AFPLnED NAT~ENAT~CS

AND

CC~ ~OTATHON ELSEVIER Applied Mathematics and Computation 100 (1999) 131-138

Two grid discretizations with backtracking of the stream function form

of the Navier-Stokes equations

X. Y e ~

Department of Mathematics and Statistics, University of' Arkansas at Little Rock, Little Rock, AR 72204, USA

A b s t r a c t

We analyze a two grid finite element method with backtracking for the stream function formulat ion of the stationary Navier-Stokes equations. This two grid method involves solving one small, nonlinear coarse mesh system, one linearized system on the fine mesh and one linear correction problem on the coarse mesh. The algorithm and error analysis are presented. © 1999 Elsevier Science Inc. All rights reserved.

Keywords." Navier-Stokes equations; Two grid discretizations

I. Introduction

We consider two grid discretization with backtracking to solve stream function formulation of the stationary Navier-Stokes equations. The advan- tages of the stream function formulation are that the incompressiblility con- dition is satisfied automatically and the pressure is not present in the weak form [1-3]. The two grid discretization techniques were originally proposed in [4] and have been extended in various directions [5-7]. These methods are based on two finite element subspaces, coarse and fine, for solving partial differential equations. The idea is to solve "hard" problems (for example, nonlinear and not symmetric positive definite) on the coarse grid and "easy" problems (linear, symmetric positive definite) on the fine grid.

l E-mail: xxye@ualr.edu.

0096-3003/99/$ - see front matter © 1999 Elsevier Science Inc. All rights reserved. PII: S0096-3003(98)00024-1

132 X. Ye / Appl. Math. Comput. 100 (1999) 131-138

The two grid methods are meaningful only when h << H where h and H are mesh sizes for fine grid and coarse grid, respectively. To increase the ratio of h and H, higher order of H in the error estimate is desired. In [7], Xu applied the backtracking techniques to the semilinear elliptic equations and improved accuracy by an extra order of H. In this paper, we apply this backtracking technique to the two grid discretizations for the stream function formulation of the Navier-Stokes equations.

We proved [6] for the Clough-Tocher triangle element that the error for Ch, the solution from the two grid method without backtracking, is

I~ -- I/yht2 ~ C( h2 ~- U 4 )

and for @*, the solution from the two grid method with backtracking is

I~ - # h ~< c ( ha + I4~) •

We consider the approximation of solutions of the equilibrium Navier- Stokes equations describing the motion of a viscous, incompressible fluid. For two dimensional flows in simply connected domains a simple formulation of the equations can be given in terms of the scalar "streamfunction" @(x,y). Thus, for a f2 planar, polygonal, simply connected domain we seek 4J(x,y) satisfying

Re-IA2~ - - ~yA~x + ~lxA~y = c u r l f in O, (1.1)

0__~@ = 0 on Of a, ~0 = O, On

Here f represents the body force, Re the Reynolds number and n the outward unit normal to f2. Once an approximate streamfunction if(x, y) is calculated the primitive variables of pressure and flow velocity can be recovered then with well-developed methods [1,2].

Let Y =H02(f2) and l" l, be the Sobolev seminorm for Hi(O) with i - - 1 , . . . , 4 . The weak form of Eq. (1.1) is: find ~ E X such that

a(~,, ~b) + b(~O, ~b, 4~) = 0e, curl <P) for all ~b E X, (1.2)

where

a(~/,, ~b) := Re-'[(~Ox~q~,= + 2~O~y~bxy + ~J~,q~) dQ, (1.3)

t2

and

bOP, 4, q~) :=/d~b(~y(Ox - ~x~by) dr2. (1.4)

f2

X.. Ye / Appl. Math. Comput. 100 (1999) 131-138

2. Two grid discretizations

133

We use the Clough-Tocher triangular element [3,8] to discretize Eq. (1.2). The Clough-Tocher triangular element specifies the function value and first derivative at the vertices of the triangle and normal derivative at the middle of the edge. Let 3-h and 3"u be regular triangulations of f2 with mesh size h and H, respectively, and X h and X H be subspaces of X associated with 9--h and c~-- H

respectively, generated by the Clough-Tocher element. The usual finite element approximation ~k h • X h is calculated by solving the

large, nonlinear system given by:

a(~h, dp) + bOPh, ~bh, 4) = Or, curl q~) for all ¢ • X h. (2.1)

It is known [6] that ~k h • X h satisfies an error estimate for the Clough-Tocher triangle element: for t~ • H4(f2) n H02(f2)

1~ b - ~kh]o + hl~b - ~bhll + h2l~ - ~/h[2 • Ch4[1~I4 • (2.2)

The solution of the nonlinear equation (2.1) can still be quite computa- tionally intensive. Therefore, this report considers the application of an at- tractive two grid finite element discretization scheme with back tracking technique.

Algorithm 1. Step 1. Solve the nonlinear system on coarse mesh for ~b ~ E XH:

a(~ n, dp H) + b(tk n, ~/~, ~b H) = (f, curl ~b H) for all c H E X H. (2.3)

Step 2. Solve the linear system on fine mesh for ~,h E Xh:

a(tk h, ~b h) + b(~bH, ~h, C~h) + b(~b h, ipH, q~h) _ b(~bn, ipH, Ch)

= (f, curl ~b H) for all q~h E X h. (2.4)

Step 3. Solve the linear system on the coarse mesh for e ~ E xH: for all •H • X H

a(e H, cH) + b(~kH, e H, q~g) + b(e H, ~H, CU) = _b(~kh _ ~bH, ~,h _ fill, ~ky).

(2.5) Step 4. Set ~" = ~b h + e ~. Let N be the finite constant,

Ib(~,C,n)[ N := sup I hlChl'th

and Ell, denote the dual norm:

(f, curl ~b) If I, := sup

134 X. Ye / Appl. Math. Comput. 100 (1999) 131-138

b(-,., .) satisfies the following inequalities [3,9,10]:

Ib(0, ~, ~)1 <~ NI0hl~12l~12, (2.6)

Ib(0, ~, ~)1 ~< c10111/21011=/=t4'h1~111/21~1'2/2- (2.7)

Lemma 2.1. The solution to Eq. (2.3) exists and satisfies 10"12~<Re[f[,. If Re2NLf[, < 1, then the solution O n to Eq. (2.3) is unique.

Proof. See [6].

Lemma 2.2. Given a solution O n to Eq. (2.3), i fRe2NLfl, < 1, then the solution to (2.4) exists uniquely and satisfies

[0hi2 ~< (Re -1 - g Re[ f I,) -I(1 + N Re z [fl ,)[ft ,- (2.8)

Proof. See [6].

Lemma 2.3. Given a solution O H to Eq. (2.3) and a solution O h to Eq. (2.4), i f Re2N[f], < 1, then Eq. (2.5) has a unique solution.

Proof. Let e H and E H be two solutions o f Eq. (2.5). Then e n - E H satisfies: for any q5 H ~ X n,

a(e H - E H, ~b H) + b(e H - E n , O n, ~,b n) + b(0 n , e n - E n, ~b n) = 0. (2.9)

Let ~b n = e N - E H in Eq. (2.9) and notice b(0 n, e H - E n, e n - E n) = 0, then

Re-1 len n 2 - E 12 - - N l e H - E " l ~ l q / ~ h <<. O.

Then

Re-1 (1 - N Re2 Lfl,)te n - E H 12<~0.2

Thus e n = E H. Since Eq. (2.5) is finite dimensional, uniqueness implies exis- tence. It is easy to see that b(0, 0, ~b) satisfies the following identity

b(@, 0, ~) = b(0, O H, ~) + b (0" , @, ~) - b (0 n, @ H, ~)

+ b (0 - O n, O - t/'H, q~). (2.10)

3. Error estimate

Let 0 be a nonsingular solution of Eq. (1.2). Then for H small enough, 0 H is within any prespecified ball about the nonsingular solution O- Thus, Eq. (2.4) linearized about 0 H is invertible for H sufficiently small. As a consequence, we have the inf-sup stability condit ions in Xh: there is a 7 > 0 such that

X. Ye / Appl. Math. Comput. 100 (1999) 131-138

inf sup a(~9, ¢) + b(~k H, ~k, 4)) + b(ff, ~H 49) ~ > 7 > 0 ,

135

(3.1)

and

inf sup a(~k, 49) + b(~k ~, ~b, 49) + b(ff, ~b H, 49) /> 7 > 0, (3.2)

(see for example Remark IV 3.1 in [3,9,10]). Eqs. (3.1) and (3.2) also hold for ~b and 49 in X u.

Let

CH(~9, 49) ~ a(~9, 49) + b(~ H, ~k, 49) + b(t~, ~9 H, 49).

Define a projection PH : X ~ X H by

C.(49,P.~)=CH(49,~) V49EX u, C EX. (3.3)

Lemma 3.1. With H small enough, PH is well defined and satisfies

- ~ t2, ( 3 . 4 ) ?H cxh

and

I~ - P - ~ l ~ ~< CHIC - PH¢I2. ( 3 . 5 )

Proof. By Eqs. (3.2) and (3.3), for ~ E X, we have

C.(¢",PH~ - ~H)= CH(OH, ~ _ cH) ]PH¢- ¢H12 ~< sup sup

~< c1¢ - ¢'12-

Then the triangle inequality gives

14 -- PHil2 ~ C inf 1~ - ~H]2. ~* eX H

To estimate ]Pn~ - ~11, the dual problem is considered: for given g ~ L2(f2), find ~ ~ H~(f2) r-~H3(f2) such that for all 49 ~ H02(O)

CH(~, 49) =: (curl 49,g). (3.6)

We shall assume that the solution ~ of Eq. (3.6) satisfies the regularity con- dition:

1t¢113 ~ cllgtlo. (3.7)

Eqs. (3.6), (3.7) and (3.3) give

136 X. Ye I Appl. Math. Comput. 100 (1999) 131-138

1P~49 - 49h = sup f~ V(PH49 -- 49)g = sup fa curl(PH49 -- 49)g Ilgll0 , Ilgllo

= sup (curI(PH49 - 49)g) =: sup CH(~, PH49 - 49)

Ifgllo , Ilgllo = sup Ct¢(¢ - PH¢,PHC~ - 49) <~ sup CI~ - P . ¢ I 2 1 P a 4 9 - 49h1~"12

Ilgllo ~ tlgllo <~ CHIPMck - 4912.

Lemma 3.2. I f ~ E H4(f2) N H02(O), and On and ~k h solve Eqs. (2.1) and (2.4) respectively, then

IO,, - ¢12 <<- CH4. (3.8)

Proof. It follows f rom Eqs. (2.1) and (2.10) that

a(~lh, 49h) + b(~lh, I~H, ~)h) jr_ b(~lH ~lh, 49h) _ b(~lH, ~H, 49h)

= Of, curl Oh) _ b(¢h _ ¢~¢, ~h -- CH ¢~).

Subtracting Eq. (2.4) f rom Eq. (3.9) gives

a(,/jh _ ¢h, qSh) + b(~h _ qjh ~b", ch) + b(~by, ¢h - ¢ h ~bh)

= _ b ( e h _ ¢H, t/jh _ ~k", c~h).

Using Eqs. (3.1), (3.10) and (2.2), we have

la(~Oh - ¢ , Oh) + b(q~ h _ ¢ , q~., ~bh) + b ( ¢ ' , qJh - ¢ , Ch)] ~< sup 49h ~h~x~ 1 1 2

Ib(¢n - fill, ~kh -- fill, 49h)1 : sup

~ f¢12

c1¢,,, - ,//-' I~ < c { I,/,,, - el~ + I¢, _ ¢,.2f2} < C{h" + H"} le l , , < CH41,1,1~.

(3.9)

(3.10)

(3.11)

Theorem 3 .3 . / f~b E H4(f2) fqH~(f2) is the solution o f Eq. (1.2) and ~h is the solution o f Eq. (2.1) such that Eq. (2.2) is satisfied, then

I~bh - ~'*[2 ~< CH5 (3.12)

and

I~ - q,*12 < C(h 2 +HS). (3.13)

x. Ye I Appl. Math. Comput. 100 (1999) 131-138 137

Proof. Using Eq. (3.3), Eq. (2.5) changes to

a (e" , Ch) + b(e" , ~ " , 0 h) + b(q / ' , e" , 0 h) = - b ( q , h - ~0", ~h _ ~/, , p , , 0 h)

V0 h ~ X h. (3.14)

Adding Eqs. (2.4) and (3.14) gives

a(~*, Ch) + b(~*, ~,H 0 h) + b(~kH, ~,,, o h ) __ b(q,n, fft¢, 0 h)

= Or, curl 0 h) - b ( ¢ h - q?', ~0 h - q,", 0 h) - b(O h - ¢ ' , ~0 h - 4 / ' , e . ¢ h - Ch)

V0 h ~ Y h. (3.15)

Subtract ing Eq. (3.15) f rom Eq. (3.9) gives

a ( ~ h - ~,*, Ch) + b(~'h - g'*, ~ , Ch) + b( ,p~ , ~'h -- ~* , Ch)

= - b ( ~ h -- ~ kH, ~h -- ~ bH, 0 h) + b(~ 9h - ~,/4, ~bh _ ~b/4, 0 h)

+ b(~O h - ip/4, ~b h

= - b ( ¢ h - q/~, Ch

- b(q.h - ¢~' , q2 + b(~b h - ~b H, ~k h

= - b ( ¢ h - q , . , ~0h

+ b(Oh _ q,H, ~?

By Eq. (3.1), we have

_ ¢ . , p H c h _ c h )

- - I/I H , O h) -[- b ( ~l h - i[i H ' ~ll h - ~ 1 H , O h )

_ ~kH Ch) + b(~bh _ ~t-t ~h _ ~bH ch)

_ ~ , , p H o h _ ch)

- - g ? , O h) - - b ( ~ h -- ¢ h , ~? _ ~ , , , Ch)

_ ¢ . , p ~ o h _ c h ) .

Vl~h - ~'12 ~ s u p 1 0 h l ~ ' { l a ( ~ h - ~* , 0 h) + b ( ~ h - if*, ~ H , 0h) 4~h ~t'h

+ b(~k H, ~b h - ~k*, 0h)l} ~< s u p 10hl~-l{Ib(~h -- ~pH gJh -- ~ h Oh)l + [b(C'h -- ~h, ¢,h _ ~ p , 0h)l

4 )h oVh "-b [b(~l h - 1~ H, ~ll h - I[IH,pHO h -- 0h)[}. (3.16)

Using Eqs. (2.6) and (2.7) and L e m m a 3.2, we estimate the terms in the right- hand side of Eq. (3.16):

[b(~h - ~/ ' , ~'h - g,h, Ch)l < Cl~'h -- ~Hl21~h -- ChlE110h[~ < CH6lChh,

Ib(Oh - tP h, 0 h -- t/J H, ch)l ~< CltPh -- Ohl21C'h -- ¢ ' h l l 0 h h ~< CH6lOhl2,

(3.17)

(3.18)

ib(~h _ q / , , g,h _ q?,, e h c h _ 0h)l

~< CIq? -qFI l /21g , h - q/~l~/21~ h - q/~I21PH0 h -- OhI11/21PHO h -- Ohl~/2

<~ c n S I O h h . (3.19)

138 X. Ye / Appl. Math. Comput. 100 (1999) 131-138

Eq. (3.19) follows from Lemma 3.1 and Eq. (2.2). Combining Eqs. (3.16)- (3.19), we have

I~h * s - q , 12 <~ C H .

The triangle inequality implies

I~ - q,*12 ~< 1~ - q'hP2 + Jq'~ - ~ ' t2 ~< C ( h~ + H s ) •

All the analysis above works for other finite element spaces. We use the Clo- ugh-Tocher element as a example to make the results clear and simple.

References

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