t6T

kyhðtÞk0 6 C B12;12

� �max06s6T

s1=2 A�1h Phqh;H

�� 0þ H2 max

06t6Tkehk1

� �; ð51Þ

where, by applying Lemmas 6 and 8 the proof is finished. h

The proof of the following theorem follows the steps of the proof of [15, Theorem 4.7].

Theorem 1. Let ðu; pÞ be the solution of (1) and (2). There exists a positive constant C such that the discrete velocity vh defined by(36) and the approximation to u over the finer grid, ~uh, satisfy the following bound:

kvhðtÞ � ~uhðtÞk0 6C

t1=2 j logðhÞjðj logðHÞjH4Þ; t 2 ð0; T�; r P 3: ð52Þ

Proof. Let us consider yhðtÞ ¼ t1=2ehðtÞ. From (37) and an easy calculation we get

yh;t þ Ahyh þ t1=2PhðuH � rehÞ ¼ t1=2Phqh;H þ1

2t1=2 eh:

Then,

yhðtÞ ¼Z t

0e�Ahðt�sÞs1=2PhðuH � rehÞdsþ

Z t

0e�Ahðt�sÞ s1=2Phqh;H þ

12s1=2 eh

� �ds:

Applying (33) we get

kA�1=2h PhðuH � rÞehÞk0 6 Ckehk0: ð53Þ

Then, using (16) we obtain
Z t
0e�Ahðt�sÞs1=2PhðuH � rehÞds

�� 06 C

Z t

0

kyhk0ffiffiffiffiffiffiffiffiffiffit � sp ds:

Applying a generalized Gronwall lemma [24, pp. 188–189], it follows that

max06t6T


Z t

0e�Ahðt�sÞs1=2Phqh;H ds

�� 0þmax

06s6t

Z t

0e�Ahðt�sÞ eh

s1=2 ds��

0

� �: ð54Þ

Applying now Lemma 2 and (17) we have

max06t6T

kyhðtÞk0 6 C j logðhÞjmax06t6T

s1=2A�1h Phqh;HðsÞ

�� 0þ B

12;12

� �max06t6T

A�1=2h ehðsÞ

�� 0

� �;

where Lemmas 6 and 9 finish the proof. h

Theorem 2. Let ðu; pÞ be the solution of (1) and (2). There exists a positive constant C such that the discrete velocity vh defined by(36) and the approximation to u over the finer grid, ~uh, satisfy the following bound for r P 3

kvhðtÞ � ~uhðtÞk1 6Ctj logðhÞjðj logðHÞjH4 þ T1=2h3Þ; t 2 ð0; T�: ð55Þ

Proof. Let us define yhðtÞ ¼ tA1=2h ehðtÞ, where ehðtÞ ¼ vhðtÞ � ~uhðtÞ. Arguing exactly as in the proof of Theorem 1, instead of

(54) we now arrive at

max06t6T


Z t

0e�Ahðt�sÞsA1=2

h Phqh;H ds��

0þmax

06t6T

Z t

0e�Ahðt�sÞA1=2

h eh ds��

0

� �:

Applying now Lemma 2 we get

max06t6T

kyhðtÞk0 6 Cj logðhÞj max06t6T

sA�1=2h Phqh;H

�� 0þmax

06t6TA�1=2

h eh

�� 0

� �;

where Lemmas 7 and 9 finish the proof. h


kA�1h ðvhðtÞ � ~uhðtÞÞk0 6 CH5; r P 4; t 2 ð0; T�:


Proof. Let us consider yhðtÞ ¼ A�1h ehðtÞ. From (37) it follows that

yhðtÞ ¼Z t

0e�ðt�sÞAh A�1



h Phqh;HðsÞds:

We first observe that arguing exactly as in the proof of [15, Lemma 4.13] we get

ke�ðt�sÞAh A�1h PhððuH � rÞehÞk0 6 C

1ffiffiffiffiffiffiffiffiffiffit � sp þ 1ffiffi

sp

� �kA�1

h ehk0 þ CH3ffiffi

sp kehk1: ð56Þ

Then,

kyhðtÞk0 6 CZ t

0


sp

� �kyhðsÞk0dsþ

Z t


h Phqh;HðsÞds�� þ Ct1=2H3 max

06s6tkehðsÞk1:

Applying now[15, Lemma 4.9] we get

max06t6T


Z t


h Phqh;HðsÞds�� þ CH3 max

06t6TkehðtÞk1: ð57Þ

For the second term on the right-hand-side above we apply Lemma 8. For the first one we use the decomposition

qh;H ¼ q1h;H þ q2

h;H; q1h;H ¼ ððuH � uÞ � rðvh � uÞÞ þ ððuH � uÞ � ruÞ:

We now argue exactly as in [15, (4.60) in Lemma 4.14], replacing one of the occurrences of z there by u� uH and making useof (26) and (27) with h replaced by H. This will allow us to obtain

max06t6T

Z t


h Phqh;HðsÞds�� 6 CH5;

which concludes the proof. h

Theorem 3. Let ðu; pÞ be the solution of (1) and (2). There exists a positive constant C such that the discrete velocity vh defined by(36) and the approximation to u over the finer grid, ~uh, satisfy the following bound:

kvhðtÞ � ~uhðtÞk0 6Ctj logðhÞj j logðHÞjH5

� �; t 2 ð0; T�; r P 4: ð58Þ

Proof. Let us define yhðtÞ ¼ tehðtÞ, where ehðtÞ ¼ vhðtÞ � ~uhðtÞ. Arguing as in the proof of Theorem 1, instead of (54) we nowarrive at

max06t6T


Z t

0e�Ahðt�sÞsPhqh;H ds

�� 0þmax

06t6T

Z t

0ke�Ahðt�sÞeh ds

�� 0

� �:

As in the proof of Theorem 2, applying now Lemma 2 to both terms on the right-hand side above we get

max06t6T


sA�1h Phqh;H

�� 0þmax

06t6TA�1

h eh

�� 0

� �;

where now Lemmas 6 and 10 finish the proof. h


kA�1=2h ðvhðtÞ � ~uhðtÞÞk0 6

Ct1=2 j logðhÞjH5; r P 4; t 2 ð0; T�:

Proof. Setting yhðtÞ ¼ t1=2A�1=2h ehðtÞ and arguing exactly as in the proof of Lemma 9, instead of (50) we now obtain

kyhðtÞk0 6

Z t

0


Z t

0e�ðt�sÞAh s1=2A�1=2


0þZ t


h

ehðsÞ2s1=2 ds

�� 0þ CH2

max06s6t

s1=2kehðsÞk1;


max06t6T


Z t

0e�ðt�sÞAh s1=2A�1=2


0þmax

06t6T

Z t


h

ehðsÞ2s1=2 ds

�� 0þ CH2 max

06t6Tt1=2kehðtÞk1:



max06t6T


� �max06s6T

ksA�1h Phqh;Hk0 þmax

06s6TkA�1

h ehk0

� �þ CH2 max

06t6Tt1=2kehk1:

For the first two terms on the right-hand-side above we apply Lemmas 6 and 10, respectively. For the last term we observethat denoting by yhðtÞ ¼ t1=2A1=2

h eh and arguing as in Theorem 1 we get

max06t6T

kyhðtÞk0 6 C j logðhÞjmax06t6T

kt1=2A�1=2h Phqh;Hk0 þ B

12;12

� �max06t6T

kehk0

� �;

so that applying now (14) and (45) to bound the first term on the right-hand side above, and the case l ¼ 0 in Lemma 8 forthe second one, the proof is completed. h

Theorem 4. Let ðu; pÞ be the solution of (1) and (2). There exists a positive constant C such that the discrete velocity vh defined by(36) and the approximation to u over the finer grid, ~uh, satisfy the following bound for r P 4:

kvhðtÞ � ~uhðtÞk1 6C

t3=2 j logðhÞj j logðhÞjH5 þ T1=2h4� �

; t 2 ð0; T�: ð59Þ

Proof. Let yhðtÞ ¼ t3=2A1=2h eh and argue as in the proof of Theorem 2 to get


t3=2A�1=2h Phqh;H

�� 0þmax

06t6Tt1=2A�1=2

h eh

�� 0

� �;

for t 2 ð0; T�. To bound the first term on the right-hand side above we apply (14) and (44), and for the second one we applyLemma 11. h

We now summarize the main results of the section in the following theorem.

Theorem 5. Let ðu; pÞ be the solution of (1) and (2). There exists a positive constant C such that the approximation to u over thefiner grid, ~uh, satisfy the following bounds for r ¼ 3;4 and t 2 ð0; T�:

kuðtÞ � ~uhðtÞk0 6C

tðr�2Þ=2 j logðhÞjj logðHÞjHrþ1 þ Ctðr�2Þ=2 hr

;


tðr�1Þ=2 j logðhÞj j logðhÞjHrþ1 þ T1=2hr� �

þ Ctðr�2Þ=2 hr�1

;

where in the last inequality we can replace the second j logðhÞj by j logðHÞj in the case r ¼ 3.

Proof. We use the decomposition u� ~uh ¼ ðu� vhÞ þ ðvh � ~uhÞ. To bound the first term we apply (11) and (28) while for thesecond we apply Theorems 1–4. h

Now, we get the error bounds for the pressure. We begin with some error estimates for the time derivative of vh � ~uh.

Lemma 12. Let ðu; pÞ be the solution of (1) and (2). Then there exists a positive constant C such that the discrete velocity vh

defined by (36) and the approximation to u over the finer grid, ~uh satisfy the following bound for r ¼ 3;4:

kvh;tðtÞ � ~uh;tðtÞk�1 6C

tðr�1Þ=2 j logðhÞj j logðhÞjHrþ1 þ hr� �

; t 2 ð0; T�: ð60Þ

In the case r ¼ 3 the second logðhÞ can be replaced by logðHÞ.

Proof. Using (37) and taking into account that keh;tk�1 6 CkA�1=2h eh;tk0 we obtain

keh;tk�1 6 kA1=2h ehk0 þ kA

�1=2h Ph ðuH � rÞeh þ qh;H

� �k0 6 kehk1 þ Ckehk0 þ kA

�1=2h Phqh;Hk0;

after using (53) in the last inequality. Applying now Theorems 1–4 together with (14) and (44) we reach (60). h

The following Lemma is proved in [15, Corollary 4.19] and Proposition 3.1 in [25] and [26] (see also [7] and [16]).

Lemma 13. Let ðu; pÞ be the solution of (1) and (2) and let ðuh; phÞ and ðvh; ghÞ the approximations defined in (18) and (19) and(22) and (23), respectively. Then, the following bound holds for r ¼ 2;3;4


kphðtÞ � pðtÞkL2=R 6C

tðr�2Þ=2 hr�1; t 2 ð0; T�; ð61Þ

kghðtÞ � pðtÞkL2=R 6C

tðr�2Þ=2 hr�1; t 2 ð0; T�: ð62Þ

Theorem 6. Let ðu; pÞ be the solution of (1) and (2). There exists a positive constant C such that the approximation to p over thefiner grid, ~ph, satisfies the following bound for t 2 ð0; T� and r ¼ 3;4:

k~phðtÞ � pðtÞkL2=R 6C

tðr�2Þ=2 hr�1 þ Ctðr�1Þ=2 j logðhÞj j logðhÞjHrþ1 þ hr

� �:

In the case r ¼ 3 the second logðhÞ can be replaced by logðHÞ.

Proof. We use the decomposition

k~ph � pkL2=R 6 k~ph � ghkL2=R þ kgh � pkL2=R:

To bound the second term on the right-hand-side above we apply (62). For the first one subtracting (22) from (20) and apply-ing the inf-sup condition (10) we obtain

bk~ph � ghkL2=R 6 k~uh � vhk1 þ kðuH � rÞehk�1 þ kqh;Hk�1 þ keh;tk�1:

We first observe that applying (31) we get kðuH � rÞehk�1 6 Ckehk0. To bound kqh;Hk�1 we apply (44). Now, the proof con-cludes applying Theorems 1–4 together with (60). h

We state in the following theorem the results that can be obtained for the mini-element.

Theorem 7. Let ðu; pÞ be the solution of (1) and (2). There exists a positive constant C such that the approximations over the finergrid computed using the mini-element, ð~uh; ~phÞ, satisfy the following bounds for t 2 ð0; T�:

k~uhðtÞ � uðtÞk0 6 CH2 þ Ch2;

k~uhðtÞ � uðtÞk1 6 Cj logðhÞjH2 þ Ch;

k~phðtÞ � pðtÞkL2=R 6 Cj logðhÞjH2 þ Ch:

Remark 2. Note that the previous bound is not optimal in the L2-norm. However, it is the best bound that it is possible toobtain with the mini-element. This lack of optimality is common to other two-grid schemes based on linear elements, see forexample, [7,18].

4. Semi-discretization in space. The second two-grid algorithm

As in the previous section we will concentrate on the approximations obtained with the Hood–Taylor mixed finite ele-ment and r ¼ 3 or r ¼ 4 and we will state at the end of the section the results that can be obtained for the mini-elementmethod with a much simpler analysis.

In the second algorithm we consider, the first level, as before, is given by the standard mixed finite-element approxima-tion to (1) and (2), that is, the solution of (18) and (19) with initial condition uHð0Þ ¼ PHðu0Þ. In the second level we solve alinearized problem on a finer grid and given ~uhð0Þ ¼ Phu0, we compute ~uhðtÞ 2 Xh;r and ~phðtÞ 2 Qh;r�1; t 2 ð0; T�, satisfying, forall /h 2 Xh;r and wh 2 Qh;r�1

ð~uh;t ;/hÞ þ ðr~uh;r/hÞ þ bðuH; ~uh;/hÞ þ bð~uh;uH;/hÞ þ ðr~ph;/hÞ ¼ ðf ;/hÞ þ bðuH;uH;/hÞ; ð63Þðr � ~uh;whÞ ¼ 0: ð64Þ

Observe that the approximation ~uh is the result of one step of Newton’s method for the Galerkin ðuh; phÞ approximation inXh;r Q h;r�1 (Eqs. (18) and (19) with H replaced by h) with ðuH; pHÞ as initial approximation. For this reason, in this sectionwe study the error eh ¼ uh � ~uh.

It is easy to obtain that

eh;t þ Aheh þ BhðuH; ehÞ þ Bhðeh;uHÞ ¼ Phqh;H; ð65Þ
where
qh;H ¼ �Fðeh;H; eh;HÞ;

where, here and in the sequel,

eh;H ¼ uH � uh:


The analysis in this section is closely related to that in the previous section. However, some extra results are needed. Weshall use the following two bounds,

k/hkL2d=ðd�1Þ 6 C /hk k1=20 A1=2

h /h

�� 1=2

0; 8/h 2 Vh;r ; ð66Þ

k/hkL1 6 C A1=2h /h

�� 1=2

0Ah/hk k1=2

0 ; 8/h 2 Vh;r ; ð67Þ

which follow from Corollary 4.4 and Lemma 4.4 in [25]. Also we shall use the following two bounds

kA�1=2h Bhðeh;H; eh;HÞk0 6 C eh;H

�� 1=20 eh;H

�� 3=21 ; ð68Þ

kA�1h Bhðeh;H; eh;HÞk0 6 C eh;H

�� 0 eh;H

�� 1; ð69Þ

with C independent of h, and where here and in the sequel Bhðvh;whÞ ¼ PhFðuh;whÞ. Both are easily obtained by duality argu-ments, the first one from (66) and the second one from (67). Notice also that as a consequence of (66) and (67) and (16) we

have that e�tAh /h

�� L1 6 Ct�3=4 /hk k0 and e�tAh A1=2

h /h

�� L2d=ðd�1Þ

6 Ct�3=4 /hk k0 so that by using duality arguments together with

these two inequalities the following two bounds easily follow

e�ðt�sÞAh Bhðeh;H; eh;HÞ��

0 6C

ðt � sÞ3=4 eh;H

�� 0 eh;H

�� 1; ð70Þ

e�ðt�sÞAh A1=2h Bhðeh;H; eh;HÞ

�� 06

C

ðt � sÞ3=4 eh;H

�� 21: ð71Þ

Lemma 14. There exists a positive constant C ¼ CðM2Þ such that

ku� uHkL1 6 CH1=2:

Proof. We will use the fact that, due to Lemma 4.3 and 4.4 in [25], and Corollary 4.4 in [25],

krPHukL6 6 C Auk k0: ð72Þ

We write u� uH ¼ ðI �PHÞuþ ðPHu� uHÞ. Applying (9), we have kPHu� uHkL1 6 CH�3=2kPhu� uHk0 6 CH1=2, where in thelast inequality we have applied (11) and (26). On the other hand, applying [25, (4.43)]

kðI �PHÞukL1 6 CkðI �PHÞuk1=2L6 rðI �PHÞuk k1=2

L6 6 CkðI �PHÞuk1=21 ð ruk kL6 þ rPhuk kL6 Þ1=2

:

Now, where, in the last inequality we have applied (5) and [25, Lemma 4.4]. Furthermore, applying (11), (5) and (72) theproof is finished. h

Lemma 15. Let ðu; pÞ be the solution of (1) and (2). Then there exists a positive constant C such that the approximations uh and ~uh

to the velocity u over the fine mesh satisfy the following bound:

kAl=2h ðuhðtÞ � ~uhðtÞÞk0 6 CH7=2�l; r P 3; l ¼ 0;1; t 2 ð0; T�: ð73Þ

Proof. Follow the arguments in the proof of Lemma 8 to obtain (47). Now, for l ¼ 1 we writes1=2kPhqh;Hk0 6 Cs1=2 eh;H

�� L1 eh;H

�� 1, so that applying Lemma 14 and (26) the proof of the case l ¼ 1 is finished. For l ¼ 0,

applying (68) we have

s1=2 A�1=2h Phqh;H

�� 06 Cs1=2 eh;H

�� 1=20 eh;H

�� 3=21 ¼ C eh;H

�� 0 eh;H

�� 1

� �1=2s1=2eh;H

�� 1;

so that applying (26) the proof is finished. h



kA�1h ðuhðtÞ � ~uhðtÞÞk0 6 CH5; r P 3; t 2 ð0; T�:

Proof. Follow the arguments in the proof of Lemma 10, but notice that now due to the terms ðeh � rÞuH and ðr � uHÞeH ,instead of (56) we have


e�ðt�sÞAh A�1h Ph ðuH � rÞehð Þ

�� 06 C


sp

� �kA�1

h ehk0 þ CH3ffiffi

sp kehk1 þ

H2ffiffisp kehk0

!:

Thus, instead of (57) we arrive at

max06t6T


Z t


h Phqh;HðsÞds�� þ C H3 max

06t6TkehðtÞk1 þ H2 max

06t6TkehðtÞk0

� �:

Thanks to Lemma 15 we have that the last two terms on the right-hand side above are bounded by CH11=2. For the first one,applying first (69) and then (27) we conclude that it is also bounded by CH5. h



kA�1=2h ðuhðtÞ � ~uhðtÞÞk0 6

Ct1=2 H5; r P 3; t 2 ð0; T�: ð74Þ

Proof. Let yhðtÞ ¼ t1=2A�1=2h ehðtÞ and follow the arguments in the proof of Lemma 9 so that instead of (48) we now have

kyhðtÞk0 6

Z t

0

Cffiffiffiffiffiffiffiffiffiffit � sp A�1

h s1=2ðBhðeh; uHÞ þ BhðuH; ehÞ��

0ds

þZ t


h Phs1=2qh;HðsÞds��

0þ 1

2

Z t

0e�ðt�sÞAh s�1=2A�1=2

h ehðsÞds��

0: ð75Þ

Now observe that by using kuHðsÞ � uðsÞkj 6 CH3�j=s1=2, instead of (49) we now have

A�1h s1=2ðBhðeh;uHÞ þ BhðuH; ehÞ

�� 06 C yhðsÞk k0 þ H3 ehk k1 þ H2 ehk k0

� �:

Thus, instead of (51) we now get

max06t6T


� �max06s6T

s A�1h Phqh;H

�� 0þmax

06s6TA�1

h ehðsÞ��

0

� �þ C H3 max

06t6Tkehk1 þ H2 max

06t6Tkehk0

� �: ð76Þ

Due to Lemma 15 we have that the last two terms on the right-hand side of (76) are oðH5Þ, and due to Lemma 16 the secondone is OðH5Þ. Finally due to (69) the first one can be bounded by C s1=2�h;H

�� 0 s1=2�h;H

�� 1, which, due to (26) is also OðH5Þ. h

Theorem 8. Let ðu; pÞ be the solution of (1) and (2). Then there exists a positive constant C such that the approximations uh and ~uh


kuhðtÞ � ~uhðtÞk0 6Ct

H5; t 2 ð0; T�; r P 3: ð77Þ

Proof. Let yhðtÞ ¼ tehðtÞ and argue as in the proof of Theorem 1 so that similarly to (54) we now get

max06t6T


Z t

0e�Ahðt�sÞsPhqh;H ds

�� 0þmax

06t6T

Z t

0e�Ahðt�sÞeh ds

�� 0

� �:

Using (70) to bound the first term on the right-hand side above, and (17) for the second one, we get

max06t6T

kyhðtÞk0 6 C T1=4 max06t6T

t1=2�h;HðtÞ��

0 t1=2�h;HðtÞ��

1 þ B12;12

� �max06t6T

t1=2A�1=2h ehðtÞ

�� 0

� �;

so that applying (26) and Lemma 17 the proof is finished. h

Theorem 9. Let ðu; pÞ be the solution of (1) and (2). Then there exists a positive constant C such that the approximations uh and ~uh


kuhðtÞ � ~uhðtÞk1 6C

tðr�1Þ=2 Hrþ1; t 2 ð0; T�; r ¼ 3;4: ð78Þ


Proof. Let yhðtÞ ¼ tðr�1Þ=2A1=2h ehðtÞ and follow the arguments in the proof of Lemma 8 so that now, instead of (46) we get

max06t6T


Z t

0e�ðt�sÞAh sðr�1Þ=2A1=2


0þ ðr � 1Þ

2

Z t

0e�ðt�sÞAh sðr�3Þ=2A1=2

h ehðsÞds��

0

� �: ð79Þ

Applying (71) to bound the first term on the right-hand side above and (17) for the second one, we have

max06t6T

kyhðtÞk0 6 C T1=4 max06t6T

t1=2�h;HðtÞ��

1 tðr�2Þ=2�h;HðtÞ��

1 þ CB12;12

� �max06t6T

tðr�2Þ=2ehðtÞ��

0

� �: ð80Þ

Due to (26) the first term on the right-hand side above is bounded by CH2Hr�1 ¼ CHrþ1. For r ¼ 4, the second one is boundedin Theorem 8. When r ¼ 3, we may write t1=2ehðtÞ

�� 0 ¼ tehðtÞk k1=2

0 ehðtÞk k1=20 so that applying Theorem 8 and Lemma 15, the

second term on the right-hand side of (80) is bounded by CH5=2H7=4 ¼ oðH4Þ h

Remark 3. For r ¼ 3 it is possible to prove the bound

kuhðtÞ � ~uhðtÞk1 6CtðH9=2j logðhÞj þ H17=4Þ; t 2 ð0; T�; r ¼ 3:

To do so, apply Lemma 2 and (68) to bound the first term on the right-hand side of (79) and the same bound as before for thesecond term.

Finally, repeating (with obvious changes) the analysis in Section 3 for the pressure, the following result is easily proved

Theorem 10. Let ðu; pÞ be the solution of (1) and (2). There exists a positive constant C such that the approximation to p over thefiner grid, ~ph, satisfy the following bound for t 2 ð0; T� and r ¼ 3;4:

k~phðtÞ � phðtÞkL2=R 6C

tðr�1Þ=2 Hrþ1; t 2 ð0; T�: ð81Þ

We now summarize the main results of the section in the following theorem.

Theorem 11. Let ðu; pÞ be the solution of (1) and (2). There exists a positive constant C such that the approximations ð~uh; ~phÞsatisfy the following bounds for r ¼ 3;4 and t 2 ð0; T�:

kuðtÞ � ~uhðtÞk0 6Ct

H5 þ Ctðr�2Þ=2 hr

;


tðr�1Þ=2 Hrþ1 þ Ctðr�2Þ=2 hr�1

;

k~phðtÞ � pðtÞkL2=R 6C

tðr�1Þ=2 Hrþ1 þ Ctðr�2Þ=2 hr�1

:

Proof. We use the decomposition u� ~u ¼ ðu� uhÞ þ ðuh � ~uÞ and apply (26) to bound the first term and Theorems 8 and 9for the second. For the pressure, using the decomposition p� ~ph ¼ ðp� phÞ þ ðph � ~phÞ and applying (61) and Theorem 10 theproof is finished. h

Finally, with a much simpler analysis, that we do not detail here for brevity, the following result can be proved

Theorem 12. Let ðu; pÞ be the solution of (1) and (2). There exists a positive constant C such that the approximations over the finergrid computed using the mini-element, ð~uh; ~phÞ, satisfy the following bounds for t 2 ð0; T�:

k~uhðtÞ � uðtÞk0 6 CH3 þ Ch2;

k~uhðtÞ � uðtÞk1 6 CH2 þ Ch;

k~phðtÞ � pðtÞkL2=R 6 CH2 þ Ch:

Remark 4. Note that Theorems 5 and 11 state the uniform convergence of optimal order of the methods for t bounded awayfrom zero. Due to the loss of regularity of the solution at t ¼ 0 the bound is singular as t approaches zero. The order of thesingularity depends on the smoothness of the solution needed to achieve the highest possible order for a given mixed finiteelement, see the bounds (7) and (8).

5. Fully discrete case

In this section we consider the fully discrete case. Let us assume that we integrate in time Eqs. (18)–(21) for the firstmethod or Eqs. (18) and (19) and (63) and (64) for the second method using the backward Euler method or the two-step


backward differentiation formula (BDF). In the case of the two-step BDF method the first step is carried out using the back-ward Euler method. We will denote by ðUn

H; PnHÞ the fully discrete Galerkin approximations to the velocity and pressure at the

time level tn ¼ nk for 0 6 n 6 N and k ¼ Dt ¼ T=N. We will denote by ðeUnh;ePn

hÞ the fully discrete approximations to the veloc-ity and pressure over the finer grid at the time level tn.

Let us denote by enH ¼ uHðtnÞ � Un

H and by ~enh ¼ ~uhðtnÞ � eUn

h the temporal errors in the approximations UnH and eUn

h respec-tively. Let us denote by pn

H ¼ pHðtnÞ � PnH and by ~pn

h ¼ ~phðtnÞ � ePnh the temporal errors in the approximations Pn

H and ePnH ,

respectively. In [16] we have proved the following error bounds. There exist constants Cl0 and k0 such that for k 6 k0 the tem-poral errors of the Galerkin approximation satisfy the following error bounds

kenHk0 þ tnkAHen

Hk0 6 Cl0

kl0

tl0�1n

; kpnHkL2ðXÞ=R 6 Cl0

kl0

tð2l0�1Þ=2n

; 1 6 n 6 N;

where l0 ¼ 1 for the Euler method and l0 ¼ 2 for the two-step BDF. Let us remark that usingken

Hk1 6 CkA1=2H en

Hk0 6 CkenHk

1=20 kAHen

Hk1=20 error bounds in the H1 norm are also obtained in a straightforward manner.

Using the same technique developed in [16] it can also be proved that analogous error bounds hold for the approxima-tions over the finer grid. More precisely, for both the first and second algorithms there exist constants Cl0 and k0 such that fork 6 k0 the temporal errors of the two-grid approximation satisfy the following error bounds

k~enhk0 þ tnkAh~en

hk0 6 Cl0

kl0

tl0�1n

; k~pnhkL2ðXÞ=R 6 Cl0

kl0

tð2l0�1Þ=2n

; 1 6 n 6 N;

where l0 ¼ 1 for the Euler method and l0 ¼ 2 for the two-step BDF. Finally, using the decompositions

uðtnÞ � eUnh ¼ ðuðtnÞ � ~uhðtnÞÞ þ ~en

h;

pðtnÞ � ePnh ¼ ðpðtnÞ � ~phðtnÞÞ þ ~pn

h;

the error bounds of the fully discrete approximations are obtained as the sum of the spatial errors (the errors in the semi-discrete approximations we have already bounded in the previous sections) plus the temporal errors.

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Optimal error bounds for two-grid schemes applied to the Navier–Stokes equations

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