A defect-correction method for the incompressible Navier–Stokes equations

A defect-correction method for theincompressible Navier–Stokes equations

W. Layton a,1, H.K. Lee b,*, J. Peterson c

a Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USAb Department of Mathematical Sciences, Clemson University, Clemson, SC 29634-1907, USA

c Department of Mathematics, Iowa State University, Ames, IA 50011, USA

Abstract

A defect-correction method for the incompressible Navier–Stokes equation with a

high Reynolds number is considered. In the defect step, the artificial viscosity parameter

r is added to the Reynolds number as a stability factor, and the residual is taken care ofin the correction step. H 1 and L2 error estimations are derived for the one-step defect-correction method, and the results of some numerical experiments are presented. These

results show that, for the driven cavity, two defect-correction steps antidiffuse the ar-

tificial viscosity approximation nearly optimally. This combination gives on a very

coarse mesh, results indistinguishable from a benchmark, very fine mesh calcula-

tion. � 2002 Elsevier Science Inc. All rights reserved.

Keywords: Defect-correction method; Navier–Stokes equations; Finite element method

1. Introduction

The defect-correction method is an iterative improvement technique which isused to increase the accuracy of a computed solution without introducing agrid refinement. In general, the basic technique can be simply described asfollows. Let F be a mapping F : X ! Y where X and Y are given normed linear

Applied Mathematics and Computation 129 (2002) 1–19www.elsevier.com/locate/amc

*Corresponding author.

E-mail addresses: [email protected] (W. Layton), [email protected] (H.K. Lee), jspeters@ias-

tate.edu (J. Peterson).

URL: http://www.math.pitt.edu/�wjl (W. Layton).1 Partially supported by NSF grants DMS 9972622, INT 9805563 and INT 9814115.

0096-3003/02/$ - see front matter � 2002 Elsevier Science Inc. All rights reserved.

PII: S0096 -3003 (01 )00026-1

spaces and suppose that our goal is to find a good approximation to x� whereF ðx�Þ ¼ 0. Suppose that we actually solve a related problem FðxÞ ¼ 0 whosesolution we denote by x0. We then find the residual or defect d0 ¼ F ðx0Þ and usethis to solve for the error or correction e0 either via the nonlinear updateFðx1Þ �Fðx0Þ ¼ d0 or the linear update F

0ðx0Þe0 ¼ d0 in order to obtain thecorrected solution x1, by setting x1 ¼ x0 þ e0; clearly the procedure can be re-peated. If the mapping F is nonlinear, then one often chooses a linearization ofF in the correction phase in order to simplify the calculations. In the case ofsolving linear systems, the well-known iterative refinement method is an ex-ample of a defect-correction technique.Defect-correction type methods have been successfully used in many set-

tings; see [12] for a survey of the technique and several examples. These types ofmethods have been studied for convection–diffusion problems by Axelsson andLayton [2], Hemker [8], and Ervin and Layton [3]. In [9] the first author initiallyinvestigated a defect-correction type method for the incompressible Navier–Stokes equations for high Reynolds numbers.This paper is organized as follows. In the remainder of this section we

present the notation that will be used throughout this paper, and in Section 2the standard weak form of the Navier–Stokes equation and a finite elementalgorithm of a defect-correction method are presented. In Section 3 results ofH 1- and L2-error estimates for the solution by the defect-correction method arestated and proved, and some numerical results are presented in Section 4. Inthese simulations, a very coarse mesh approximation is compared to the veryfine mesh, benchmark calculations of [4]. As expected, the usual Galerkinmethod fails for high enough Re and the artificial viscosity Galerkin approx-imation is overly diffuse. As expected, the defect-correction updates are muchbetter. It is, however, quite surprising that the second correction is so good asto be indistinguishable from the fine mesh results of [4].

1.1. Notation

Let X be an open, bounded region in R2 or R3 with a Lipschitz continuousboundary C. Throughout this paper we will use the standard notation forSobolev spaces (see [1]). The Sobolev space of real-valued functions withsquare integrable derivatives of order up to r equipped with the usual norm isdenoted by k � kr. We will denote H 0ðXÞ by L2ðXÞ and the standard L2-innerproduct by ð�; �Þ. Also HrðXÞ will denote the space of vector-valued functionseach of whose n components belong to HrðXÞ and the dual space of HrðXÞ willbe denoted by H�rðXÞ. Of particular interest to us will be the constrainedspaces

L20ðXÞ ¼ w 2 L2ðXÞZ

XwdX

��

¼ 0�;

2 W. Layton et al. / Appl. Math. Comput. 129 (2002) 1–19

H10ðXÞ ¼ fv 2 H1ðXÞ j v ¼ 0 on Cg;

and

V ¼ fv 2 H10ðXÞ j divv ¼ 0g:

For simplicity, we will denote H10ðXÞ and L20ðXÞ by

X ¼ H10ðXÞ and Q ¼ L20ðXÞ

throughout this paper. We will also make use of the semi-norm juj1 ¼ kruk0for u 2 H1ðXÞ.

2. A defect-correction type method for the Navier–Stokes equations

In this section we present a defect-correction type method for the steady-state incompressible Navier–Stokes equations which is of use in calculating thesolution at higher values of the Reynolds number. The scheme entails thecalculation of the nonlinear Navier–Stokes equations with an added artificialviscosity term on a relatively coarse grid and the correction of this solution onthe same grid using a linearized defect-correction technique. The method in-corporates the artificial viscosity term as a stabilizing factor, making both thenonlinear system easier to resolve and the linearized system easier to precon-dition or solve by other iterative means.

2.1. The incompressible Navier–Stokes equations

We are interested in approximating the solution of the steady-state, in-compressible Navier–Stokes equations using finite element methods. In par-ticular, we consider the primitive variable formulation in terms of the velocity uand the pressure p given by

� 1

ReDuþ u � ruþrp ¼ f in X;

divu ¼ 0 in X;

u ¼ 0 on C;

ð2:1Þ

where Re is the Reynolds number and f 2 H�1ðXÞ is the given body force perunit mass. In the sequel we will represent the inverse Reynolds number by m.Here, for simplicity, we have imposed a no-slip boundary condition; otherboundary conditions may be handled by standard techniques, see [5,7].We consider the following standard weak formulation of (2.1); see [5,13].

Seek ðu; pÞ 2 X� Q such that

W. Layton et al. / Appl. Math. Comput. 129 (2002) 1–19 3

ma0ðu; vÞ þ a1ðu; u; vÞ þ bðv; pÞ ¼ ðf; vÞ 8v 2 X;

bðu; qÞ ¼ 0 8q 2 Q;ð2:2Þ

where

a0ðu; vÞ ¼Z

Xru � rvdX 8u; v 2 X; ð2:3Þ

a1ðu;w; vÞ ¼1

2

ZXu � rw � vdX

��Z

Xu � rv � wdX

�8u; v;w 2 X;

ð2:4Þ

and

bðu; qÞ ¼ �Z

XqdivudX 8u 2 X; q 2 Q: ð2:5Þ

For analysis purposes, it is useful to restate (2.2) into the following equivalentproblem: seek u 2 V satisfying

ma0ðu; vÞ þ a1ðu; u; vÞ ¼ ðf; vÞ 8v 2 V: ð2:6Þ

In order to discretize, we choose finite dimensional subspaces Xh � X,Qh � Q and define the discrete problem corresponding to (2.2) as follows: seekðuh; phÞ 2 Xh � Qh such that

ma0ðuh; vhÞ þ a1ðuh; uh; vhÞ þ bðvh; phÞ ¼ ðf; vhÞ 8vh 2 Xh;

bðuh; qhÞ ¼ 0 8qh 2 Qh:ð2:7Þ

If we define the space of discretely divergence-free functions as

Vh ¼ fvh 2 Xh j bðvh; qhÞ ¼ 0 8qh 2 Qhg; ð2:8Þ

then the discrete problem corresponding to (2.6) can be written as: seek uh 2 Vh

satisfying

ma0ðuh; vhÞ þ a1ðuh; uh; vhÞ ¼ ðf; vhÞ 8vh 2 Vh: ð2:9Þ

As usual, Vh 6�V; we define a measure of the ‘‘angle’’ between the spaces V andVh to be

H ¼ supvh2Vhjvh j1¼1

infv2V

jv� vhj1: ð2:10Þ

This particular weak form of the Navier–Stokes equations has been exten-sively studied both from a theoretical and computational viewpoint. It is wellknown, see [5, 13], that the bilinear and trilinear forms defined in (2.3)–(2.5)satisfy certain continuity conditions. In particular we have that

ja0ðu; vÞj6 juj1jvj16 kuk1kvk1 8u; v 2 X; ð2:11Þ


ja1ðu; v;wÞj6N juj1jvj1jwj16Nkuk1kvk1kwk1 8u; v;w 2 X; ð2:12Þ

and

jbðv; qÞj6ffiffiffin

pjvj1kqk0 8v 2 X; q 2 Q: ð2:13Þ

In addition, the bilinear form a0ð�; �Þ is coercive on X� X, i.e.,

ja0ðv; vÞjP jvj21 8v 2 X ð2:14Þ

and a1ð�; �; �Þ satisfies the propertya1ðu; v; vÞ ¼ 0 8u; v 2 X ð2:15Þ

due to our definition of the form given in (2.4).Using the fact that the form bð�; �Þ satisfies the well-known stability property

supv2X

bðv; qÞkvk1

P bkqk0 ð2:16Þ

for all q 2 Q with b > 0, one can show that for each solution u of (2.6) thereexists a p such that the pair ðu; pÞ satisfies (2.2). Uniqueness of the solution canbe guaranteed if, in addition,

f ¼ m � Nmkfk�1 > 0: ð2:17Þ

Moreover, the solution can be bounded by

juj161

mkfk�1: ð2:18Þ

Similar existence and uniqueness results hold for the approximate problem ifthe approximating spaces are chosen so that the form bð�; �Þ satisfies the sta-bility condition over the subspaces, i.e.,

supvh2Xh

bðvh; qhÞkvhk1

P bhkqhk0 ð2:19Þ

for all qh 2 Qh with bh > 0. In the case that ðu; pÞ and ðuh; phÞ denote the uniquesolutions of (2.2) and (2.7), respectively, standard error estimates can be de-rived for ku� uhk1 and kp � phk0. Again see [5,13] for details.

2.2. A defect-correction type method

We propose a method which allows us to solve the nonlinear system (2.7) ona coarser mesh than one uses when employing standard finite element tech-niques; the coarse-mesh solution is then corrected using the same grid. Themethod we propose is a defect-correction-type method in which a linearizedcorrection step is used. The defect-correction method which we consider


incorporates an artificial viscosity parameter r as a stabilizing factor in thesolution algorithm. For a fixed value of the grid parameter h the method re-quires the solution of one nonlinear system and a few, usually one or two,linear correction steps. The proposed method is described in the followingparagraphs.We consider the following problem which is identical to (2.7) except for an

artificial viscosity term. Let ðuh0; ph0Þ 2 ðXh;QhÞ satisfy

mð þ rhÞa0ðuh0; vhÞ þ a1ðuh0; uh0; vhÞ þ bðvh; ph0Þ ¼ ðf; vhÞ 8vh 2 Xh;

bðuh0; qhÞ ¼ 0 8qh 2 Qh:ð2:20Þ

The resulting residual or defect Rðuh0; ph0Þ for the momentum equation is definedby

ðRðuh0; ph0Þ; vhÞ ¼ ðf; vhÞ � ma0ðuh0; vhÞ � a1ðuh0; uh0; vhÞ � bðvh; ph0Þ: ð2:21Þ

We let the error or correction ð�h0; qh0Þ satisfy the linear problem

mð þ rhÞa0ð�h0; vhÞ þ a1ðuh0; �h0; vhÞ þ a1ð�h0; uh0; vhÞ þ bðvh; qh0Þ

¼ ðRðuh0; ph0Þ; vhÞ 8vh 2 Xh;

bð�h0; qhÞ ¼ �bðuh0; qhÞ 8qh 2 Qh:

ð2:22Þ

We expect uh1 ¼ uh0 þ �h0, ph1 ¼ ph0 þ qh

0 to be a better approximation to uh than uh0.

To obtain an equation for ðuh1; ph1Þ we use the definition for the residual (2.21) torewrite the linear problem for the correction; we obtain

mð þ rhÞa0ðuh1; vhÞ þ a1ðuh0; uh1; vhÞ þ a1ðuh1; uh0; vhÞ þ bðvh; ph1Þ

¼ ðf; vhÞ þ rha0ðuh0; vhÞ þ a1ðuh0; uh0; vhÞ 8vh 2 Xh;

bðuh1; qhÞ ¼ 0 8qh 2 Qh:

ð2:23Þ

In general, the algorithm can be described as follows.

Algorithm 2.1. Step 1. Solve the nonlinear system (2.20) for ðuh0; ph0Þ.

Step 2. For j ¼ 0; 1; 2; . . . compute the corrected solution from

mð þ rhÞa0ðuhjþ1; vhÞ þ a1ðuhj ; uhjþ1; vhÞ þ a1ðuhjþ1; uhj ; vhÞ þ bðvh; phjþ1Þ

¼ ðf; vhÞ þ rha0ðuhj ; vhÞ þ a1ðuhj ; uhj ; vhÞ 8vh 2 Xh; ð2:24Þ

bðuhjþ1; qhÞ ¼ 0 8qh 2 Qh:

For each j the residual is given by

R uhj ;phj

� ; vh

� ¼ f; vh �

� ma0 uhj ; vh

� � a1 uhj ;u

hj ; v

h�

� b vh;phj�

ð2:25Þ


and the correction ð�hj ; qhj Þ satisfies the linear system

ðm þ rhÞa0ð�hj ; vhÞ þ a1ðuhj ; �hj ; vhÞ þ a1ð�hj ; uhj ; vhÞ þ bðvh; qhj Þ

¼ Rðuhj ; phj Þ; vh�

8vh 2 Xh;

bð�hj ; qhÞ ¼ �bðuhj ; qhÞ 8qh 2 Qh:

ð2:26Þ

In the numerical computations we have typically found that one or two cor-rection steps is adequate.Note that if the iteration converges, then from (2.26) we have that

Rðuhj ; phj Þ ! 0 so that our original equation (2.7) is satisfied. However, we stressthat for a convection-dominated problem (i.e., small m) the result after only oneor two updates provides an approximate solution with better accuracy andquality if the process is iterated to convergence. We thus consider only the errorafter one or two steps.

3. Error estimates

In this section we show that one correction in our defect-correction algo-rithm increases the formal order of the method by one power of h up to theorder of the best approximation in the space. The first theorem gives the errorin the H1-seminorm for the velocity while the second gives the error in the L2-norm of the velocity. TheH1 error estimate is a sharper version of a result in [9]and is presented here for completeness.The theorem shows, for example, that if one implements the Taylor–Hood

element in which quadratic elements are used for the velocities and linear el-ements for the pressure, then performing one correction step increases the H1

estimate for the velocity from OðhÞ to Oðh2Þ. The second update then increasesthe L2 rate of convergence from Oðh2Þ to Oðh3Þ.We begin this section by stating the H 1- and L2- error estimates for the initial

solution uh0. These results follow the standard finite element estimates for theNavier–Stokes equations, the only difference being the contribution from theadded artificial viscosity term. Since the problem for ðuh0; ph0Þ has the artificialviscosity term rha0ð�; �Þ, we obtain an OðhÞ term in the estimate. We state theresult in the following lemma.

Lemma 3.1. Assume that ðu; pÞ and ðuh0; ph0Þ denote the unique solutions of (2.2)and (2.20), respectively. Then there exist constants C1 and C2, independent of h,such that

ju� uh0j16C1 inf~uuh2Vh

ju� ~uuhj1 þ C2H inf~pph2Qh

kp � ~pphk0 þrmf

hkfk�1; ð3:1Þ

where H is defined by (2.10), f is defined by (2.17),


C1 ¼ 1þ1

fm

�þ 2N

mkfk�1

�; and C2 ¼

ffiffiffin

p

f:

If the solution to the linearized adjoint problem corresponding to (2.2) issufficiently regular, then one can obtain an estimate for e0 ¼ u� uh0 in the L

2-norm. In particular, the linearized adjoint problem is to seek ðw; kÞ satisfying

� 1

ReDw� u � rwþ w � ðruÞT þrk ¼ e0 in X;

r � w ¼ 0 in X;ZX

kdX ¼ 0; w ¼ 0 on oX:

ð3:2Þ

We make the assumption that the solution to (3.2) is H2-regular; in particular,we assume that there is a constant Cw ¼ Cwðu; mÞ such that

kwk2 þ kkk16Cwke0k0: ð3:3Þ

The variational formulation of (3.2) is to seek ðw; kÞ 2 ðX;QÞ such that

ma0ðv;wÞ þ a1ðu; v;wÞ þ a1ðv; u; fwÞ þ bðv; kÞ ¼ ðe0; vÞ 8v 2 X;

bðw; qÞ ¼ 0 8q 2 Q:ð3:4Þ

The following result for approximating w by functions in Vh is standard; see [5]:

inf~wwh2Vh

krðw� ~wwhÞk06 1

�þ

ffiffiffin

p

bh

�inf~wwh2Xh

krðw� ~wwhÞk0;

where bh is the constant in the stability condition (2.19) for bð�; �Þ. If we assume,for example, that w 2 H2ðXÞ, then the best approximation property of thespace Xh gives

inf~wwh2Xh

krðw� ~wwhÞk06Chkwk26Chke0k0:

Combining these results yields

inf~wwh2Vh

krðw� ~wwhÞk06C 1

�þ

ffiffiffin

p

bh

�hke0k0 ¼ Cahke0k0: ð3:5Þ

With these assumptions, the estimate for ke0k0 can be obtained in an analogousmanner to the standard estimate for ku� uhk0.

Lemma 3.2. Assume that ðu; pÞ and ðuh0; ph0Þ denote the unique solutions of (2.2)and (2.20), respectively. Then there exist constants C3 and C4, independent of h,such that


ku� uh0k06 h C3ju

� uh0j1 þ C4N ju� uh0j21 þ

ffiffiffin

pC4 inf

~pph2Qhkp � ~pphk0

!

þ h2C4rmkfk�1; ð3:6Þ

where

C3 ¼ Ca m

�þ 2N

mkfk�1

�þ c1Cw

ffiffiffin

p; and C4 ¼ Ca þ c2Cw:

We now turn to estimating the error in the corrected solution uh1 and thesolution u of (2.2). We first estimate je1j1, where e1 ¼ u� uh1. This result gives usthat the H 1-error after one correction is one power of h better than the H 1-errorin the initial solution uh0 up to the power of h in the best approximation in thegiven subspace.

Theorem 3.1. Let ðu; pÞ, ðuh0; ph0Þ, and ðuh1; ph1Þ denote the unique solutions to (2.2),(2.20), and (2.23), respectively. Then there exist constants C5 and C6, independentof h, such that

ju� uh1j16 C5

�þ rh

f

�inf~uuh2Vh

ju� ~uuhj1 þ C6 inf~pph2Qh

kp � ~pphk0 þrfhju� uh0j1

þ Nfju� uh0j

21;

where

C5 ¼ 1þ1

fm

�þ 2N

mkfk�1

�; C6 ¼

ffiffiffin

p

f;

and f is given by (2.17).

Proof.We begin by subtracting the equation for ðuh1; ph1Þ given in (2.23) from theequation for ðu; pÞ given in (2.2) where we have written the latter equationholding over the appropriate subspaces. We obtain

ma0ðe1; vhÞ � rha0ðuh1; vhÞ þ a1ðu; u; vhÞ

� a1ðuh0; uh1; vhÞ � a1ðuh1; uh0; vhÞ þ bðvh; p � ph1Þ

¼ �rha0ðuh0; vhÞ � a1ðuh0; uh0; vhÞ 8vh 2 Xh;

bðu� uh1; qhÞ ¼ 0 8qh 2 Qh:

Since Vh � Xh the first equation holds over all vh 2 Vh. Adding and subtractingappropriate terms in the above expression yields


ðm þ rhÞa0ðe1; vhÞ � rha0ðe1; vhÞ � rha0ðuh1; vhÞ þ a1ðu; u; vhÞ � a1ðuh0; uh1; vhÞ� a1ðuh0; u; vhÞ þ a1ðuh0; u; vhÞ � a1ðuh1; uh0; vhÞ þ a1ðu; uh0; vhÞ � a1ðu; uh0; vhÞþ bðvh; p � ph1Þ¼ �rha0ðuh0; vhÞ � a1ðuh0; uh0; vhÞ 8vh 2 Vh

which implies

ðm þ rhÞa0ðe1; vhÞ � rha0ðu; vhÞ þ a1ðu; u; vhÞ þ a1ðuh0; e1; vhÞ � a1ðuh0; u; vhÞþ a1ðe1; uh0; vhÞ � a1ðu; uh0; vhÞ

¼ �rha0ðuh0; vhÞ � bðvh; p � ph1Þ � a1ðuh0; uh0; vhÞ 8vh 2 Vh:

Thus the equation for the error can be written as

ðm þ rhÞa0ðe1; vhÞ þ a1ðuh0; e1; vhÞ þ a1ðe1; uh0; vhÞ¼ rha0ðe0; vhÞ � a1ðe0; e0; vhÞ � bðvh; p � ph1Þ 8vh 2 Vh:

We now let ~uuh be an arbitrary element in Vh, ~pph an arbitrary element of Qh

and add and subtract appropriate terms to get

ðm þ rhÞa0ð~uuh � uh1; vhÞ þ a1ðuh0; ~uuh � uh1; v

hÞ þ a1ð~uuh � uh1; uh0; v

hÞ¼ dðm þ rhÞa0ð~uuh � u; vhÞ þ rha0ðe0; vhÞ þ a1ðuh0; ~uuh � u; vhÞ

þ a1ð~uuh � u; uh0; vhÞ � a1ðe0; e0; vhÞ � bðvh; p � ~pphÞ

� bðvh; ~pph � ph1Þ 8vh 2 Vh:

We now set vh ¼ ~uuh � uh1 2 Vh, use property (2.15), and (2.11)–(2.14) toobtain the bound

m

þ rh� Nkruh0k0�krðuh1 � ~uuhÞk0

6 m

þ rhþ 2Nkruh0k0�krðu� ~uuhÞk0 þ rhkre0k0 þ Nkre0k20

þffiffiffin

pkp � ~pphk0:

Again, due to the uniqueness assumption on uh0 we have thatm þ rh� Nkruh0k0P f > 0. We now take the infimum over all ~uuh 2 Vh and~pph 2 Qh and use the bound on the solution uh0 to obtain the final result

juh1 � ~uuhj16 ~CC5

�þ r

fh�inf~uuh2Vh

ju� ~uuhj1 þ C6 inf~pph2Qh

kp � ~pphk0 þrfhju� uh0j1

þ Nfju� uh0j

21;

where

~CC5 ¼1

fm

�þ 2N

mkfk�1

�and C6 ¼

ffiffiffin

p

f:


Combining this result with the fact that e1 ¼ ðu� ~uuhÞ � ðuh1 � ~uuhÞ we get thedesired result. �

As in the estimate for ku� uh0k0 we need the linearized dual problem. Spe-cifically, we seek ðw; kÞ satisfying (3.2) with e0 replaced by e1. We again makethe assumption that the solution to this dual problem isH2-regular, i.e., there isa constant Cw ¼ Cwðu; mÞ such that

kwk2 þ kkk16Cwke1k0: ð3:7Þ

The variational formulation is to seek ðw; kÞ 2 ðX;QÞ such that

ma0ðv;wÞ þ a1ðu; v;wÞ þ a1ðv; u;wÞ þ bðv; kÞ ¼ ðe1; vÞ 8v 2 X;

bðw; qÞ ¼ 0 8q 2 Q:ð3:8Þ

Analogous to (3.5) we have the approximation result

inf~wwh2Vh

krðw� ~wwhÞk06C 1

�þ

ffiffiffin

p

bh

�hke1k0 ¼ hCake1k0 ð3:9Þ

for w 2 H2ðXÞ. The L2-estimate is given in the following theorem and the prooffollows the usual duality argument.

Theorem 3.2. Let u be the solution of (2.2) and uh1 the solution to (2.23). Let uh0 bethe solution of (2.20). Suppose further that (3.7) and (3.9) hold. Then there areconstants C7 and C8, independent of h, such that

ku� uh1k06 h C7ju"

� uh1j1 þ rhC8ju� uh0j1 þffiffiffin

pC8 inf

~pph2Qhkp � ~pphk0

þ rhC8ju� uh1j1 þ 2NC8ju� uh1j1ju� uh0j1 þ NC8ju� uh0j21

#: ð3:10Þ

Proof. To prove this result we use a duality argument to get a bound for ke1k0.We begin by setting v ¼ e1 ¼ u� uh1 2 X in (3.8) to get

ke1k20 ¼ ma0ðe1;wÞ þ a1ðu; e1;wÞ þ a1ðe1; u;wÞ þ bðe1; kÞ: ð3:11Þ

We then add and subtract the terms ma0ðe1; ~wwhÞ, a1ðe1; u; ~wwhÞ, and a1ðu; e1; ~wwhÞwhere ~wwh is an arbitrary element of Vh to obtain

ke1k20 ¼ ma0ðe1;w� ~wwhÞ þ a1ðu; e1;w� ~wwhÞ þ a1ðe1; u;w� ~wwhÞþ ma0ðe1; ~wwhÞ þ a1ðu; e1; ~wwhÞ þ a1ðe1; u; ~wwhÞ þ bðe1; kÞ:

The weak form of the Navier–Stokes equations given in (2.2) holdsfor all ~wwh 2 Xh � X,qh 2 Qh � Qandsowemay subtract these fromthe equationsfor ðuh1; ph1Þ given in (2.23) with vh ¼ ~wwh to obtain the Galerkin orthogonalitycondition


ma0ðe1; ~wwhÞ þ a1ðu; u; ~wwÞh � a1ðuh0; uh1; ~wwhÞ � a1ðuh1; uh0; ~wwhÞþ a1ðuh0; uh0; ~wwhÞ þ bð~wwh; p � ph1Þ þ rha0ðuh0; ~wwhÞ � rha0ðuh1; ~wwhÞ

¼ 0 8 ~wwh 2 Xh: ð3:1Þ

Note that since Vh � Xh the above condition holds for all ~wwh 2 Vh. We want touse this condition in our expression for ke1k20; to this end, we rewrite theGalerkinorthogonality condition by adding and subtracting appropriate terms to get

ma0ðe1; ~wwhÞ � rha0ðe0; ~wwhÞ þ rha0ðe1; ~wwhÞ þ a1ðe1; u; ~wwhÞ þ a1ðuh1; u; ~wwhÞþ a1ðe0; uh1; ~wwhÞ � a1ðu; uh1; ~wwhÞ þ a1ðe1; uh0; ~wwhÞ � a1ðu; uh0; ~wwhÞþ a1ðuh0; uh0; ~wwhÞ þ a1ðu; u; ~wwÞh � a1ðu; u; ~wwhÞ þ bð~wwh; p � ph1Þ

¼ 0 8 ~wwh 2 Vh:

A regrouping of terms gives that

ma0ðe1; ~wwhÞ þ a1ðe1; u; ~wwhÞ þ a1ðu; e1; ~wwhÞ¼ rha0ðe0; ~wwhÞ � rha0ðe1; ~wwhÞ � a1ðuh1; u; ~wwhÞ � a1ðe0; uh1; ~wwhÞ

� a1ðe1; uh0; ~wwhÞ þ a1ðu; uh0; ~wwhÞ � a1ðuh0; uh0; ~wwhÞ þ a1ðu; u; ~wwhÞ� bð~wwh; p � ph1Þ:

This is now substituted into the expression for ke1k20 and appropriate termsadded and subtracted to obtain

ke1k20 ¼ ma0ðe1;w� ~wwhÞ þ a1ðu; e1;w� ~wwhÞ þ a1ðe1; u;w� ~wwhÞ þ bðe1; kÞþ rha0ðe0; ~wwhÞ � rha0ðe1; ~wwhÞ þ rha0ðe1;wÞ � rha0ðe1;wÞ� a1ðuh1; u; ~wwhÞ � a1ðe0; uh1; ~wwhÞ þ a1ðe0; u; ~wwhÞ � a1ðe0; u; ~wwhÞ� a1ðe1; uh0; ~wwhÞ þ a1ðe1; u; ~wwhÞ � a1ðe1; u; ~wwhÞ þ a1ðu; uh0; ~wwhÞ� a1ðuh0; uh0; ~wwhÞ þ a1ðu; u; ~wwhÞ � bð~wwh; p � ph1Þ

¼ mð þ rhÞa0ðe1;w� ~wwhÞ þ a1ðu; e1;w� ~wwhÞ þ a1ðe1; u;w� ~wwhÞþ bðe1; kÞ þ rha0ðe0; ~wwhÞ � rha0ðe1;wÞ þ a1ðe0; e1; ~wwhÞþ a1ðe1; e0; ~wwhÞ � a1ðe0; e0; ~wwhÞ � bð~wwh; p � ~pphÞ � bð~wwh; ~pph � ph1Þ

¼ ðm þ rhÞa0ðe1;w� ~wwhÞ þ a1ðe1; u;w� ~wwhÞ þ a1ðu; e1;w� ~wwhÞ� rha0ðe1;wÞ � rha0ðe0;w� ~wwhÞ þ rha0ðe0;wÞ� a1ðe0; e1;w� ~wwhÞ � a1ðe1; e0;w� ~wwhÞ þ a1ðe0; e0;w� ~wwhÞþ a1ðe1; e0;wÞ þ a1ðe0; e1;wÞ � a1ðe0; e0;wÞ þ bðe1; kÞþ bðw� ~wwh; p � ~pphÞ � bðw; p � ~pphÞ;


where ~pph is an arbitrary element ofQh.Wemust nowbound the terms on the right-hand side. From the continuity conditions (2.11)–(2.13) we immediately have

ke1k206 ðmh

þ rhÞkre1k0 þ 2Nkre1k0kruk0 þ 2Nkre1k0kre0k0

þ Nkre0k20ikrðw� ~wwhÞk0 þ 2Nkre1k0kre0k0krwk0

þ Nkre0k20krwk0 þffiffiffin

pkre1k0kkk0 þ rh kre1k0krwk0

�þ kre0k0krðw� ~wwhÞk0 þ kre0k0krwk0

þ

ffiffiffin

pkrðw�

� ~wwhÞk0kp � ~pphk0 þ krwk0kp � ~pphk0:

We can use (3.9) to bound krðw� ~wwhÞk0 in terms of ke1k0. To bound the termskkk0 and krwk0 we first use the inverse inequalities

kkk06 c1hkrkk0 and krwk06 c2hkwk2 for k 2 H1ðXÞ;w 2 H2ðXÞ

and then use (3.7). Applying these bounds and taking the infimum over all~wwh 2 Vh, we obtain

ke1k206 hke1k0 Ca ðm�h

þ rhÞje1j1 þ 2N je1j1juj1 þ 2N je1j1je0j1 þ N je0j21

þ rhje0j1 þffiffiffin

pkp � ~pphk0

þ c2Cw 2N je1j1je0j1

�þ N je0j21

þ rhje1j1 þ rhje0j1 þffiffiffin

pkp � ~pphk0

þ

ffiffiffin

pje1j1c1Cw

i:

Thus

ke1k06 h C7je1j1

"þ C8rhje0j1 þ C8

ffiffiffin

pinf~pph2Qh

kp � ~pphk0 þ rhC8je1j1

þ 2NC8je1j1je0j1 þ NC8je0j21

#;

where

C7 ¼ Ca m

�þ 2N

mkfk�1

�þ c1Cw

ffiffiffin

p; C8 ¼ Ca þ c2Cw: �

4. Numerical results

In this section we present some numerical results for the implementation ofthe defect-correction method proposed in Section 2 for the well-known driven


cavity problem. We chose this problem because there are numerous results inthe literature available for comparison.In calculating solutions of the Navier–Stokes equations for large values of

the Reynolds number, one usually incorporates a continuation method. In sucha method one first obtains a solution at some low value of the Reynoldsnumber, such as Re ¼ 1 and then uses a continuation method to advance thesolution in the Reynolds number. Computationally one sees that as the Rey-nolds number increases, the attraction ball for the solution decreases. Hencerelatively small steps in the Reynolds number must be taken which results incalculating the solution of the nonlinear system (2.7) for many values of theReynolds number. In addition, one needs a sufficiently refined grid to accu-rately approximate the solution at high Reynolds numbers.Let the region X be the unit square where

X ¼ fðx; yÞ j 0 < x < 1; 0 < y < 1g:

Our problem is to approximate the solution to the Navier–Stokes equations inX when we impose the normal component of the velocity to be zero on oX andthe tangential component to be zero except along y ¼ 1 where it is set to one.To define the finite element spaces we divide the region X into rectangles and

divide each rectangle into two triangles. On each triangle, we use the Taylor–Hood element, i.e., piecewise quadratic functions for the velocity and piecewiselinear functions for the pressure. This choice is known to satisfy the stabilitycondition given in (2.19); see [6]. In addition, a nonuniform grid in both di-rections is employed.Numerical experiments were performed for Reynolds numbers 1000 and

3200 so that we may compare with the results published in [4]. We first considerthe solution of our model problem using the standard finite element approachwhere no artificial viscosity is used. We employ the same method of approxi-mation that we use in the defect-correction experiments, i.e., the Taylor–Hoodelement with a nonuniform grid. For the Reynolds number 1000, 11, 15, and 19grid points are adapted. For 3200 a standard continuation method with in-crement 400 is used with the starting Reynolds number 400. However, using 21,26, and 31 grid points, the standard finite element method incorporated withthe continuation method does not generate approximate solutions, since iter-ations for the nonlinear system do not converge. It is necessary to refine thegrid and reduce the increment of Reynolds number. In Fig. 1 we present nu-merical results for the horizontal and vertical components of the velocity atReynolds number 1000, computed using standard finite elements with 11, 15,and 19 grid points in each directions; in addition, the numerical results ob-tained in [4] are indicated on the plot by a small star. From this figure one cansee that 19 or more points in each direction is necessary to capture the behaviorof the solution at Reynolds number 1000.


We now proceed to describe the results of our experiments using the defect-correct method for the driven cavity problem. In the following plots we willcompare the horizontal component of the velocity obtained using the methodwith the results published in [4]. Similar results hold for the vertical componentof the velocity. We first compare the results obtained at the Reynolds number1000 using a different number of correction steps with the optimal artificialviscosity parameter 0.01, which was found by various precomputations. InFig. 2 we graph the results obtained by the defect correction method with 19grid points using from no corrections to using two corrections. As can be seenfrom the figure, two corrections appear adequate.We next compare the results at Reynolds number 3200 by 26 grid points

with viscosity parameter r ¼ 0:01. The results of the numerical experiments are

Fig. 1. Horizontal and vertical velocity for Re ¼ 1000.

Fig. 2. Horizontal and vertical velocity by defect-correction method for Re ¼ 1000 (r ¼ 0:01).


illustrated in Fig. 3; in the plot we have plotted the results after zero, one, andtwo corrections, and the results published in [4].The defect-correction method is computationally more efficient for a very

high Reynolds number like 3200 since the systems can be solved on a coarsermesh than is usually possible due to the added artificial viscosity term.In implementing the defect-correction method discussed in this paper, sev-

eral variations can be considered. One of the possibilities may be a combina-tion with a multi-level method. (See [10].) The multi-level method can beapplied in the step 1 (defect step) for efficient computing of the nonlinearsystem. The multi-level approach may be also used in the correction step toincrease accuracy of the solution of the linear equation with a slight modifi-cation of the algorithm. See [11] for more details.

Appendix A. Proof of H1 estimate for u� uh0

Subtracting (2.20) from (2.2) with v ¼ vh and adding and subtracting theterm a1ðuh0; u; vhÞ we have

ma0ðu� uh0; vhÞ � rha0ðuh0; vhÞ þ a1ðu� uh0; u; v

hÞ þ a1ðuh0; u� uh0; vhÞ

þ bðvh; p � ph0Þ ¼ 0 8vh 2 Xh;

bðu� uh0; qhÞ ¼ 0 8qh 2 Qh:

From (2.8), we have that Vh � Xh and so the first equation holds for allvh 2 Vh. We let ~uuh be an arbitrary element of Vh, ~pph an arbitrary element of Qh,n an arbitrary element of V and add and subtract appropriate terms to get

Fig. 3. Horizontal and vertical velocity by defect-correction method for Re ¼ 3200 (r ¼ 0:01).


ma0ð~uuh � uh0; vhÞ þ a1ð~uuh � uh0; u; v

hÞ þ a1ðuh0; ~uuh � uh0; vhÞ

¼ �ma0ðu� ~uuh; vhÞ þ rha0ðuh0; vhÞ � a1ðu� ~uuh; u; vhÞ � a1ðuh0; u� ~uuh; vhÞ� bðvh � n; p � ~pphÞ � bðvh; ~pph � ph0Þ 8vh 2 Vh; qh 2 Qh:

We now choose vh ¼ ~uuh � uh0 2 Vh, use the coercivity of a0ð�; �Þ, the continuityof the forms given in (2.11)–(2.13), property (2.15), and take the infimum overall n 2 V to arrive at

m

� Nkruk0�krð~uuh � uh0Þk0

6 mkrðu� ~uuhÞk0 þ rhkruh0k0 þ Nkrðu� ~uuhÞk0 kruk0

þ kruh0k0�

þffiffiffin

pkp � ~pphk0 infn2V

krð~uuh � uh0 � nÞk0krð~uuh � uh0Þk0

:

We have that

infn2V

krð~uuh � uh0 � nÞk0krð~uuh � uh0Þk0

6H:

Also note that since we have assumed Nkfk�1 < m2 for uniqueness andkruk06 ð1=mÞkfk�1, then m � Nkruk0 > 0; thus

fkrð~uuh � uh0Þk06 m

� Nkruk0�krð~uuh � uh0Þk0;

where f ¼ m � ðN=mÞkfk�1 > 0 since we are assuming uniqueness. Using thebounds for the solutions u and uh0 and taking the infimum over all ~uu

h 2 Vh andall ~pph 2 Qh we obtain

krðuh0 � ~uuhÞk06 ~CC1 inf~uuh2Vh

krðu� ~uuhÞk0 þ C2H inf~pph2Qh

kp � ~pphk0 þ hrfm

kfk�1;

where

~CC1 ¼1

fm

�þ 2N

mkfk�1

�and C2 ¼

ffiffiffin

p

f:

Combining this result with the fact that

krðu� uh0Þk06 krðu� ~uuhÞk0 þ krð~uuh � uh0Þk0concludes proof.

Appendix B. Proof of L2 estimate for u� uh0

To prove this result we use a duality argument to get a bound for ke0k0. Webegin by setting v ¼ e0 ¼ u� uh0 2 X in the dual problem to get


ke0k20 ¼ ma0ðe0;wÞ þ a1ðu; e0;wÞ þ a1ðe0; u;wÞ þ bðe0; kÞ:

We then add and subtract the terms ma0ðe0; ~wwhÞ, a1ðe0; u; ~wwh), and a1ðu; e0; ~wwh)where ~wwh is an arbitrary element of Vh to obtain

ke0k20 ¼ ma0ðe0;w� ~wwhÞ þ ma0ðe0; ~wwhÞ þ a1ðu; e0;w� ~wwhÞ þ a1ðu; e0; ~wwhÞ

þ a1ðe0; u;w� ~wwhÞ þ a1ðe0; u; ~wwhÞ þ bðe0; kÞ:

The weak form of the Navier–Stokes equations given in (2.2) holds for all~wwh 2 Xh � X, qh 2 Qh � Q and so we may subtract these from the equations forðuh0; ph0Þ with vh ¼ ~wwh to obtain the Galerkin orthogonality condition

ma0ðe0; ~wwhÞ þ a1ðu; u; ~wwÞh � a1ðuh0; uh0; ~wwhÞ þ bð~wwh; p � ph0Þ

� rha0ðuh0; ~wwhÞ ¼ 0 8 ~wwh 2 Xh: ðB:1Þ

Note that since Vh � Xh the above condition holds for all ~wwh 2 Vh. Adding andsubtracting a1ðuh0; u; ~wwh) gives

ma0ðe0; ~wwhÞ þ a1ðe0; u; ~wwhÞ þ a1ðuh0; e0; ~wwhÞ

¼ rha0ðuh0; ~wwhÞ � bð~wwh; p � ph0Þ 8 ~wwh 2 Vh:

This is now substituted into the expression for ke0k20 to obtain

ke0k20 ¼ ma0ðe0;w� ~wwhÞ þ a1ðu; e0;w� ~wwhÞ þ a1ðe0; u;w� ~wwhÞ

� a1ðe0; e0;w� ~wwhÞ þ a1ðe0; e0;wÞ þ bðe0; kÞ � rha0ðuh0;w� ~wwhÞ

þ rha0ðuh0;wÞ þ bðw� ~wwh; p � ~pphÞ � bðw; p � ~pphÞ;

where ~pph 2 Qh is arbitrary. We now proceed to bound the right-hand side usingthe continuity conditions on the forms and the regularity of the dual solution.We have

ke0k206 mkre0k0krðw� ~wwhÞk0 þ 2Nkre0k0kruk0krðw� ~wwhÞk0

þ Nkre0k20 krðw�

� ~wwhÞk0 þ krwk0þ

ffiffiffin

pkre0k0kkk0

þ rhkruh0k0krðw� ~wwhÞk0 þ rhkruh0k0krwk0þ

ffiffiffin

pkrðw� ~wwhÞk0kp � ~pphk0 þ

ffiffiffin

pkrwk0kp � ~pphk0

6 mkre0k0Cahke0k0 þ 2Nkre0k0kruk0Cahke0k0þ Nke0k20 Cahke0k0

þ krwk0

�þ

ffiffiffin

pkrðe0Þk0kkk0

þ rhkruh0k0Cahke0k0 þ rhkruh0k0krwk0þ

ffiffiffin

pCahke0k0kp � ~pphk0 þ

ffiffiffin

pkrwk0kp � ~pphk0;


where we have used (3.5). Now using the inverse inequalities

kkk06 c1hkrkk0 and krwk06 c2hkwk2 for k 2 H1ðXÞ; w 2 H2ðXÞ

and the bound for kwk2 þ jkj1 from (3.7), we have

ke0k206 mkre0k0Cahke0k0 þ 2Nkre0k0kruk0Cahke0k0þ Nkre0k20 Cahke0k0

þ c2hCwke0k0

�þ

ffiffiffin

pkre0k0c1hCwke0k0

þ rhkruh0k0Cahke0k0 þ rhkruh0k0c2hCwke0k0þ

ffiffiffin

pCahke0k0kp � ~pphk0 þ

ffiffiffin

pc2hCwke0k0kp � ~pphk0

and thus

ke0k06 h mCa

�þ 2NCa

1

mkfk�1 þ c1Cw

ffiffiffin

p �kre0k0

þ h ðNCa

�þ c2CwÞkre0k20

þ h

ffiffiffin

pCa

þ

ffiffiffin

pc2Cw

�kp � ~pphk0

þ h2 rCað þ rc2CwÞ1

mkfk�1:

Now using the bound kruk06 ð1=mÞkfk�1 and taking the infimum over all~pph 2 Qh gives the desired result.

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A defect-correction method for the incompressible Navier–Stokes equations