Chebyshev pseudospectral-finite element method for the three-dimensional unsteady Navier–Stokes equation

Applied Mathematics and Computation 101 (1999) 209±244

Chebyshev pseudospectral-®nite element method for the three-dimensional unsteady Navier±Stokes equation Jing-yu Hou, Ben-yu Guo

*

Shanghai University of Science and Technology, Shanghai 201800, China

Abstract A mixed Chebyshev pseudospectral-®nite element method is developed for solving the three-dimensional evolutionary Navier±Stokes equation. The generalized stability and the convergence of the scheme are proved strictly. The numerical results presented show the advantages of this mixed method. Ó 1999 Published by Elsevier Science Inc. All rights reserved. AMS Classi®cation: 65N30; 76D99 Keywords: Three-dimensional Navier±Stokes equation; Chebyshev pseudospectral-®nite element approximation

1. Introduction There are many studies on the numerical solutions of nonlinear partial di€erential equations arising in ¯uid mechanics. Early studies mainly concerned the ®nite di€erence method, e.g., see [1,2]. Also the ®nite element method has been used in computational ¯uid dynamics, see [3±6]. As we know, the accuracy of both the above methods is limited. But the precision of the spectral method increases as the smoothness of the genuine solution increases. So it has been applied successfully to numerical experiments of ¯uid ¯ow, see

*

Corresponding author. Present address: President Oce, Shanghai University, Jiading Campus, Jiading, Shanghai, China. 0096-3003/99/$ ± see front matter Ó 1999 Published by Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 9 8 ) 0 0 0 4 1 - 1

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[7]. In particular, the pseudospectral method is easier to be performed and saves a lot of calculation time. Hence it becomes more and more popular in this ®eld. Recently, several mixed methods appeared for semi-periodic problems. For example, the Fourier spectral-®nite di€erence method and the Fourier spectral-®nite element method, e.g., see [8±15]. On the other hand, for keeping the convergence of ``in®nite order'', some authors provided mixed Fourier± Chebyshev spectral and Fourier±Chebyshev pseudospectral methods, see [16± 18]. But most practical problems are non-periodic. Whereas the sections of domains might be rectangular in certain directions, such as the ¯uid ¯ow in a cylindrical container, for solving such problems, the Chebyshev spectral-®nite element approximation was proposed, see [19]. It is also preferable to use the Chebyshev pseudospectral-®nite element approximation, see [20,21]. In this paper, we develop a mixed Chebyshev pseudospectral-®nite element method for solving the three-dimensional unsteady Navier±Stokes equation. Comparing with the Chebyshev spectral-®nite element method, this approach can be easier performed easilier and saves some work. In particular, it is easy to deal with the nonlinear terms. The outline of this paper is as follows. We construct the scheme and present the numerical results in Sections 2 and 3, respectively. In Section 4, we list some lemmas. Finally we strictly prove the generalized stability and convergence of the scheme in Sections 5 and 6. The main idea and the technique for theoretical analysis in this paper are also applicable to other nonlinear problems.

2. The scheme Let Q be a convex polygon in R2 with the coordinate x ˆ …x1 ; x2 †, and I ˆ fy=jyj < 1g, X ˆ f…x; y†=x 2 Q; y 2 Ig. The boundary of X is denoted by @X, and the boundary of Q is denoted by C. The velocity vector and the pressure are denoted by U ˆ …U1 ; U2 ; U3 † and P , respectively. m > 0 is the kinetic viscosity. U0 …x; y† and f …x; y; t† are given functions. Let T > 0; @t ˆ @=@t, @1 ˆ @=@x1 , @2 ˆ @=@x2 , @3 ˆ @=@y. We consider the following problem: 8 2 > < @t U ‡ d…U ; U † ‡ rP ÿ mr U ˆ f ; in X  …0; T Š; …1† r2 P ‡ U…U † ˆ r
f ; in X  ‰0; T Š; > : in X [ @X; U jtˆ0 ˆ U0 …x; y†; where d…U ; V † ˆ @1 …V1 U † ‡ @2 …V2 U † ‡ @3 …V3 ; U †; U…U † ˆ

3 X l;qˆ1

…@l Uq @q Ul ÿ @l Ul @q Uq †:

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211

Indeed, Eq. (1) is one of the representations of the Navier±Stokes equation, see [1,2]. As indicated in [22], it is equivalent to the usual representation for the Navier±Stokes equation, provided that the second derivatives of the velocity exist. Suppose that the boundary is a non-slip wall and so U ˆ 0 on @X. There is no boundary condition for the pressure generally. But if we use the second formula of Eq. (1) to evaluate the value of the pressure P , we need a nonstandard boundary condition. Sometimes, we can assume approximately that …@P =@n†j@X ˆ 0, as in [1]. In addition, for ®xing the value of pressure, we require that for all t 6 T , Z P …x; y; t† dx dy ˆ 0: X

It is obvious that for each time t and U , the second formula of Eq. (1) is a Neuman problem for P . It can be veri®ed from the boundary condition of U that this Neuman problem is consistent. This model is often used in ¯uid dynamics. Since the derivation of the second formula of Eq. (1) implies the incompressible condition, this model helps us to avoid the dicult job of constructing the trial function space satisfying the incompressibility and Babuska±Brezi condition, in calculations. Now we are going to derive the scheme. Firstly let D be an interval (or a domain) in R …or R2 ; R3 †. L2 …D†; H r …D† and H0r …D† …r > 0† denote the usual Hilbert spaces with the usual inner products and norms. We also de®ne L20 …D† ˆ

8 < :

u 2 L2 …D†

9 =

,Z D

2 ÿ1=2

Let x…y† ˆ …1 ÿ y †

and

Z …u; v†x;I ˆ

u dD ˆ 0 : ;

uvx dy;

kuk2x;I ˆ …u; u†x;I ;

I

L2x …I†

ˆ fu…y†=u is measurable on I and kukx;I < 1g:

Furthermore Z …u; v†x ˆ

uvx dx dy;

2

kukx ˆ …u; u†x ;

X

L2x …X† ˆ fu…x; y†=u is measurable on X and kukx < 1g: Let ax …u; v† ˆ …ru; r…vx††L2 …X† . Then the weak formulation of Eq. (1) is to ®nd U 2 X 3 …X† and P 2 Y …X† for all 0 < t 6 T such that

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(

…@t U ‡ d…U ; U † ‡ rP ; v†x ‡ max …U ; v† ˆ …f ; v†x ; ax …P ; w† ˆ …U…U † ÿ r
f ; w†x ; 8w 2 X~ …X†

8v 2 X 3 …X†;

…2†

where X~ …X† ˆ fu=u; @1 u; @2 u; @3 u 2 L2x …X†g; X …X† ˆ X~ …X† \ fu=u ˆ 0 on @Xg;    @u 2 ~ Y …X† ˆ X …X† \ L0 …X† \ u ˆ 0 for jyj ˆ 1 : @y We can verify the existence and uniqueness of the local solution of Eq. (2), under some conditions on f and U0 . Next we construct the trial function spaces. For the Chebyshev approximation, let N be any positive integer and PN …I† denote the set of all algebraic polynomials of degree 6 N , de®ned on I. Let VN …I† ˆ fu…y† 2 PN …I†=u…ÿ1† ˆ u…1† ˆ 0g;    du du …ÿ1† ˆ …1† ˆ 0 : WN …I† ˆ u…y† 2 PN …I† dy dy For the ®nite element approximation, we decompose Q into M triangles Qm (1 6 m 6 M), Q ˆ [M mˆ1 Qm . Assume that the triangulation is a regular family satisfying the inverse assumption, see [23]. Let hm be the diameter of Qm and qm be the supremum of the diameters of the circles inscribed in Qm . Then max1 6 m 6 M hm =qm 6 d, d being a positive constant independent of the decomposition of the domain Q. Moreover for any integer k P 1, let k S~h …Q† ˆ fu…x†=ujQm 2 Pk …Qm †; 1 6 m 6 Mg; k Shk …Q† ˆ S~h …Q† \ H01 …Q†:

Furthermore we take the spaces XNk ;h …X† and YNk ;h …X† as the trial function spaces for the velocity and the pressure respectively, de®ned as XNk ;h …X† ˆ Shk …Q† VN …I†; k YNk ;h …X† ˆ ‰…S~h …Q† \ H 1 …Q†† WN …I†Š \ L20 …X†:

Besides k k X~ N ;h …X† ˆ …S~h …Q† \ H 1 …Q†† VN …I†:

Now let fy …j† ; x…j† g be the nodes and weights of Gauss±Lobatto integration, i.e., ( ; 0 6 j 6 N; y …j† ˆ cos jp N p x…0† ˆ x…N † ˆ 2N ; x…j† ˆ Np ; 1 6 j 6 N ÿ 1:

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213

It is well known that for any g 2 P2N ÿ1 …I†, Z g…y†x…y† dy ˆ

N X g…y …j† †x…j† :

…3†

jˆ0

I

The discrete inner products and norms are de®ned as follows, …u; v†N ;x ˆ

N X u…y …j† †v…y …j† †x…j† ; jˆ0

Z

…u; v†N ;h;x ˆ

…u; v†N ;x dx;

2

kukN ;x ˆ …u; u†N ;x ; 2

kukN ;h;x ˆ …u; u†N ;h;x :

Q

 ! PN …I† be the interpolation operator, i.e., Let Pc : C…I† Pc u…y …j† † ˆ u…y …j† †; 0 6 j 6 N ; k and Pkh : C…Q† ! S~h …Q† \ H 1 …Q† be the piecewise Lagrange interpolation of degree k over each Qm ; 1 6 m 6 M. Furthermore let PN ;h : L2x …X† ! XNk ;h …X† be the L2x …X†-orthogonal projection. Let s be the step size of time t, and    T _ ; Rs ˆ R_ s [ f0g; Rs ˆ t ˆ ls=1 6 l 6 s 1 ut …t† ˆ …u…t ‡ s† ÿ u…t††: s For approximating the nonlinear terms d…U ; U † and U…U † in Eq. (2), we de®ne dc …u; v† ˆ

3 X

@j Pc …vj u†;

jˆ1

Uc …u† ˆ

3 X

‰Pc …@l uq @q ul † ÿ Pc …@l ul @q uq †Š:

l;qˆ1

For approximating the term ax …U ; v†; let aN ;h;x …u; v† ˆ

2 X jˆ1

…@j u; @j v†N ;h;x ÿ …@32 u; v†N ;h;x :

Let k P 0, u and p denote the approximations to U and P , respectively. Then the Chebyshev pseudospectral-®nite element method for solving Eq. (2) is to 3 k ®nd u 2 …XNk‡k ;h …X†† and p 2 YN ;h …X† such that 8 …ut ‡ dc …u; u† ‡ rp; v†N ;h;x ‡ maN ;h;x …u ‡ rsut ; v† ˆ …f ; v†N ;h;x ; > > > > < 8v 2 …X k‡k …X††3 ; t 2 R_ s ; N ;h …4† k > > aN ;h;x …p; w† ˆ …Uc …u† ÿ r
f ; w†N ;h;x ; 8w 2 X~ N ;h …X†; t 2 Rs ; > > : u…0† ˆ PN ;h U0 ;

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where r is a parameter and 0 6 r 6 1. If r ˆ 0, then Eq. (4) is an explicit scheme for u. Otherwise, it is an implicit scheme and so we have to solve a linear system to evaluate the velocity at each time t 2 Rs . We can verify the existence and uniqueness of the local solution of Eq. (4) under certain conditions.

3. Numerical results In this section, we provide some numerical results of scheme (4). Let Q ˆ ‰0; 1Š  ‰0; 1Š. We take the following test functions: 2

U1 …x; y; t† ˆ ÿA eBt x21 x2 …x1 ÿ 1† …x2 ÿ 1†…2x2 ÿ 1†…y 3 ÿ y†; 2

U2 …x; y; t† ˆ ÿA eBt x1 x22 …x1 ÿ 1†…2x1 ÿ 1†…x2 ÿ 1† …y 3 ÿ y†; 2

U3 …x; y; t† ˆ A eBt x1 x2 …x1 ÿ 1†…x2 ÿ 1†…2x1 ÿ 1†…2x2 ÿ 1†…y 2 ÿ 1† ; P …x; y; t† ˆ A eBt …2x31 ÿ 3x21 ‡ 0:5†…2x32 ÿ 3x22 ‡ 0:5†…y 3 ÿ 3y†: We use Chebyshev pseudospectral-®nite element scheme (CPSFE) (4) with k ˆ 1. The domain Q is divided uniformly into rectangular subdomains with mesh size h ˆ 1=M in x1 and x2 directions. For comparison, we also consider the ®nite element scheme (FEM). In this case, the interval I is divided with the length h ˆ 2=N  ; U ˆ …U1 ; U2 ; U3 † is approximated by a quadratic ®nite element scheme and P by a linear ®nite element scheme. For describing the errors of numerical solutions, let I^x ˆ fxj =xj ˆ jh; 1 6 j 6 M ÿ 1g; for CPSFE and FEM;   jp I^y ˆ yj =yj ˆ cos ; 1 6 j 6 N ÿ 1 ; for CPSFE; N  1 6 j 6 N  ÿ 1g; I^y ˆ fyj =yj ˆ ÿ1 ‡ jh;

for FEM;

^ ˆ f…x; y†=x 2 I^x  I^x ; y 2 I^y g X and P3 E…U …t†† ˆ P E…P …t†† ˆ

P

qˆ1

2 ^ jUq …x; y; t† ÿ uq …x; y; t†j …x;y†2X P3 P 2 ^ jUq …x; y; t†j qˆ1 …x;y†2X

^ …x;y†2X

P

jP …x; y; t† ÿ p…x; y; t†j ^ …x;y†2X

jP …x; y; t†j

2

2

!1=2 :

!1=2 ;

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215

Table 1 The errors of scheme (4) and FEM, m ˆ 0:001 t 0.5 1.0 1.5 2.0 2.5

Scheme (4), k ˆ 1

Scheme FEM

E…U…t††

E…P …t††

E…U…t††

E…P …t††

0:1717  10ÿ3 0:2770  10ÿ3 0:3936  10ÿ3 0:5247  10ÿ3 0:6736  10ÿ3

0:8900  10ÿ3 0:9163  10ÿ3 0:9426  10ÿ3 0:9690  10ÿ3 0:9954  10ÿ3

0:2113  10ÿ2 0:4125  10ÿ2 0:6147  10ÿ2 0:8203  10ÿ2 0:1028  10ÿ1

0:7190  10ÿ2 0:7549  10ÿ2 0:7919  10ÿ2 0:8301  10ÿ2 0:8692  10ÿ2

Table 2 The errors of scheme (4) and FEM, m ˆ 0:0001 t 0.5 1.0 1.5 2.0 2.5

Scheme(4), k ˆ 1

Scheme FEM

E…U …t††

E…P …t††

E…U …t††

E…P …t††

0:1687  10ÿ3 0:3325  10ÿ3 0:4967  10ÿ3 0:6622  10ÿ3 0:8293  10ÿ3

0:8899  10ÿ3 0:9167  10ÿ3 0:9435  10ÿ3 0:9704  10ÿ3 0:9973  10ÿ3

0:2123  10ÿ2 0:4273  10ÿ2 0:6424  10ÿ2 0:8583  10ÿ2 0:1076  10ÿ1

0:7190  10ÿ2 0:7564  10ÿ2 0:7935  10ÿ2 0:8319  10ÿ2 0:8717  10ÿ2

Table 3 The errors of scheme (4), M ˆ N ˆ 10; s ˆ 0:001; k ˆ 1; m ˆ 0:001 t 0.5 1.0 1.5 2.0 2.5

E…U …t††

E…P …t†† ÿ4

0:6172  10 0:8564  10ÿ4 0:1121  10ÿ3 0:1419  10ÿ3 0:1757  10ÿ3

0:2213  10ÿ3 0:2271  10ÿ3 0:2330  10ÿ3 0:2389  10ÿ3 0:2448  10ÿ3

Table 4 The errors of scheme (4), M ˆ N ˆ 10; s ˆ 0:001; k ˆ 1; m ˆ 0:0001 t

E…U …t††

E…P …t††

0.5 1.0 1.5 2.0 2.5

0:6104  10ÿ4 0:9825  10ÿ4 0:1356  10ÿ3 0:1731  10ÿ3 0:2111  10ÿ3

0:2211  10ÿ3 0:2273  10ÿ3 0:2332  10ÿ3 0:2395  10ÿ3 0:2457  10ÿ3

In calculations, we take A ˆ 0:2; B ˆ 0:1, k ˆ 1; N ˆ 4, M ˆ N  ˆ 10 and s ˆ 0:005; r ˆ 0. Tables 1 and 2 show the numerical results of schemes CPSFE and FEM. We ®nd that scheme CPSFE gives much better results than scheme FEM, even with relatively small N . Besides, we also take k ˆ 1; M ˆ N ˆ 10; s ˆ 0:001; r ˆ 0 and use scheme (4) with k ˆ 1. The corresponding results

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are shown in Tables 3 and 4, from which we know that when N is increased and s is decreased, much better results can be obtained. These numerical results also show the convergence of the scheme CPSFE.

4. Some lemmas In this section, we list some notations and lemmas which will be used later. Firstly, we introduce some Sobolev spaces with the weight function x…y†. For any integer r P 0, set !1=2

r r X

d v 2

kvkr;x;I ˆ jvjm;x;I ; jvjr;x;I ˆ r ; dy x;I mˆ0 Hxr …I† ˆ fv=kvkr;x;I < 1g: For any real r > 0, the space Hxr …I† is de®ned by the complex interpolation between the spaces Hx‰rŠ …I† and Hx‰r‡1Š …I†. Furthermore C01 …I† ˆ fv=v is infinitely differentiable on I and has compact supportg: r …I† denotes the closure of C01 …I† in Hxr …I†. Besides L1 …I† is the space of H0;x essentially bounded functions with the norm k
k1;I . Next, let B be a Banach space with the norm k
kB and D be a bounded open domain in R2 with the boundary satisfying locally Lipschitz conditions. Let c c c ˆ …c1 ; c2 † 2 N2 ; jcj ˆ c1 ‡ c2 ; z ˆ …z1 ; z2 † 2 D and @ c u ˆ @ c1 ‡c2 u=@z11 @z22 . De®ne

L2 …D; B† ˆ fu…z†: D ! B=u is strongly measurable and kukL2 …D;B† < 1g; C…D; B† ˆ fu…z†: D ! B=u is strongly continuous and kukC…D;B† < 1g where

0

kukL2 …D;B† ˆ @

Z

11=2 kuk2B dzA

;

kukC…D;B† ˆ max ku…z†kB : z2D

D

For any integer s P 0, de®ne H s …D; B† ˆ fu…z†=kukH s …D;B† < 1g equipped with the semi-norm and norm jujH s …D;B† ˆ

k@zs ukL2 …D;B† ;

kukH s …D;B† ˆ

s X juj2H m …D;B†

!1=2 :

mˆ0

For real s > 0, we de®ne H s …D; B† by the complex interpolation as before. Now, we introduce the non-isotropic space

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Hxr;s …X† ˆ L2 …Q; Hxr …I†† \ H s …Q; L2x …I††;

217

r; s P 0

with the norm 2

2

2

kukHxr;s …X† ˆ kukL2 …Q;Hxr …I†† ‡ kukH s …Q;L2x …I†† : Also let Mxr;s …X† ˆ Hxr;s …X† \ H sÿ1 …Q; Hx1 …I††;

r; s P 1;

5=3 Ar;s …Q; Hx1 …I†† \ H s‡1 …Q; Hxr …I†† \ H s …Q; Hxr‡1 …I††; x …X† ˆ H

Yxr;s;d …X†

ˆ

Hxr;s …X†

\H

1‡d

…Q; Hx‰r=2Š‡1=2‡d …I††

\H

‰s=2Š‡1‡d

r; s P 0;

…Q; Hx1=2‡d …I††;

r; s P 0; d > 0; r;s;d Y1;x …X† ˆ Hxr‡1;s‡1 …X† \ H 1 …Q; Hxr …I†† \ H s …Q; Hx1 …I†† \ H 1‡d …Q; Hx‰r=2Š‡3=2‡d …I†† \ H ‰s=2Š‡1‡d …Q; Hx3=2‡d …I†† \ H 2‡d …Q; Hx‰r=2Š‡1=2‡d …I†† \ H ‰s=2Š‡2‡d …Q; Hx1=2‡d …I††; r; s P 0; d > 0: r;s …X† Their norms are de®ned in the way similar to k
kHxr;s …X† . Moreover let H0;x r;r 1 r;s r;r r be the closure of C0 …X† in Hx …X†. If r ˆ s, then Hx …X† ˆ Hx …X†, H0;x …X† ˆ r …X† and denote their semi-norm and norm by j
jr;x and k
kr;x , respecH0;x tively, etc. In addition, we denote by L1 …I†; L1 …X† and W 1;1 …X† the usual Sobolev spaces with the norms k
k1;I , k
k1 and k
k1;1 , etc. For the product 3 Hilbert spaces (e.g., …L2x …X†† ), we still use the notations …:; :†x , k
kx and k
kHxr;s …X† , etc. For simplicity of statements, let s ˆ min…s; k ‡ 1† and denote throughout this paper by c; a positive constant independent of h; N ; s and any function, which may be di€erent in di€erent cases. For some lemmas, we require that there exist a suitably big positive constant c1 and a positive constant c2 such that

c1 hÿ4=3 6 N 6 c2 hÿ4=3 :

…5†

 for all t 2 Rs , then Lemma 1. If u…x; y; t† 2 L2 …Q† C…I† 2

2

2…u…t†; ut …t††N ;h;x ˆ …ku…t†kN ;h;x †t ÿ skut …t†kN ;h;x : This lemma can be proved in the same way as in the proof of Lemma 1 of [17]. Let PN denote the Chebyshev truncated operator. We have that (see [7]) ku ÿ PN ukLqx …I† 6 crN ;q N ÿm kukWxm;q …X† ;

m P 0; 1 6 q 6 1;

…6†

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with  rN ;q ˆ

1 ‡ ln N

if q ˆ 1 or q ˆ 1;

1

otherwise:

Lemma 2 (Lemma 1 of [21]). If u 2 L2x …X† and v 2 L2 …Q†  PN …I†, then p kvkx 6 kvkN ;h;x 6 2kvkx ; j…u; v†N ;h;x ÿ …u; v†x j 6 c…ku ÿ PN ÿ1 ukx ‡ ku ÿ Pc ukx †kvkx : 1 …X†, we have Lemma 3 (Lemma 2 of [21]). For any u 2 H0;x 2

aN ;h;x …u; u† P 14kuk1;x : k

Lemma 4 (Lemma 6 of [19]). For any u 2 S~h …Q† PN …I† with k P 1, we have p N kukx : kuk1 6 c h Moreover if u 2 H 1 …Q; L2x …I††, then kPN uk1 6 cN jln hj1=2 kukH 1 …Q;L2x …I†† : k 1=2 Remark 1. For any u 2 Hx1 …X† \ X~ N ;h …X†, kuk1 6 cN jln hj kukH 1 …Q; L2x …I††. r~;~ s Lemma 5 (Lemma 2 of [19]). Let r~ ˆ min…r; 1†, ~s ˆ min…s; 1† and u 2 H0;x …X† r;s \ Hx …X† with r; s P 0, then

ku ÿ PN ;h ukx 6 c…N ÿr ‡ hs†kukHxr;s …X† : Lemma 6 (Lemma 6 of [21]). If u 2 H b …Q; Hxr …I†† with b P 0; r > 1=2 and 0 6 a 6 r, then ku ÿ Pc ukH b …Q;Hxa …I†† 6 cN 2aÿr kukH b …Q;Hxr …I†† : In order to get better error estimation, we introduce the projection 1 1 …X† ! XNk ;h …X† such that for any u 2 H0;x …X†, PN ;h : H0;x ax …PN ;h u ÿ u; v† ˆ 0;

8v 2 XNk ;h …X†;

1 1 …X† ! XNk ;h …X† such that for any u 2 H0;x …X† and the projection PNc ;h : H0;x

aN ;h;x …PNc ;h u; v† ˆ ax …u; v†;

8v 2 XNk ;h …X†:

Then we have the following lemma.

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219

1 Lemma 7. Let condition (5) hold and k P 1. If u 2 H0;x …X† \ Mxr;s …X† with r; s P 1, then

ku ÿ PN ;h uk1;x 6 c…N 1ÿr ‡ hsÿ1 †kukMxr;s …X† ; ku ÿ PNc ;h uk1;x 6 c…N 1ÿr ‡ hsÿ1 †kukMxr;s …X† : If in addition u 2 Mxr‡1=4;s …X†, then ku ÿ PN ;h ukx 6 c…N ÿr ‡ hs†kukM r‡1=4;s …X† ; x

ku ÿ

PNc ;h ukx

6 c…N

ÿr

s

‡ h †kukM r‡1=4;s …X† : x

Proof. The ®rst and the third conclusions come from Lemma 4 of [19]. Next by an argument similar to the proof of Lemma 7 of [20], we get ku ÿ PNc ;h uk1;x 6 c…N 1ÿr ‡ hsÿ1 †kukMxr;s …X† : Finally, following the same lines as in the last part of the proof of Lemma 7 in [20], we get ku ÿ PNc ;h ukx 6 c…N ÿ1 ‡ h†…N 3=4ÿr ‡ hsÿ1 †kukM r‡1=4;s …X† x

which completes the proof. Lemma 8 (Lemma 5 of [19]). If u 2 Hxr …I† with r > 1=2, then kPN uk1;I 6 ckukr;x;I : We know from Theorem 3.1.5 of [23] that for any u 2 H s …Q† with s > 1, ku ÿ Pkh ukl;1;Q 6 chsÿlÿ1 jujs;Q ;

l ˆ 0; 1:

…7†

1 …X† \ Mx3=2;5=3 …X† \ H b …Q; Hxa …I†† Lemma 9. Let condition (5) hold. If u 2 H0;x with a > 1=2; b > 1, then

kPNc ;h uk1 6 ckukM 3=2;5=3 …X†\H b …Q;H a …I†† : x

x

1 …X† \ Aa;b If in addition u 2 H0;x x …X† with a > 1=2; b > 1; then

kPNc ;h uk1;1 6 ckukAa;b : x …X† Proof. We have kPNc ;h uk1 6 kPNc ;h u ÿ Pkh PN uk1 ‡ kPkh PN uk1 : By (5) and (6), Lemmas 4 and 7 and Theorem 3.2.1 of [23], we have

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kPNc ;h u

ÿ

Pkh PN uk1

p N …kPNc ;h u ÿ ukx ‡ ku ÿ PN ukx ‡ ku ÿ Pkh ukx † 6c h p N ……N ÿ5=4 ‡ h5=3 †kukM 3=2;5=3 …X† ‡ N ÿ5=4 kukL2 …Q;H 5=4 …I†† 6c x x h 5=3 ‡ h kukH 5=3 …Q;L2x …I†† † 6 ckukM 3=2;5=3 …X† : x

By Lemma 8 and (7), kPkh PN uk1 6 ckPN ukH b …Q;C…I†† 6 ckukH b …Q;Hxa …I†† : Thus the ®rst conclusion follows. Next, we prove the second one. We de®ne the operator Ph1 : H01 …Q† ! Shk …Q† such that for any u 2 H01 …Q†, 2 X …@j …Ph1 u ÿ u†; @j v†L2 …Q† ˆ 0; jˆ1

8v 2 Shk …Q†:

Then we have (see [13]) ku ÿ Ph1 ukl;Q 6 chsÿl jujs;Q ;

s P 1; 0 6 l 6 1:

…8†

1 1 …I† ! VN …I† such that for any u 2 H0;x …I†, Also we de®ne the operator PN1 : H0;x

…@3 …PN1 u ÿ u†; @3 …vx††L2 …I† ˆ 0;

8v 2 VN …I†:

We have (see (9.5.17) of [7]) ku ÿ PN1 ukl;x;I 6 cN lÿr kukr;x;I ;

0 6 l 6 1; r P 1:

…9†

Now, we begin to estimate kPNc ;h uk1;1 . In fact, jPNc ;h uj1;1 6 jPNc ;h u ÿ PN ;h uj1;1 ‡ jPN ;h uj1;1 :

…10†

By Lemma 4, we have

p N c ÿ 6c …11† jP u ÿ PN ;h uj1;1 : h N ;h Let # be the identity operator. Then by Lemma 3 and the de®nitions of PNc ;h and PN ;h , and the fact that jPNc ;h u

PN ;h uj1;1

j…z; v†N ;x ÿ …z; v†x;I j ˆ jzN vN j; where zq ˆ

2 pcq

Z

1 ÿ1

z…y†Tq …y†x…y† dy;

8z; v 2 PN …I†;

c0 ˆ 2; cq ˆ 1

for q P 1;

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221

Tq …y† being the Chebyshev polynomial of order q, we have that 2

kPNc ;h u ÿ PN ;h uk1;x 6 4aN ;h;x …PNc ;h u ÿ PN ;h u; PNc ;h u ÿ PN ;h u† ˆ 4ax …PN ;h u; PNc ;h u ÿ PN ;h u† ÿ 4aN ;h;x …PN ;h u; PNc ;h u ÿ PN ;h u† ˆ4

2 X ‰…@j PN ;h u; @j …PNc ;h u ÿ PN ;h u††x jˆ1

ÿ …@j PN ;h u; @j …PNc ;h u ÿ PN ;h u††N ;h;x Š: Therefore kPNc ;h u ÿ PN ;h uk1;x 6 c

2 X jˆ1

k…# ÿ PN ÿ1 †@j PN ;h ukx

2 X 6 c …k…# ÿ PN ÿ1 †…@j PN ;h u ÿ @j Ph1 PN1 u†kx jˆ1

‡ k…# ÿ PN ÿ1 †@j Ph1 PN1 ukx †:

…12†

Furthermore 2 X k…# ÿ PN ÿ1 †…@j PN ;h u ÿ @j Ph1 PN1 u†kx 6 cjPN ;h u ÿ Ph1 PN1 uj1;x :

…13†

jˆ1

Let w ˆ PN ;h u ÿ Ph1 PN1 u. We have from Lemma 3 and the de®nition of PN ;h that kPN ;h u ÿ Ph1 PN1 uk21;x 6 4ax …PN ;h u ÿ Ph1 PN1 u; PN ;h u ÿ Ph1 PN1 u† ˆ 4ax …u ÿ Ph1 PN1 u; PN ;h u ÿ Ph1 PN1 u† ˆ4

2 X …@j …u ÿ Ph1 PN1 u†; @j w†x jˆ1

‡ 4…@3 …u ÿ Ph1 PN1 u†; @3 …wx††L2 …X† : Moreover 2 X jˆ1

…@j …u ÿ Ph1 PN1 u†; @j w†x ˆ

2 X jˆ1

……@j …u ÿ PN1 u†; @j w†x

‡ …@j …PN1 ÿ Ph1 PN1 †u; @j w†x † ˆ

2 X jˆ1

…@j …u ÿ PN1 u†; @j w†x ;

…@3 …u ÿ Ph1 PN1 u†; @3 …wx††L2 …X† ˆ …@3 …u ÿ Ph1 u†; @3 …wx††L2 …X† ‡ …@3 …Ph1 u ÿ PN1 Ph1 u†; @3 …wx††L2 …X† ˆ …@3 …u ÿ Ph1 u†; @3 …wx††L2 …X† :

222

Thus

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  2 kPN ;h u ÿ Ph1 PN1 uk1;x 6 c ku ÿ PN1 ukH 1 …Q;L2x …I†† ‡ ku ÿ Ph1 ukL2 …Q;Hx1 …I†† kwk1;x

and so

  kPN ;h u ÿ Ph1 PN1 uk1;x 6 c…N ÿr ‡ hs† kukH 1 …Q;Hxr …I†† ‡ kukH s…Q;Hx1 …I†† :

…14†

Similarly, 2 2 X X k…# ÿ PN ÿ1 †@j Ph1 PN1 ukx 6 k…# ÿ PN ÿ1 †…@j Ph1 PN1 u ÿ @j PN1 u†kx jˆ1

jˆ1

‡

2 X jˆ1

k…# ÿ PN ÿ1 †@j PN1 ukx

and 2 X k…# ÿ PN ÿ1 †…@j Ph1 PN1 u ÿ @j PN1 u†kx 6 ck…# ÿ PN ÿ1 †PN1 ukH 1 …Q;L2x …I†† jˆ1

6 cN ÿr kukH 1 …Q;Hxr …I†† ; 2 X k…# ÿ PN ÿ1 †@j PN1 ukx 6 cN ÿr kukH 1 …Q;Hxr …I†† :

So

jˆ1

2 X k…# ÿ PN ÿ1 †@j Ph1 PN1 ukx 6 cN ÿr kukH 1 …Q;Hx1 …I†† :

…15†

jˆ1

Thus we have from (12)±(15) that

  kPNc ;h u ÿ PN ;h uk1;x 6 c…N ÿr ‡ hs† kukH 1 …Q;Hxr …I†† ‡ kukH s…Q;Hx1 …I††

and (11) leads to

  jPNc ;h u ÿ PN ;h uj1;1 6 c kukH 1 …Q;H 5=4 …I†† ‡ kukH 5=3 …Q;Hx1 …I†† : x

…16†

Now, we turn to estimate the term jPN ;h uj1;1 . We have that jPN ;h uj1;1 6 jPN ;h u ÿ Ph1 PN1 uj1;1 ‡ jPh1 PN1 uj1;1 p N  jP u ÿ Ph1 PN1 uj1;x ‡ jPh1 PN1 uj1;1 : 6c h N ;h From (14), we get p   N  jPN ;h u ÿ Ph1 PN1 uj1;x 6 c kukH 1 …Q;H 5=4 …I†† ‡ kukH 5=3 …Q;Hx1 …I†† : x h

…17†

…18†

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223

We have also that jPh1 PN1 uj1;1 6

2 X jˆ1

k@j Ph1 PN1 uk1 ‡ k@3 Ph1 PN1 uk1 :

…19†

For j ˆ 1; 2; k@j Ph1 PN1 uk1 6 k@j Ph1 PN1 u ÿ @j Ph1 PN uk1 ‡ k@j Ph1 PN u ÿ @j Pkh PN uk1 ‡ k@j Pkh PN uk1 ; p N 1 1 kPh …PN u ÿ PN u†kH 1 …Q;L2x …I†† k@j Ph1 PN1 u ÿ @j Ph1 PN uk1 6 c h p   N 6c kPN1 u ÿ ukH 1 …Q;L2x …I†† ‡ ku ÿ PN ukH 1 …Q;L2x …I†† h 6 ckukH 1 …Q;H 5=4 …I†† ; x

k@j Ph1 PN u

ÿ

@j Pkh PN uk1

6 ch

ÿ1

k@j Ph1 PN u

ÿ @j Pkh PN ukL2 …Q;C…I††

 6 chÿ1 kPh1 PN u ÿ PN ukH 1 …Q;C…I††  ‡kPkh PN u ÿ PN ukH 1 …Q;C…I††

6 ckPN ukH 2 …Q;C…I†† 6 ckukH 2 …Q;Hxa …I†† ; k@j Pkh PN uk1 6 k@j Pkh PN u ÿ @j PN uk1 ‡ kPN …@j u†k1 6 ckPN ukH 2 …Q;C…I†† ‡ ck@j ukC…Q;Hxa …I†† 6 ckukH 2 …Q;Hxa …I†† ‡ ck@j ukH b …Q;Hxa …I†† 6 ckukH b‡1 …Q;Hxa …I†† : So 2 X k@j Ph1 PN1 uk1 6 ckukH 1 …Q;H 5=4 …I††\H b‡1 …Q;H a …I†† : jˆ1

x

x

…20†

On the other hand, k@3 PN1 Ph1 uk1 6 kPh1 …@3 PN1 u† ÿ Pkh …@3 PN1 u†k1 ‡ kPkh …@3 PN1 u† ÿ Pkh PN …@3 u†k1 ‡ kPkh PN …@3 u†k1 :

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Moreover kPh1 …@3 PN1 u†

ÿ

Pkh …@3 PN1 u†k1

p N 1 6c kPh …@3 PN1 u† ÿ Pkh …@3 PN1 u†kx h p N 6c …kPh1 …@3 PN1 u† ÿ @3 PN1 ukx h ‡ kPkh …@3 PN1 u† ÿ @3 PN1 ukx † 6 ck@3 PN1 ukH 5=3 …Q;L2x …I†† 6 ckPN1 ukH 5=3 …Q;Hx1 …I†† 6 ckukH 5=3 …Q;Hx1 …I†† ;

p kPkh …@3 PN1 u† ÿ Pkh PN …@3 u†k1 6 c N kPkh …@3 PN1 u ÿ PN …@3 u††kC…Q;L2x …I†† p 6 c N k@3 PN1 u ÿ @3 ukH b …Q;L2x …I††  ‡ kPN …@3 u† ÿ @3 ukH b …Q;L2x …I†† 6 ckukH b …Q;H 3=2 …I††

and

x

kPkh PN …@3 u†k1 6 ckPN …@3 u†kH b …Q;C…I†† 6 ckukH b …Q;Hxa‡1 …I†† : Thus k@3 PN1 Ph1 uk1 6 ckukH 5=3 …Q;Hx1 …I††\H b …Q;Hxa‡1 …I†† :

…21†

We obtain from (19)±(21) that for a > 1=2 and b > 1, jPh1 PN1 uj1;1 6 ckukH 5=3 …Q;Hx1 …I††\H b‡1 …Q;Hxa …I††\H b …Q;Hxa‡1 …I†† :

…22†

Thus from (17), (18) and (22), jPN ;h uj1;1 6 ckukAa;b : x …X†

…23†

Finally, the combination of estimates (10), (16) and (23) leads to the second conclusion. Lemma 10 (Lemma 15 of [21]). If u; v 2 Yxr;s;d …X† with r; s P 0 and d > 0; then kuvkHxr;s …X† 6 ckukYxr;s;d …X† kvkYxr;s;d …X† : k

Lemma 11. For all u 2 …S~h …Q† \ Hx1 …Q††  PN …I†; we have 2

2

juj1;x 6 …2N 4 ‡ cd hÿ2 †kukx : This lemma can be proved in the way similar to the proof of Lemma 7 in [17]. 1 …X†; then Lemma 12 (Lemma 3 of [21]). If u 2 L2x …X† and v 2 H0;x

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225

j…u; @3 …vx††L2 …X† j 6 2kukx k@3 vkx : k

k

Lemma 13. Let u 2 …S~h …Q† \ H 1 …Q†† WN …I† and g 2 …S~h …Q†\H 1 …Q†† PN …I† satisfy the following equation aN ;h;x …u; v† ˆ …g; v†N ;h;x ;

k

8v 2 X~ N ;h …X†:

…24†

If Eq. (5) holds, then juj1;x 6 ckgkx : Proof. We ®rst introduce some preliminary knowledge. Let g ˆ …1 ÿ y 2 †1=2 . De®ne the spaces L2g …I† and Hgr …I† in the same way as L2x …I† and Hxr …I†, etc. From Lemma 3.1 of [24], we know that for any n 2 L2g …I† and k > 0, there exists a unique function w 2 Hg2 …I† such that ( 2 Lw ˆ ÿ ddyw2 ‡ kw ˆ n in I; …25† dw …ÿ1† ˆ dw …1† ˆ 0: dy dy Let H ÿs …I† be the dual space of H s …I†. From Eq. (25), we have 2

2

2

kwkH 1 …I† ‡ kkwkL2 …I† 6 cknkH ÿ1 …I† : On the other hand, if n 2 L2 …I†, then w 2 H 2 …I†. By multiplying Eq. (25) by w; ÿd2 w=dy 2 and integrating the results by parts, we obtain that 2

2

2

kwkH 2 …I† ‡ kkwkH 1 …I† 6 cknkL2 …I† : By the theory of interpolation of spaces, we have that if n 2 H ÿs …I† with 0 < s < 1, then w 2 H 2ÿs …I† and 2

2

2

kwkH 2ÿs …I† ‡ kkwkH 1ÿs …I† 6 cknkH ÿs …I† :

…26†

Since for any real s P 0; Hgs …I† 2 H sÿ1=4 …I† (Theorem 4.2 of [24]), we get kwk2H 7=4 …I† ‡ kkwk2H 3=4 …I† 6 cknk2H ÿ1=4 …I† 6 cknk2g;I :

…27†

Moreover for any real s P 1=4; H s …I† 2 Hxsÿ1=4 …I† (Theorem 4.1 of [24]), and thus (27) leads to 2

2

2

2

kwk3=2;x;I ‡ kkwk1=2;x;I 6 cknkg;I ˆ ckLwkg;I :

…28†

Next, we consider the following eigenvalue problem, that is to ®nd / 2 k S~h …Q† \ H 1 …Q† and k such that  Z Z Z Z  @/ @z @/ @z k dx ˆ k ‡ /z dx; 8z 2 S~h …Q† \ H 1 …Q†: @x1 @x1 @x2 @x2 Q

Q

…29†

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Then there exists a normalized L2 …Q†-orthogonal eigenfunction system f/l …x†g; l ˆ 1; 2; . . . ; Mh (see [25]). The corresponding eigenvalues are ranged as 0 ˆ k0 < k1 6 k2 6


6 kMh : Let /0 …x†  1. By Eq. (29), we have  Z Z  @/l @/q @/l @/q dx ˆ kl dlq ; ‡ @x1 @x1 @x2 @x2

l; q ˆ 0; 1; . . . ; Mh ;

Q

and so by the inverse inequality (see [23]), kl ˆ j/l j21;Q 6 chÿ2 k/l k2Q ˆ chÿ2 :

…30†

Now, we turn to prove the lemma. Since u 2 k g 2 …S~h …Q† \ H 1 …Q†† PN …I†, we put uˆ

Mh X ul …y†/l …x†; lˆ0

gˆ

Mh X lˆ0

k …S~h …Q†

\ H 1 …Q†† WN …I† and

gl …y†/l …x†;

where ul …y† 2 WN …I†; gl …y† 2 PN …I†. From Eq. (24), we have ! ! Mh Mh 2 X X X @/l @ 2 ul ul ; @j v ÿ / …x†; v @xj @y 2 l jˆ1 lˆ1 lˆ1 x N ;h;x ! Mh X k ˆ gl …y†/l …x†; v ; 8u 2 X~ …X†: lˆ1

N ;h

…31†

N ;h;x

Firstly, we consider the case with l 6ˆ 1. Take v ˆ …1 ÿ y 2 †zl /l …x† in Eq. (31) with zl 2 PN ÿ2 …I†. Then  2  d ul ; z ˆ …gl …y†; …1 ÿ y 2 †zl †N ;x : kl …ul …y†; …1 ÿ y 2 †zl †N ;x ÿ l dy 2 g;I Clearly Lul ˆ ÿd2 ul =dy 2 ‡ kul . Thus   d 2 ul …Lul ; zl †g;I  ÿ 2 ‡ kl ul ; zl ˆ …gl ; zl †g;I ‡ E1 …zl † ‡ E2 …zl † dy g;I

…32†

with E1 …zl † ˆ kl …ul ; …1 ÿ y 2 †ul †x;I ÿ kl …ul ; …1 ÿ y 2 †zl †N ;x ; E2 …zl † ˆ …gl ; …1 ÿ y 2 †zl †N ;x ÿ …gl ; …1 ÿ y 2 †zl †x;I : Let P~N be the L2x …I†-orthogonal projection, and zl ˆ P~N ÿ2 Lul . Then we have zl ˆ Lul ÿ kl …ul ÿ P~N ÿ2 ul † 2 PN ÿ2 …I†:

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227

Thus Eq. (32) reads 2 kLul kg;I ÿ kl …Lul ; ul ÿ P~N ÿ2 ul †g;I ˆ …gl ; Lul †g;I ÿ kl …gl ; ul ÿ P~N ÿ2 ul †g;I ‡ E1 …P~N ÿ2 Lul † ‡ E2 …P~N ÿ2 Lul †:

…33†

Furthermore, since …1 ÿ y 2 †zl 2 PN …I†, we get from (9.3.5) of [7] that E1 …P~N ÿ2 Lul †jj 6 ckl k…1 ÿ y 2 †zl kx;I kul ÿ P~N ÿ1 ul kx;I 6 ckl kzl kg;I kul ÿ P~N ÿ1 ul kx;I 6 ckl kul ÿ P~N ÿ1 ul kx;I …kLul kg;I ‡ kl kul ÿ P~N ÿ2 ul kg;I †: jE2 …P~N ÿ2 Lul †j 6 ckgl kx;I k…1 ÿ y 2 †zl ÿ P~N ÿ1 …1 ÿ y 2 †zl kx;I 6 ckgl kx;I k…1 ÿ y 2 †zl kx;I 6 ckgl kx;I kzl kg;I 6 ckgl kx;I …kLul kg;I ‡ kl kul ÿ P~N ÿ2 ul kg;I †: By substituting the above estimations into Eq. (33), we obtain 1 2 2 2 2 kLul kg;I 6 ckgl kx;I ‡ ck2l …kul ÿ P~N ÿ1 ul kx;I ‡ kul ÿ P~N ÿ2 ul kg;I †: 2

…34†

We know from (28) that 2

2

2

kul k3=2;x;I ‡ kl kul k1=2;x;I 6 ckLul kg;I : Moreover, we have from (34) that 2

2

2

kul k3=2;x;I ‡ kl kul k1=2;x;I 6 ckgl kx;I ‡ ck2l N ÿ3 kul k3=2;x;I : According to (30), 1 ÿ ck2l N ÿ3 P 1 ÿ chÿ4 N ÿ3 . Furthermore thanks to condition (5), we have 1 ÿ chÿ4 N ÿ3 P a > 0 and so 2

2

2

kul k3=2;x;I ‡ kl kul k1=2;x;I 6 ckgl kx;I : Hence for l 6ˆ 0; 2

2

2

jul j1;x;I ‡ kl kul kx;I 6 ckgl kx;I :

…35†

Next we consider the case with l ˆ 0. By taking v ˆ ÿ…1 ÿ y 2 †@y2 u0 /0 …x† in Eq. (31), we get 2

k@y2 u0 kg;I ˆ ÿ…g1 ; …1 ÿ y 2 †@y2 u0 †N ;x 6 ckg1 kN ;x k…1 ÿ y 2 †@y2 u0 kN ;x 6 ckg1 kx;I k…1 ÿ y 2 †@y2 u0 kx;I 6 ckg1 kx;I k@y2 u0 kg;I : Thus k@y2 u0 kg;I 6 ckg1 kx;I . Moreover, since u0 2 WN …I†; we have from H older inequality that

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Z

2

ju0 j1;x;I ˆ

2

Z

I

6

0 x…y†@

I

Z

Z

Z

x…y†…@y u0 † dy ˆ

x…y† I

10

…@s2 u0 †2 g…s† dsA@

y

ÿ1

Z

I

@s2 u0 …s† ds

2 dy

1

x…s† dsA dy 6 ck@y2 u0 kg;I :

I

Therefore ju0 j21;x;I 6 ckg1 k2x;I :

…36†

Finally we get from Eqs. (35) and (36) that 2

juj1;x ˆ

Mh X 2 2 2 …jul j1;x;I ‡ kl kul kx;I † 6 ckgkx lˆ0

which completes the proof. Lemma 14 (Lemma 4.16 of [2]). Suppose that (i) Z is a non-negative function de®ned on Rs ; (ii) D1 ; D2 ; D3 and q are non-negative constants; (iii) H …Z† is a function such that if Z 6 D3 , then H …Z† 6 0; (iv) for all t 2 Rs and t > 0, X …D1 Z…t0 † ‡ D2 Z 2 …t0 † ‡ H …Z…t0 †††; Z…t† 6 q ‡ s t0 2Rs t0 6 tÿs

(v) Z…0† 6 q and q e2D1 t1 6 min…D1 =D2 ; D3 † for some t1 2 Rs . Then for all t 2 Rs and t 6 t1 , we have Z…t† 6 q e2D1 t :

5. The generalized stability This section is dedicated to the generalized stability of scheme (4). Assume that u…0† and f …t† have the errors u~…0† and f~…t† which induce the errors of u…t† and p…t†, denoted by u~…t† and p~…t†, respectively. Then they satisfy the following equation u; u ‡ u~† ‡ dc …u; u~† ‡ r~ p; v†N ;h;x ‡ maN ;h;x …~ u ‡ rs~ ut ; v† …~ ut ; v†N ;h;x ‡ …dc …~ ˆ …f~; v†N ;h;x ;

3

8v 2 …XNk‡k ;h …X†† ;

p; w† ˆ …Uc …~ u† ‡ Uc …u; u~† ÿ r
f~; w†N ;h;x ; aN ;h;x …~

…37† k 8w 2 X~ N ;h …X†;

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229

where Uc …u; u~† ˆ

3 X

‰Pc …@l u~q @q ul ‡ @l uq @q u~l † ÿ Pc …@q u~q @l ul ‡ @l u~l @q uq †Š:

l;qˆ1

Let e > 0, and m be a positive constant which will be determined later. By taking v ˆ 2~ u…t† ‡ ms~ ut …t† in the ®rst formula of Eq. (37), we have from Lemmas 1±3 that m mrms2 2 2 2 2 k~ ut k1;x uk1;x ‡ ut kN ;h;x ‡ k~ …k~ ukN ;h;x †t ‡ s…m ÿ 1 ÿ e†k~ 4 2 6 X ‡ 2mrsaN ;h;x …~ ut ; u~† ‡ mmsaN ;h;x …~ u; u~t † ‡ Fj jˆ1

  sm2 kPc f~k2x 6 2k~ uk2x ‡ 2 ‡ 2e

…38†

where F2 ˆ …dc …u; u~†; 2~ u ‡ ms~ ut †N ;h;x ;

u; u†; 2~ u ‡ ms~ ut †N ;h;x ; F1 ˆ …dc …~ F3 ˆ 2…dc …~ u; u~†; u~†N ;h;x ; p; u~†N ;h;x ; F5 ˆ 2…r~

F4 ˆ ms…dc …~ u; u~†; u~t †N ;h;x ;

F6 ˆ ms…r~ p; u~t †N ;h;x :

We know from (3) that 2mrsaN ;h;x …~ ut ; u~† ˆ A1 ‡ A2 ‡ A3 ; mmsaN ;h;x …~ u; u~t † ˆ B1 ‡ B2 ‡ B3 ; where A1 ˆ 2mrs

2 X jˆ1

…@j u~t ; @j u~†N ;h;x ;

A2 ˆ 2mrs…@y u~t ; @y u~†x ; 2

A3 ˆ 2mrs…@y u~t ; yx u~†x ;

B1 ˆ mms

2 X …@j u~; @j u~t †N ;h;x ; jˆ1

B2 ˆ mms…@y u~; @y u~t †x ; B3 ˆ mms…@y u~; yx2 u~t †x :

It is easy to verify that 2  m X 2 2 ‰…k@x u~kN ;h;x †t ÿ sk@x u~t kN ;h;x Š; A1 ‡ B1 ˆ ms r ‡ 2 jˆ1  m 2 2 A2 ‡ B2 ˆ ms r ‡ ‰…k@y u~kN ;h;x †t ÿ sk@y u~t kN ;h;x Š: 2

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J. Hou, B. Guo / Appl. Math. Comput. 101 (1999) 209±244

By Lemma 1 of [26], we have jA3 j ‡ jB3 j 6 P3

Let juj21;N ;h;x ˆ

 m m m 2 2 r‡ k@y u~kx ‡ 4ms2 r ‡ k@y u~t kx : 4 2 2

jˆ1

k@j uk2N ;h;x . Then (38) reads

m 2 2 2 …k~ ukN ;h;x †t ‡ s…m ÿ 1 ÿ e†k~ ut kN ;h;x ‡ …4 ÿ m ÿ 2r†j~ uj1;x 8 6   X m 2 mrms2 m 2 2 k~ ut k1;x ÿ 5ms2 r ‡ ‡ ms r ‡ Fj …j~ uj1;N ;h;x †t ‡ j~ ut j1;N ;h;x ‡ 4 2 2 jˆ1   sm2 2 2 kPc f~kx : 6 2k~ u kx ‡ 2 ‡ …39† 2e Now, we estimate jFj j. Firstly from (3) and Green's formula, F1 ˆ

2 X …@j Pc …uj u~†; 2~ u ‡ ms~ ut †N ;h;x ‡ …@3 Pc …u3 u~†; 2~ u ‡ ms~ ut †x jˆ1

ˆÿ

2 X …Pc …uj u~†; 2@j u~ ‡ ms@j u~t †N ;h;x ÿ …Pc …u3 u~†; @3 ……2~ u ‡ ms~ ut †x††L2 …X† : jˆ1

So by Lemmas 2 and 12, jF1 j 6

2 X kPc …uj u~†kN ;h;x k2@j u~ ‡ ms@j u~t kN ;h;x ‡ 2kPc …u3 u~†kx j2~ u ‡ ms~ ut j1;x jˆ1

6 ckuk1 k~ ukx …2j~ uj1;x ‡ msj~ ut j1;x † 6

em 2 emms2 c…m ‡ 1† j~ ut j21;x ‡ j~ uj1;x ‡ kuk21 k~ uk2x : 6 8 em

Similarly, jF2 j 6

em 2 emms2 c…m ‡ 1† j~ ut j21;x ‡ j~ uj1;x ‡ kuk21 k~ uk2x : 6 8 em

Next, we have F3 ˆ ÿ2

2 X …Pc …~ uj u~†; @j u~†N ;h;x ÿ 2…Pc …~ u3 u~†; @3 …~ ux††L2 …X† : jˆ1

J. Hou, B. Guo / Appl. Math. Comput. 101 (1999) 209±244

231

By Lemmas 4 and 12, 1=2

jF3 j 6 ck~ uk1 k~ ukx j~ uj1;x 6 cN j ln hj 6

2

k~ ukx j~ uj1;x

em 2 cN j ln hj 2 2 uj1;x : j~ uj ‡ k~ ukx j~ 8 1;x em

Similarly, jF4 j 6

emms2 cm 2 2 2 j~ ut j1;x ‡ uj1;x : N j ln h jk~ ukx j~ 6 em

We apply Lemma 13 to the second formula of Eq. (37) and obtain j~ pj1;x 6 c…kUc …~ u†kx ‡ kUc …u; u~†kx ‡ kPc …r
f~†kx †: Moreover



X

3

u†kx ˆ ‰Pc …@l u~q @q u~l † ÿ Pc …@l u~l @q u~q †Š kUc …~

l;qˆ1

x p N 2 6 cj~ uj1;1 j~ uj1;x 6 c j~ uj1;x ; h uj1;x : kUc …u; u~†kx 6 cjuj1;1 j~

Thus

and

p N 2 uj1;x ‡ j~ uj1;x ‡ kPc …r
f~†kx † j~ pj1;x 6 c…juj1;1 j~ h jF5 j 6 cj~ pj1;x k~ u kx

  em 2 1 2 cN 2 2 2 2 u kx ‡ 6 j~ k~ ukx j~ uj1;x ‡ ckPc …r
f~†kx : uj ‡ c 1 ‡ juj1;1 k~ 8 1;x em emh2

By Lemma 11, pj1;x k~ ut k x jF6 j 6 msj~ 6 esk~ ut k2x ‡ ‡

cm2 s 2 cm2 s juj1;1 ‡ j~ kPc …r
f~†k2x uj21;x e e

cms2 N 2 2 …2N 4 ‡ cd hÿ2 †k~ ukx j~ uj1;x : eh2

Let jjj
jjj1;1 ˆ maxt2Rs ku…t†k1;1 , etc. By substituting the above estimations into Eq. (39), we have

232

J. Hou, B. Guo / Appl. Math. Comput. 101 (1999) 209±244

  m 15 ÿ m ÿ 2r ÿ 4e j~ uj21;x ‡ s…m ÿ 1 ÿ ‡ 8 4   rm m 2 em  2 ÿ 5…2r ‡ m† ÿ ‡ ms r ‡ …j~ uj1;N ;h;x †t ‡ ms2 j~ ut j1;x 2 4 2

…k~ uk2N ;h;x †t

2e†k~ ut k2x

6 M1 k~ uk2x ‡ B…k~ ukx †j~ uj21;x ‡ G1 where c 2 2 ‰…1 ‡ m†jjjujjj1 ‡ jjjujjj1;1 Š; em m cm2 s jjjujjj21;1 B…k~ ukx † ˆ ÿ ‡ 32 e    c cN 1 2 4 ÿ2 …1 ‡ m†N j ln hj ‡ 2 ‡ m s…2N ‡ cd h † k~ ‡ uk2x ; em eh m     m2 s cm2 s 2 ~ kPc f kx ‡ c 1 ‡ kPc …r
f~†k2x : G1 ˆ 2 ‡ 2e e M1 ˆ c ‡

Using Lemma 11 again, we obtain h i  em rm  2 2 ÿ ut kx …k~ ukN ;h;x †t ‡ s m ÿ 1 ÿ 2e ÿ ms 5…2r ‡ m† ‡ …2N 4 ‡ cd hÿ2 † k~ 2 4    m 15 m 2 ÿ m ÿ 2r ÿ 4e j~ uj21;x ‡ ms r ‡ ‡ …j~ uj1;N ;h;x †t 8 4 2 2

2

6 M1 k~ ukx ‡ B…k~ ukx †j~ uj1;x ‡ G1 :

…40†

Let e be suitably small and l suitably large. Suppose that 1
: ms…2N 4 ‡ cd hÿ2 † < ÿ l 5 ‡ 2e ÿ r4

…41†

We take  mˆ

33 ‡ 2e ‡ 10rms…2N 4 ‡ cd hÿ2 † 32



1 1ÿ l

2

ÿ1 :

Then the coecient of the term k~ ut kx in (40) is not less than s=32. Obviously m6

33 10r
‡ 2e ‡ ÿ 32 l 5 ‡ 2e ÿ r4

! 1ÿ

1 l

ÿ1 :

J. Hou, B. Guo / Appl. Math. Comput. 101 (1999) 209±244

So if

 l>

 ÿ1 10r 7 79 ‡ ; ÿ 2r ÿ 4e ÿ 2r ÿ 6e 5 ‡ 2e ÿ r4 2 32

233

…42†

2

then the coecient of the term j~ uj1;x in (40) is not less than m=32. Thus Eq. (40) reads  s m 2 m 2 2 2 ut kx ‡ j~ uj1;x ‡ ms r ‡ …k~ ukN ;h;x †t ‡ k~ …j~ uj1;N ;h;x †t 32 32 2 …43† 2 2 6 M1 k~ ukx ‡ B…k~ ukx †j~ uj1;x ‡ G1 : Let

s X …sk~ ut …t0 †k2x ‡ mj~ u…t0 †j21;x †; 32 t0 2Rs t0
E…t† ˆ k~ u…t†k2x ‡

t0 2Rs t0
By summing (43) for t 2 Rs , we have X 2 E…t† 6 q…t† ‡ s …M1 E…t0 † ‡ B…E…t0 ††j~ u…t0 †j1;x †: t0 2Rs t0
Finally we use Lemma 14 to get the following conclusion. Theorem 1. Assume that (i) Conditions (41) and (42) hold; (ii) for certain suitably small positive constant c3 ; sjjjujjj1;1 < c3 m; (iii) there exist positive constants d1 and d2 depending only on jjjujjj1;1 and m such that for some t1 2 Rs , h2 : N Then for all t 2 Rs ; t 6 t1 we have q…t1 † ed1 t1 6 d2

E…t† 6 q…t† ed1 t : Remark 2. Theorem 1 tells us that scheme (4) possesses generalized stability. The details of this kind of stability can be found in [2,27]. Clearly, the stability depends not only on the scheme, but also on the data errors. 6. The convergence In this section, we deal with the convergence of scheme (4). Let the operator 1 PN1 be the same as in Section 4. Also let P~h be the operator such that for any v 2 H 1 …Q†,

234

J. Hou, B. Guo / Appl. Math. Comput. 101 (1999) 209±244 2 Z Z X jˆ1

1

@j …u ÿ P~h u†@j Z dx ˆ 0;

k

8Z 2 S~h …Q† \ H 1 …Q†;

…44†

Q

Z Z

1

…u ÿ P~h u† dx ˆ 0:

…45†

Q

It can be veri®ed that for any v 2 H s …Q† with s P 1, 1

ku ÿ P~h ukH l …Q† 6 chsÿl kukH S~…Q† ; In fact, by the de®nition of 1 2 ju ÿ P~h ujH 1 …Q† ˆ

ˆ

2 X jˆ1 2 X jˆ1

1 P~h ,

0 6 l 6 1; l 6 s:

we have that for any v 2

…46† k S~h …Q†

\ H 1 …Q†;

1 1 …@j …u ÿ P~h u†; @j …u ÿ P~h u††L2 …Q†

1

…@j …u ÿ P~h u†; @j …u ÿ v††L2 …Q† 1

6 ju ÿ P~h ujH 1 …Q† ju ÿ vjH 1 …Q† :

So

1 ju ÿ P~h ujH 1 …Q† 6

k

inf

v2S~h …Q†\H 1 …Q†

ju ÿ vjH 1 …Q†

6 ku ÿ Pkh ukH 1 …Q† 6 chsÿ1 kukH s…Q† : Due to Eq. (45), we have from Poincare inequality that 1

ku ÿ P~h ukH 1 …Q† 6 chsÿ1 kukH s…Q† : P2 For l ˆ 0, let ax …w; z† ˆ jˆ1 …@j w; @j z†L2 …Q† . Let /g be the solution of problem 8 ax …/ ; v† ˆ …g; v†L2 …Q† ; 8v 2 H 1 …Q†; > > > : Q

It is obvious that 8 2 X > > > < ÿ @j2 /g ˆ g;

in Q;

…48† > > @/ > : g ˆ 0; on C: @n We know from Lax±Milgram lemma that there is a unique solution of Eq. (48). Furthermore by multiplying Eq. (47) by @j /g ; @j2 /g and integrating the results by parts, we have j/g jH s …Q† 6 ckgkL2 …Q† ; s ˆ 1; 2. Then the second formula of Eq. (47) and Poincare inequality lead to k/g kH 2 …Q† 6 ckgkL2 …Q† . Next, by taking 1 v ˆ u ÿ P~h u in Eq. (47), we obtain jˆ1

J. Hou, B. Guo / Appl. Math. Comput. 101 (1999) 209±244

235

1 1 ax …/g ; u ÿ P~h u† ˆ …g; u ÿ P~h u†L2 …Q† : 1 From the de®nition of P~h , we have 1 ax …/h ; u ÿ P~h u† ˆ 0;

k 8/h 2 S~h …Q† \ H 1 …Q†;

and thus 1 1 ax …/g ÿ /h ; u ÿ P~h u† ˆ …g; u ÿ P~h u†L2 …Q† :

…49†

As we know, 1 k~ u ÿ P~h u~kL2 …Q† ˆ sup

1 j…u ÿ P~h u; g†L2 …Q† j

kgkL2 …Q†

g2L2 …Q†

:

By Eq. (49) 1 1 j…u ÿ P~h u; g†L2 …Q† j ˆ jax …/g ÿ /h ; u ÿ P~h u†j 1

6 ju ÿ P~h ujH 1 …Q† j/g ÿ /h jH 1 …Q† 1

6 ju ÿ P~h ujH 1 …Q†

k

inf

/h 2S~h …Q†\H 1 …Q†

j/g ÿ /h jH 1 …Q†

1 6 ju ÿ P~h ujH 1 …Q† j/g ÿ Pkh /g jH 1 …Q† 1 6 chju ÿ P~h ujH 1 …Q† j/g jH 2 …Q†

6 chskvkH s~…Q† kgkL2 …Q† : Thus 1

ku ÿ P~h ukL2 …Q† 6 chskukH s~…Q† : Then conclusion (46) follows from the interpolation of spaces. In order to get better error estimation, we ®rst compare the numerical result k …u; p† to U  ˆ PNc ;h U and P  2 …S~h …Q† \ H 1 …Q†† PN …I†, de®ned as Z y @P 1 1 …x; s† ds: P~h PN1 ÿ1 P  ˆ P~h P …x; ÿ1† ‡ @s ÿ1 Let U~ ˆ u ÿ U  and P~ ˆ p ÿ P  . Then by Eqs. (2) and (4), we have 8 …U~ t ‡ dc …U~ ; U  ‡ U~ † ‡ dc …U  ; U~ † ‡ rP~; v†N ;h;x ‡ maN ;h;x …U~ ‡ rsU~ t ; v† > > > > > 5 > P 3 > > ˆ Aj …v†; 8v 2 …XNk‡k > ;h …X†† ; < jˆ1 8 > P k > > ~; w† ˆ …Uc …U~ † ‡ U …U  ; U~ †; w† … P ‡ Aj …w†; 8w 2 X~ N ;h …X†; a > N ;h;x c N ;h;x > > jˆ6 > > > : ~ c U …0† ˆ PN ;h U0 ÿ PN ;h U0 ;

236

J. Hou, B. Guo / Appl. Math. Comput. 101 (1999) 209±244

where A1 …v† ˆ …@t U ; v†x ÿ …Ut ; v†N ;h;x ; A2 …v† ˆ …d…U ; U †; v†x ÿ …dc …U  ; U  †; v†N ;h;x ; A3 …v† ˆ ÿmrsaN ;h;x …Ut ; v†;

A4 …v† ˆ …rP ; v†x ÿ …rP  ; v†N ;h;x ; A5 …v† ˆ …f ; v†N ;h;x ÿ …f ; v†x ; A6 …w† ˆ ax …P ; w† ÿ aN ;h;x …P  ; w†; A7 …w† ˆ ÿ…U…U †; w†x ‡ …Uc …U  †; w†N ;h;x ; A8 …w† ˆ …r
f ; w†x ÿ …r
f ; w†N ;h;x : We now turn to estimate jAj …v†j; 1 6 j 6 5. Let s be the same as before and ~s ˆ min…k ‡ k ‡ 1; s†. Take v ˆ 2U~ in Aj …v†; 1 6 j 6 5. We obtain from Lemmas 2 and 7 that for r; s P 1, jA1 …U~ †j ˆ j…@t U ; U~ †x ÿ …Ut ; U~ †N ;h;x j 6 j…@t U ÿ Ut ; U~ †x j ‡ j…Ut ÿ Ut ; U~ †x j ‡ j…Ut ; U~ †x ÿ …Ut ; U~ †N ;h;x j 6 kU~ kx ‡ c…N ÿ2r ‡ h2~s †kU kC1 …0;T ;M r‡1=4;~s …X†† ‡ cskU kH 2 …t;t‡s;L2x …X†† : 2

2

2

x

Next we have A2 …U~ † ˆ B1 ‡ B2 ‡ B3 ‡ B4 with B1 ˆ …d…U ; U †; U~ †x ÿ …dc …U ; U †; U~ †x ; B2 ˆ …dc …U ; U †; U~ † ÿ …dc …U ; U †; U~ † x

N ;h;x ;

B3 ˆ …dc …U ; U †; U~ †N ;h;x ÿ …dc …U  ; U †; U~ †N ;h;x ; B4 ˆ …dc …U  ; U †; U~ †N ;h;x ÿ …dc …U  ; U  †; U~ †N ;h;x : From Lemmas 6 and 10, we have that for r > 1=2 and d > 0, jB1 j ˆ

3 X jˆ1

6

2 X jˆ1

j…@j …# ÿ Pc †…Uj U †; U~ †x j

j……# ÿ Pc †…Uj U †; @j U~ †x j ‡ j……# ÿ Pc †…U3 U †; @3 …U~ x††L2 …X† j

6 cN ÿr

3 X kUj U kL2 …Q;Hxs …I†† jU~ j1;x jˆ1

6 cN kU kY r;0;d …X† jU~ j1;x : ÿr

2

x

J. Hou, B. Guo / Appl. Math. Comput. 101 (1999) 209±244

237

By (3), Lemmas 2, 6 and 10, jB2 j ˆ

2 X jˆ1

ˆ

2 X jˆ1

6c

j…@j Pc …Uj U †; U~ †x ÿ …@j Pc …Uj U †; U~ †N ;h;x j j…Pc …Uj U †; @j U~ †x ÿ …Pc …Uj U †; @j U~ †N ;h;x j

2 X jˆ1

k…# ÿ PN ÿ1 †Pc …Uj U †kx jU~ j1;x

6 cN ÿr

2 X kUj U kL2 …Q;Hxr …I†† jU~ j1;x jˆ1

6 cN kU k2Y r;0;d …X† jU~ j1;x : ÿr

x

By Lemmas 6 and 7, jB3 j 6

2 X jˆ1

j…Pc …Uj U † ÿ Pc …Uj U †; @j U~ †N ;h;x j

‡ j…Pc …U3 U † ÿ Pc …U3 U †; @3 …U~ x††L2 …X† j 6 ckU k1 jU~ j1;x

3 X jˆ1

kPc …Uj ÿ Uj †U kx

6 c…N ÿr ‡ hs~†kU k1 kU kM r‡1=4;~s …X† jU~ j1;x : x

Similarly jB4 j 6 c…N ÿr ‡ hs~†kU  k1 kU kM r‡1=4;~s …X† jU~ j1;x : x

Thus 2 jA2 …U~ †j 6 c…N ÿr ‡ hs~†…kU kYxr;0;d …X† ‡ …kU k1 ‡ kU  k1 †kU kM r‡1=4;~s …X† †jU~ j1;x : x

It is easy to prove that jA3 …U~ †j ˆ mrsjaN ;h;x …Ut ; U~ †j ! X 2   ~ ~ ˆ mrs @j Ut ; @j U ‡ …@3 Ut ; @3 …U x††L2 …X† jˆ1 N ;h;x 6 cmrs…kUt ÿ Ut k1;x ‡ kUt k1;x †jU~ j1;x 6 cmrskU k 1 jU~ j : 1 C …0;T ;Hx …X††

1;x

We have A4 ˆ D1 ‡ D2 with D1 ˆ …rP ; U~ †x ÿ …rP  ; U~ †x ;

D2 ˆ …rP  ; U~ †x ÿ …rP  ; U~ †N ;h;x :

238

J. Hou, B. Guo / Appl. Math. Comput. 101 (1999) 209±244

We have from trace theorem and the de®nition of P  that for j ˆ 1; 2, 1 1 j@j …P ÿ P  †; U~ j †x j 6 c…k…# ÿ P~h †P …x; ÿ1†kx ‡ k…# ÿ P~h PN1 ÿ1 †@3 P kx †jU~ j1;x

6 c…hskP …x; ÿ1†kH s…Q;L2x …I††

‡ …N ÿr ‡ hs†kP kH s…Q;Hx1 …I†\H 1 …Q;Hxr‡1 …I†† †jU~ j1;x

6 c…N ÿr ‡ hs†kP kH s…Q;Hx1 …I††\H 1 …Q;Hxr‡1 …I†† jU~ j1;x : On the other hand, 1

j@3 …P ÿ P  †; U~ 3 †x j 6 k…# ÿ P~h PN1 ÿ1 †@3 P kx kU~ kx 1 1 6 …k…# ÿ P~h †@3 P kx ‡ kP~h …# ÿ PN1 ÿ1 †@3 P kx †kU~ kx 6 c…hskP kH s…Q;H 1 …I†† ‡ N ÿr kP kH 1 …Q;H r‡1 …I†† †kU~ kx : x

x

The above statements bound the value of jD1 j. Similarly, by (3), X 2  ~  ~ jD2 j ˆ ……@j P ; U j †x ÿ …@j P ; U j †N ;h;x † jˆ1 6 ck…# ÿ PN ÿ1 †P  kx jU~ j1;x 6 c…k…# ÿ PN ÿ1 †P kx ‡ kP  ÿ P kx †jU~ j1;x 6 c…N ÿr ‡ hs†kP k s 1 jU~ j r‡1 1 H …Q;Hx …I††\H …Q;Hx …I††

1;x :

So jA4 …U~ †j 6 c…N ÿr ‡ hs†kP kH s…Q;Hx1 …I††\H 1 …Q;Hxr‡1 …I†† …kU~ kx ‡ jU~ j1;x †: Clearly jA5 …U~ †j ˆ j…f ; U~ †N ;h;x ÿ …f ; U~ †x j 6 c…k…# ÿ Pc †f kx ‡ k…# ÿ PN ÿ1 †f kx †kU~ kx 6 cN ÿr kf kL2 …Q;H r …I†† kU~ kx : x

It is a little dicult to bound jA6 …w†j. We ®rst have A6 …w† ˆ ax …P ; w† ÿ aN ;h;x …P  ; w† ˆÿ

2 X ……@j2 P ; w†x ‡ …@j P  ; @j w†N ;h;x † ÿ …@32 …P ÿ P  †; w†x : jˆ1

On the other hand, for j ˆ 1; 2, …@j2 P ; w†x ˆ …@j2 P …x; ÿ1†; w†x ‡

Z

y ÿ1

@3P ds; w @s@x2j

 ; x

J. Hou, B. Guo / Appl. Math. Comput. 101 (1999) 209±244

239

1 …@j P  ; @j w†N ;h;x ˆ …@j P~h P …x; ÿ1†; @j w†N ;h;x   Z y @P 1 ds; @j w ‡ @j P~h PN1 ÿ1 @s ÿ1 N ;h;x Z y  @2P 1 ˆ …@j P …x; ÿ1†; @j w†N ;h;x ‡ PN ÿ1 ds; @j w @s@xj ÿ1 N ;h;x ! Z y 3 @ P ˆ ÿ…@j2 P …x; ÿ1†; w†N ;h;x ÿ PN1 ÿ1 ds; w : @s@x2j ÿ1 N ;h;x

Thus A6 …w† can be rewritten as A6 …w† ˆ E1 ‡ E2 ‡ E3 , where E1 ˆ

2 X

0

Z

@

jˆ1

E2 ˆ

2 X jˆ1

!

y

@3P PN1 ÿ1 ds; w @s@x2j ÿ1

Z ÿ

N ;h;x

y ÿ1

@3P ds; w @s@x2j

! 1 A; x

……@j2 P …x; ÿ1†; w†N ;h;x ÿ …@j2 P …x; ÿ1†; w†x †;

E3 ˆ ÿ…@32 …P ÿ P  †; w†x : We now estimate jE1 j. Clearly E1 ˆ …1† E1;j

…2†

E2;j ˆ

…1† jˆ1 …E1;j

…2†

‡ E2;j † where

! @3P ÿ #† ds; w ; @s@x2j ÿ1 x ! ! Z y Z y 3 @ P @3P 1 1 PN ÿ1 ds; w ÿ PN ÿ1 ds; w : @s@x2j @s@x2j ÿ1 ÿ1 Z

ˆ

P2

y

…PN1 ÿ1

N ;h;x

x

Furthermore …1†

jE1;j j 6 ck…# ÿ PN1 ÿ1 †

@3P k kwkx @s@x2j x

6 cN ÿr kP kH 2 …Q;Hxr‡1 …I†† kwkx ; Z y @3P …2† jE2;j j 6 ck…# ÿ PN1 ÿ1 † PN1 ÿ1 dskx kwkx @s@x2j ÿ1

Z y

Z y 3



@3P @ P 1 1

6c …PN ÿ1 ÿ #† ds ‡ …# ÿ PN ÿ1 † ds

2 2 @s@xj ÿ1 ÿ1 @s@xj x x !

Z y 3

@ P …# ÿ PN1 ÿ1 † ds ‡

PN ÿ1

kwkx 2 @s@x ÿ1 j x

240

J. Hou, B. Guo / Appl. Math. Comput. 101 (1999) 209±244

0

Z

ÿr @ 6 cN kP kH 2 …Q;Hxr‡1 …I††

y ÿ1

1

@3P ds ‡ kP kH 2 …Q;Hxr‡1 …I†† Akwkx @s@x2j L2 …Q;H r …I†† x

ÿr

6 cN kP kH 2 …Q;Hxr‡1 …I†† kwkx : Thus jE1 j 6 ckP kH 2 …Q;Hxr‡1 …I†† kwkx : We now estimate jE2 j. By the trace theorem, jE2 j 6 c

2 X jˆ1

…k…# ÿ Pc †@j2 P …x; ÿ1†kx ‡ k…# ÿ PN ÿ1 †@j2 P …x; ÿ1†kx †kwkx

6 cN ÿr

2 X k@j2 P …x; ÿ1†kL2 …Q;Hxr …I†† kwkx jˆ1

ÿr

6 cN kP kH 2 …Q;Hxr‡1 …I†† kwkx : Finally by (46), 1

jE3 j 6 k@3 …# ÿ P~h PN1 ÿ1 †@3 P kx kwkx 1 1 6 …k…# ÿ P~h †@32 P kx ‡ kP~h …@3 …# ÿ PN1 ÿ1 †@3 P †kx †kwkx 1

6 c…k…# ÿ P~h †@32 P kx ‡ hk…# ÿ PN1 ÿ1 †@3 P kH 1 …Q;Hx1 …I†† ‡ k…# ÿ PN1 ÿ1 †@3 P kL2 …Q;Hx1 …I†† †kwkx 6 c…hsk@32 P kH s…Q;L2x …I†† ‡ hN 3=4ÿr k@3 P kH 1 …Q;H r‡1=4 …I†† x

‡ N ÿr k@3 P kL2 …Q;Hxr‡1 …I†† †kwkx 6 c…N ÿr ‡ hs†kP kH s…Q;H 2 …I††\H 1 …Q;H r‡5=4 …I††\L2 …Q;H r‡2 …I†† kwkx : x

x

x

Thus jA6 …w†j 6 c…N ÿr ‡ hs†kP kH s…Q;H 2 …I††\H 2 …Q;H r‡1 …I††\H 1 …Q;H r‡5=4 …I††\L2 …Q;H r‡2 …I†† kwkx : x

We have A7 …w† ˆ K1 ‡ K2 ‡ K3 with K1 ˆ …Uc …U  †; w†N ;h;x ÿ …Uc …U †; w†N ;h;x ; K2 ˆ …Uc …U †; w†N ;h;x ÿ …Uc …U †; w†x ; K3 ˆ …Uc …U †; w†x ÿ …U…U †; w†x :

x

x

x

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Furthermore

X 3

‰…P …@ U  @ U  † ÿ Pc …@l Uq @q Ul † jK1 j ˆ c

l;qˆ1 c l q q l

kwkx ÿ…Pc …@l Ul @q Uq † ‡ Pc …@l Ul @q Uq ††Š

N ;h;x

6c

3 X

…kPc …@l Uq @q …Ul ÿ Ul †† ‡ Pc …@l …Uq ÿ Uq †@q Ul †kN ;h;x

l;qˆ1

‡

!

3 X

kPc …@l Ul @q …Uq l;qˆ1

ÿ Uq †† ‡

Pc …@l …Ul

ÿ Ul †@q Uq †kN ;h;x kwkx

6 c…N ÿr ‡ hs~ÿ1 †…kU  k1;1 ‡ kU k1;1 †kU kMxr‡1;~s …X† kwkx : Also we have jK2 j 6 ck…# ÿ PN ÿ1 †Uc …U †kx kwkx 6 c…kUc …U † ÿ U…U †kx ‡ k…# ÿ PN ÿ1 †U…U †kx †kwkx : Moreover kUc …U † ÿ U…U †kx 6

3 X ÿ l;qˆ1

6 cN ÿr

k…# ÿ Pc †…@l Uq @q Ul †kx ‡ k…# ÿ Pc †…@l Ul @q Uq †kx




3   X k@l Uq @q Ul kL2 …Q;Hxr …I†† ‡ k@l Ul @q Uq kL2 …Q;Hxr …I††

l;qˆ1

2

6 cN ÿr kU kY r;0;d …X† ; 1;x

2

k…# ÿ PN ÿ1 †U…U †kx 6 cN ÿr kU kY r;0;d …X† : 1;x

Thus 2

jK2 j 6 cN ÿr kU kY r;0;d …X† kwkx : 1;x

We get the bound for jK3 j which is the same as the bound for jK2 j. So 2

jA7 …w†j 6 c…N ÿr ‡ hs~ÿ1 †……kU  k1;1 ‡ kU k1;1 †kU kMxr‡1;s …X† ‡ kU kY r;0;d …X† †kwkx : 1;x

Evidently jA8 …w†j 6 cN ÿr kf kL2 …Q;Hxr‡1 …I††\H 1 …Q;Hxr …I†† kwkx :

242

J. Hou, B. Guo / Appl. Math. Comput. 101 (1999) 209±244

Now, we take v ˆ msU~ t in Aj …v†; 1 6 j 6 5. Then we get that cm2 s ÿ2r 2 2 msjA1 …U~ t †j 6 eskU~ t kx ‡ …N ‡ h2s †kU kC1 …0;T ;M r‡1=4;~s …X†† x e 2 2 cm s kU k2H 2 …t;t‡s;L2x …X†† ; msjA2 …U~ t †j 6 cms…N ÿr ‡ hs†…kU k2Y r;0;d …X† ‡ x e ‡ …kU k1 ‡ kU  k1 †kU kM r‡1=4;~s …X† †jU~ t j1;w ; x

msjA3 …U~ t †j 6 cmmrs2 kU kC1 …0;T ;Hx1 …I†† jU~ t j1;x ;

msjA4 …U~ t †j 6 cms…N ÿr ‡ hs†kP kH s…Q;Hx1 …I††\H 1 …Q;Hxr‡1 …I†† …kU~ t kx ‡ jU~ t j1;x †; msjA5 …U~ t †j 6 cmsN ÿr kf kL2 …Q;Hxr …I†† kU~ t kx :

Moreover Lemmas 5, 7 and 11 lead to kU~ …0†kx 6 c…N ÿ2r ‡ h2~s †kU0 kM r‡1=4;~s …X† ; 2

2

x

sjU~ …0†j21;x 6 cs…2N 4 ‡ cd hÿ2 †kU~ 0 k2x 6 cs…2N 4 ‡ cd hÿ2 †…N ÿ2r ‡ h2~s †kU0 k2M r‡1=4;~s …X† : x



for a > 1=2 and b > 1. Furthermore if condition Besides, kU k1;1 6 ckU kAa;b x …X† (41) holds, then for r > 5=4 and s > 8=3,  2 h 2 ÿ2r 2 s 2…~ sÿ1† : s ‡N ‡h ‡h ˆo N Finally, by an argument similar to the proof of Theorem 1, we obtain the following conclusion. Theorem 2. Let U and u be the solutions of Eqs. (2) and (4), respectively. Assume that (i) k P 1; condition (5) and condition (i) of Theorem 1 hold; (ii) for r > 5=4; s > 8=3 and a > 1=2 and b > 1; r;0;d U 2 C…0; T ; Mxr‡1;~s …X† \ W 1;1 …X† \ Aa;b x …X† \ Y1;x …X††

\ C 1 …0; T ; Mxr‡1=4;~s …X†† \ H 2 …0; T ; L2x …X††; (iii) for r > 5=4 and s > 5=3, P 2 C…0; T ; H s…Q; Hx2 …I†† \ H 2 …Q; Hxr‡1 …I†† \ H 1 …Q; Hxr‡2 …I†††; f 2 C…0; T ; L2 …Q; Hxr‡1 …I†† \ H 1 …Q; Hxr …I†††: Then there exists a positive constant d3 depending only on m and the norms of U and P in the spaces mentioned above such that for all t 6 T , kU …t† ÿ u…t†kx 6 d3 …s ‡ N ÿr ‡ hs ‡ hs~ÿ1 †:

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