Pointwise error estimates for a streamline diffusion

IMA Journal of Numerical Analysis (1997) 17, 29-59

Pointwise error estimates for a streamline diffusion scheme on a Shishkin mesh for a convection-diffusion problem WEN G U O AND MARTIN STYNESI

Department of Mathematics, University College, Cork, Ireland [Received 5 July 1994 and in revised form 1 December 1995]

1. Introduction

The streamline diffusion method is a finite element method introduced by Hughes & Brooks (1979) in the context of stationary convection-diffusion problems. Mathematical analyses of the method have been performed by Johnson & Navert (1981), Johnson et al (1987) and Niijima (1990) for stationary problems. Navert (1982) extended the method to time-dependent convection-diffusion problems and obtained local L2 error estimates of order k + 1/2, for piecewise polynomial finite elements of degree k, in smooth regions (i.e., regions away from any layers). Many other authors have applied the streamline diffusion method to various problems (see the references in Johnson (1987), Roos et al (1996)). Nevertheless, the extensive literature dealing with this method contains no proof of pointwise convergence, uniformly in the diffusion parameter, inside a boundary layer. In the present paper we shall improve the results just mentioned for the problem:

7, u(x,0) = uo(x)

(1.1)

for 0 «;c « 1,

where Q = (0,1) X (0, T], E is a small positive parameter, a > 0 is a constant, and u0 e L2[0, 1], / E L2(fi). Here for simplicity we have taken the coefficients of the differential equation to be constant. The solution u of (1.1) has in general a boundary layer near the side x = 1 of Q. While many practical problems involve two or more space dimensions, we have restricted our attention to one t e-mail address: [email protected] © Oxford University Press 1997

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We analyze a streamline diffusion scheme on a special piecewise unform mesh for a model time-dependent convection-diffusion problem. The method with piecewise linear elements is shown to be convergent, independently of the diffusion parameter, with a pointwise accuracy of almost order 5/4 outside the boundary layer and almost order 3/4 inside the boundary layer. Numerical results are also given.

30

WEN GUO AND MARTIN STYNES


dimension because even in this setting the analysis is lengthy. Since a time variable is also present, our arguments have a two-dimensional flavour and we expect that they can be extended to the multi-dimensional case. We shall give pointwise error analyses for the streamline diffusion method both outside and inside the bounday layer at x = 1. We obtain convergence, uniformly in e, at nodes inside the layer by introducing a special piecewise uniform mesh that resolves part of the boundary layer. Our analysis shows that when the streamline diffusion method is combined with this special mesh, it retains its usual accuracy in smooth regions. In the case of piecewise linear finite elements, the pointwise error bound is almost order 5/4 away from layers and almost order 3/4 near and inside the boundary layer. (The lower order of accuracy is due to the overlap of the layer and the coarse part of the mesh; compare the interpolation error bounds of (3.17) and (3.18) below.) Our analysis uses the techniques of Niijima (1990) and of Johnson et al (1987), who considered an elliptic problem on a quasiuniform mesh. In contrast we deal here with a parabolic problem on a highly nonuniform mesh. The arguments of Johnson et al (1987), Niijima (1990) cannot be immediately applied on this mesh; in particular, we need careful nonstandard interpolation error estimates. Indeed, our approach leads to a slight sharpening of Niijima's results—see Remark 4.1 below. The idea of using a piecewise uniform mesh to guarantee accurate numerical results inside the boundary layer is due to Shishkin (1990). His analysis is set in a finite difference framework and in particular seems applicable only to difference schemes that satisfy a discrete maximum principle. It is therefore inappropriate for the streamline diffusion method; an alternative approach, such as that presented here, is needed. The Shishkin mesh is remarkable in two ways: first, it resolves part but not all of the boundary layer, yet still yields convergence that is uniform in e; second, despite the fact that there is an abrupt change in mesh size, this does not destabilize the difference scheme. The fine part of the mesh is suitably aligned to give some resolution of the boundary layer, while outside that layer the mesh orientation is unimportant. (The analysis of the streamline diffusion finite element method on coarse meshes that are aligned along the streamlines is considered in Zhou (1995).) The piecewise uniform Shishkin mesh is specified a priori; thus, instead of pursuing a more complex adaptive mesh approach, we demonstrate in our analysis and numerical results that pointwise accurate numerical results can be obtained inside outflow boundary layers using simple meshes. An outline of the paper is as follows: in Section 2 we introduce a special piecewise uniform mesh and construct a streamline diffusion scheme on this mesh. Section 3 discusses the properties of our finite element space and analyzes the interpolation errors. In Section 4 we define a discrete Green's function G associated with the scheme and estimate it. Our main uniform convergence results are given in Section 5. Finally, Section 6 presents some numerical results. Throughout the paper, C will denote a generic positive constant, not necessarily the same at each occurrence, that is independent of e and of any mesh used.

STREAMLINE DIFFUSION SCHEME ON A SHISHKIN MESH

31

2. Mesh and scheme Let N and M be two positive integers, satisfying max {N/M, M/N}« C.

(2.1)

We assume that N is even and N>4.

(2.2)

Let A e (0,1/2) denote a mesh transition parameter, which may depend on N and e, and will be specified in Section 3. We write Q = fli U £22 with £2X = (0, 1 - A) X (0, T] and Q2 = G\QU Introduce a set of mesh points {(*,, tj) E Q: i = 0,..., JV and ; = 0,..., M} with =

*''

( 2 ( 1 - \)N~H, 11 - A + 2AAT'(i -Nil),

for i = 0,..., N/2, foii = N/2 + l,...,N,

(

' '

tj = TAT1/,

for; = 0,..., M.

(2.4)

By drawing lines through these mesh points parallel to the x- and f-axes, Qx and Q2 are each partitioned into MN/2 rectangles. Divide each rectangle into two triangles by drawing the diagonal of the rectangle which runs from northwest to southeast (here, as is customary, we have taken the x-axis running west to east and the f-axis south to north). This yields a triangulation of Qi denoted by £2?, for / = 1, 2. Each of these i2f is a uniform triangulation by means of right-angled triangles r, with base

r2(l-A)Ar'

ferret,

1

fortE^,

"'-hAAT

( 1 5 )

and altitude K = TM~X

for all

T E JQ",

(2.6)

N

where Q = Q? U Q%. Since we are interested in the singularly perturbed case, we assume throughout that £=sAT\ (2.7) Our next aim is to formulate a time-stepping procedure for (1.1) so that the discrete solution can be computed successively on a sequence of time levels. On each time slab Sj = [0, 1] X (*,._,, tj]

for j = 1,...,M,

(2.8)

we define a finite element space Vf by V,- {v s C(Sj): v(0, t) = i/(U) = 0 V* e (t,.}, t,], v\z is linear Vr E Q" such that r ° c Sj},

(2.9)

where C(Sj) denotes the space of continuous functions on St and r° is the interior of T.


and

32


We also introduce the streamline derivative wp for all differentiable functions w by defining we = awx + w,. We shall apply the streamline diffusion method (Navert (1982), Johnson (1987)) to the problem (1.1) on each slab 5; successively, imposing the initial value at f = /,-_, weakly and the boundary condition strongly. To this end, we introduce the finite element space on £2: V = {ve L\Q): v \s, e Vj for / = 1,..., M],

(2.10)

and define, for each v e Vjt for; = l,...,M and 0=sx =£1, v+(x, ?,_,) = lim v(x, r,_, + s),

(2.11)

j + 0

v~(x, tj)= lim v(x, tj+s).

(2.12)

We also use the notation of (2.11) and (2.12) for those functions in C(5y) for which the indicated limits exist. Notation. For all measurable D £ Q, set

Vv,w s L2(O),

(v, w)D = [ [ vw dr dr

(2.13)

D

(2.14) For ;'= 1,...,M, set Aj(D)

= {(x,t)sD:t

= tj},

(2.15)

and define

)J.D = f

Vv,we L2(Aj),

w(JC, 0)w(x, 0 dx

VveL2(Aj).

H.D=(v,v)lD

(2.16) (2.17)

When D = Q, we omit D from the notation. We now formulate our streamline diffusion method as follows. For / = 1,..., M, find U eV such that e(Ux, vx)s, + (U0+U,v + pvjs, + (U+> w+>;-. = (/, V + f.)VP)S, +

(U ', v );-l

(2.18)

where we set (U-, i>+>0 = («o- v+ >o

VveV1

(2.19)

and

p = p(x) = \ 10

X)N~l

for x e (0, 1-A), otherwise.

(2.20)


s—»-0


33

2.1 Scheme (2.18)-(2.20) is essentially the same as that given in Navert (1982), Johnson (1987). The only difference is that we take p = 0 on Q2. This is because later A is chosen quite small, which implies that the mesh in Q2 is very fine in the x-direction. Consequently our scheme is not upwinded on Q2. REMARK

REMARK 2.2 For each j , (2.18)-(2.20) is equivalent to a linear system of equations. Since the space V is denned independently on each slab with no continuity requirements from one slab to the next, the solution U will in general have jumps across each time level tj.

Define H\(Q) = {w e Hl(Q): w+(-,tj)

and w~(-, tt) exist

and lie in L2(Aj) for; = 0,...,M},

By summation of (2.18) over all St, we get the following discrete analogue of (1.1). Find UeV such that B(U,v) = (f,v + pvp) + (u0,v+)0

VveV,

(2.21)

where for all w, v e H\(Q) we set B(w, v) = e(wx, vx) + (w0 + w,v + pv0) + 2

+ IIPÎII>+ \\V\\2D + 2 \[v]\fD. 7=0

When D = £2 we omit D from the above norms. The following theorem states a stability inequality for (2.21) that also guarantees the existence and uniqueness of the discrete solution U. THEOREM

2.1

If U is the solution of (2.21), then


where we set w'(x, 0) = 0 and w+(x, T) = 0 for 0 =£x « 1. In order to write (2.18)-(2.20) in a compact form suitable for analysis we introduce the jump [v] of v e H\(£2) across each time level by defining, for 0=Sx=sl, [v](x, tj) = (v+- v~)(x, tj), for j = 0,..., M.

34


Proof. From (2.22), we have B(U, U) = e \\UX\\2 + (U0> U)+ \\p%f+ \\U\\2 + (U, pUfi) + t

([U], U+)j.

(2.24)

7=0

An integration yields (Up, U) + Y

(Wl U+)j = | E iy_,}

y=o

/= l

7=0

1=1

j=0

(2-25)

The Cauchy-Schwarz inequality and the arithmetic-geometric mean inequality give \(U,pU^\^\{\\U\\2+\\pUp\\2}, (2.26) since p « 2AT1 =s 1. Thus from (2.24)-(2.26),

^(f/.^^lllli/lll2.

(2.27)

On the other hand, (2-28) Taking v = U in (2.21), the desired result follows from (2.27) and (2.28).

•

3. Properties of V and the interpolation error

In this section, we shall discuss inverse and interpolation properties of our finite element space V that will be used in the sequel. We also specify the transition parameter A of the mesh. First of all, we consider some properties of V. LEMMA

3.1 For any v e V, we have

(i) for

Te

C2f, 1 =£ q *£p =s oo and / = 0, 1, ||w|k

(3-5)

by the Cauchy-Schwarz inequality. Noting that we deduce (3.3). It remains to show (3.4). From the definition of vp, it is sufficient to prove that l,...,M.

(3.6)

In fact, for each T in 5, n £22, by (3.2) and (3.3),

Thus

since 5, D i2^ contains N triangles. This implies (3.6). 3.2 Let p s (1, «>]. Assume that w E W^(fl). Let x e ft". Let the linear function which interpolates to w at the nodes of T. Then LEMMA

• JV;

, 1

II (w - w ), II i^-w * C { / / t II M ^ l l ^ ^ + ^llw,, 11^)}, C{Hr | k « | | ^ ( T ) + AT l l ^ l l ^ , ) } ,

denote (3.7) (3.8) (3.9) (3.10)


snn2

36


Proof. Let f denote the reference triangle with vertices at (0,0), (1,0) and (0,1). Let F be a one-to-one linear function that maps f onto r. Set ()

()

Vzef.

Then it is well known that

where (w)7 is the linear function that interpolates to iv at the vertices of f, and i , f are the variables used in x. We observe that

and

The desired estimates (3.8) and (3.9) now follow by transforming the above to integrals over x. Finally, using the Bramble-Hilbert lemma (Ciarlet (1980)),

Again transforming to x, one obtains (3.10). This completes the proof.

•

An immediate inference from Lemma 3.2 is the following interpolation error result on those triangles x where the solution is smooth. THEOREM

3.1 For any x e £2", if

||M||C2(T) ^

C, then (3.12)

and II(« -K / )*III.-XT)

+

IK" - " ' k l k - M ^ C A T 1 .

(3.13)

In order to obtain a satisfactory pointwise error bound in our later estimates, we shall require the local L" interpolation error for a solution with typical boundary layer behaviour to be at least first order in Q^ and second order in £22A calculation, based on (3.7) and (3.23) below, then motivates the choice (3.14)


then we transform (3.11) to integrals over x to get (3.7). Next, using Lemma 2.3 from Kfi&k (1991), we obtain


37

with the constant a chosen to satisfy 0 < a «£ a. We shall assume from now on that (3.14) holds. Note that this choice of A implies (see (2.5)) that the mesh in the A:-direction is very fine in Q2 and coarse in fl,. Note also that the boundary layer at x = 1 is typically of width O(eln(l/e)) and in practice one usually has £~*>N; consequently (see (3.14)) the mesh resolves only part of the layer. The following theorem gives interpolation error bounds on each triangle x where the solution exhibits boundary layer behaviour. THEOREM

3.2 For any x e Q", assume that du

• C{l + £-' exp (-a(l - x)/e)}

V(x, t) e x

(3.15)

and that for / + / « 2 w e have dx'dt>y"

•Ce'

V(x, 0 £ T.

(3.16)

Then (i) if x e Q], we have (3.17) (ii) if T e Q2> w ^ have (3.18) and (3.19) Proof. Without loss of generality, we assume that the vertices of x are (JC,-_I, tm), (xh tm-x) and (*,_,, rm_i). Thus on t, we have

u'(x, t) = u(x,_,,

x, t) + u(Xi, tm^)2(x, t)

(3.20)

where (3-21)

, 0 = (x, - x)/Ht - (r - tm-,)IK. Clearly

2 Ux, t) =

and

0 «£ 4>i(x, i)=£ 1 on x.

(3.22)


di+iu .

38


By (3.20) and (3.22), for {x, f ) e r w e get \(u - u')(x, t)\

= \u(x, t)(4n + 2 + 4>3)(x,') ~ «'(*. 01 =£ \u(x, 0 - «(*,_,, tm)\ + \u(x, t) - u(x,, *„_,)! + \u(x, t) =S \U(X, t) - U(X,-U 01 + |K(*, : -1, 0 - "(*,-l> 'm)l

II(OC,_J, rm_,)

+ \u(x, t) - u(x,, 01 + \u(xh 0 - «(**, *«-i)l + \U(X, 0 - «(JC/-1, 01 + I«(JC.--1, 0 - "(*/-l> 'm-l)l

c{ P [1 + e"1 exp (-o(l - £)/£)] df + f" ds),

by (3.15) and (3.16), (3.23)

since T e i2f means that x , « 1 - A. This, together with the choice (3.14) of A, implies the result of part (i). To prove part (ii), we note that for i £ ^ , HT = 2\N~1 =4a~1eN~1 \nN. Using this and (3.16) in (3.7) and (3.8), we obtain (3.18) and (3.19). D 4. Discrete Green's function

Let (x*, t*) be a mesh node in Q. The discrete Green's function G e V associated with (JC*, t*) is defined by B(x,G) = X(x*,t*)

\/XeV,

(4.1)

where we recall that G(x, t) = 0 for t > T. From (2.27), G is well defined. In this section we shall, derive a global estimate for G in the energy norm ||| • ||| and prove that G is negligible outside a narrow region that extends upstream from (x*, t*). This region is defined by A, = {(*, 0 e £2 : 0 < x =s x* + KQ(Tp In N,\x-at-(x*-

at*)\ =s Koav In W}, (4.2)

where Ko is a positive constant that is independent of e, N and M. We choose Ko in the proof of Theorem 4.2 below; ap and av will be given in (4.13) and (4.14) respectively. We start by introducing a cut-off function with exponential decay. Set -at-(x*-at*)\)\ ( (x*-at*-(x-at) where (4.4)

By some elementary calculations, one can easily prove


C{AT> + exp ( - a ( l - x,)/e)} ! + exp {-akle)),


LEMMA

39

4.1 For (o(x, t) defined in (4.3), we have

(i) 00 onQ; (iii) for each r e G", if ap & Hz and an 3= K, then max w/min w =s C, r

max |a)^|/min \(op\ ss C;

t

I T

(iv) for all / and m, d'+mw(x, t)

: Cap'o--ma>(x, t)

on

(v) for all / s= 1 and all m, on Q;

(vi) on any triangle r* that contains (x*, t*), co(x, t) 5* C; (vii) a)(x, t) « CN~Ko on We shall first derive a global estimate on G in a weighted energy norm, defined by

n!

l(-)c ir

M

(4.5) ;=0

This estimate will be obtained by proving the following three lemmas. 4.2 If a^^yN~} and independent of N, M and e, LEMMA

ye, then for •y&l sufficiently large and (4.6)

=e

^ ) G, G x )

to

(4.7)


dl+mo)(x, t) dp'i

40


Integrating by parts gives

/=)

l \ (O

Ij

\ W '

Ij-i

Substituting this into (4.7), we get

-) G, Gx) y=0

Oil x

I

(4.8)

e\ —

2

+cr- 2 || w -*C|| 2 } )

(4.9)

using Lemma 4.1(iv). Similarly, using (2.20), (4.10) Finally,

(4.11) since (2.20) and (2.2) imply p < \. Collecting (4.9)-(4.11) into (4.8), we get

with 6 = Tm.x{eap\ eav2, N 'tr^'}. Using e 1 sufficiently large, independent of N, M and e. Then \'

G

where (G/a))' is the interpolant from V to G/a>. Proof. For convenience we set

Then the Cauchy-Schwarz inequality gives \B(E, C ) | « e ||o.l£x|| ||ft»-*Gx||

(4.15)

Note that there is no term ||OJ iG^||x32 in IIIWEftf+ 2

W ^ e - ' c r - 2 + AT2*"2 + N-3(7j3 + N " 1 ^ 1 + N'3^4) \\\G\\\l (4.39) from (4.13) and (4.14), where the first bound above is used when e « N~2 and the second when N~2 < e =s A?"1.


We may now bound the terms in (4.21). From (4.23), (4.35) and (4.36), using Lemma 4.1 (iii), we have

46


Finally,

7=1

N + 1)N" 3 {A 4 (^2 + L% + L 2 ,

+ A22(L (L22, + L\ L\ + + Ll) Ll) |||G|||U |||G|||U ++ L L2 W^ =£ Ce~lN-4\2 In2 N{L2, + L22 + L23 + L2, using 1 =s e- 1 ^- 1 , A « C and (4.19), ^ CAT4 In4N{a~2 + a~02 + ea? + eN2*? + a~4 + N2a'2} |||G|||2, recalling (3.14) and (4.34),

since crp = yN'1 and av > yN~* from (4.14). Substituting (4.38)-(4.40) into (4.21), we get

The desired result follows on choosing y sufficiently large, independently of N, M and e. D We are now in a position to present the main results of this section. 4.1 Assume that ap and av are chosen as in (4.13) and (4.14) respectively. Then THEOREM

where when (x*, t*) e I31; u . Ll, otherwise. Proof. The first inequality can be easily obtained by using Lemma 4.1(i). To prove the second inequality, we take % ~ (G/w)' in (4.1) and get

s=r

But by Lemmas 4.2 and 4.4,

Now Lemma 4.3 yields the desired result. Using Theorem 4.1 we can derive our second bound on G.

D


(4-40)

STREAMLINE DIFFUSION SCHEME ON A SH1SHKIN MESH

47

4.2 Assume that the hypotheses of Theorem 4.1 hold. Then for each nonnegative integer v, there exists a positive constant C = C(v) such that

THEOREM

u w ,

(4.41)

and + e |G|w. (fi2 ^)« CAT",

(4.42)

where we have used the usual notation for the Sobolev space Wi and its associated seminorm and norm. Proof. Define Q c Q by (4.2) with Ko replaced by K0/2. Assume without loss of generality that Q^ is a mesh domain, by enlarging it slightly where necessary. Given v, choose tfo = 4(v + 3). (4.43) on Q\£^. Hence ,

(4.44)

by Theorem 4.1. Then (4.41) follows using the inverse estimate (3.1). Next, let (x1, t') e f22\A) be arbitrary. Suppose first that x' - at' =£ x* - at * - Koav In N. Then we have

|G(jr\ r')l = I f

Gx(s, f) ds + f

Gx(s, f) ds

« CN{|| Gx || r , n n i + VA IIGx|| TjnaJ, since Gx is piecewise constant, where T} is the union of those mesh triangles that contain the line segment {(x, t') e Q\Qo : O^x =sx'}. Note that by our supposition Now suppose instead that x'-

at'» x*-at*+

Koav In N.

Then

where T2 £ i2\f2£ is the union of those mesh triangles that contain the line segment {(x, t') e QXQQ : x' « x « 1}. Hence for (J:', t') e Q2\ô we always have

l

^ + Veln^N \\Gx\\n^,

by (4.44). This proves the LX(Q2\QO) estimate in (4.42).

using (3.1),


Then by Lemma 4.1(vii), w =£ CN'Kori^CN'2iv+3)

48


It remains to show the e \G\wno2\Oo) estimates. We may assume that (x', t') does not lie on the boundary of any triangle. Suppose that (x', t') lies in the triangle T0. NOW GX and G, are constant on r0. Hence |G x (x\ f)\ + \G,(x', t')\ = [meas (T0)]-*(l|Gx||to + ||G ( || To )« Ce-'AT", using (3.2), the fact that r 0 £ £2\£%>, and (4.44).

•

4.1 In obtaining the global energy norm estimate of G in Theorem 4.1, we sharpened Niijima's approach (Niijima (1990)) for elliptic problems. Using the ideas above, one can improve the global L2-estimates and //'-estimates of the Green's function given in (Niijima (1990), Lemma 2.2) by removing all In factors there. REMARK

5. Pointwise error estimates In this section we shall estimate the nodal error between the exact solution u and our computed solution U. In order to derive a nodal error formula suitable for an analysis under weak assumptions on u, we need the following lemma. 5.1 For any w e V and any mesh subdomain D c Q there exist Pw e V and a positive constant C, independent of N, M and e, such that LEMMA

Pw = w

onD,

(5.1) (5.2) (5.3)

(1, \(Pw)~\)j =sC ||vv|!z.^(/3>,

for j = 1,...,M,

(5-4)

where |-|w'(n) denotes the usual seminorm in W\(Q). Proof. For each / e {1,...,M} let {y can be bounded by \vtj\ = K,l « II w ||L.(Tiv)

for some r° ; £ £) n 5,.

(5.7)

From (5.5)-(5.7) we get HWIIZ.-CÔD>,

(5.8)

since || 0,-JII^XD « CN'1 for all i and ;, and there are at most four terms in the sum. Summing (5.8) over all; we get (5.9) Now (5.2) follows easily. Note that for all / and i, \

(5.10)

Using (5.10), one can prove analogously to the above that \Pw\w\ia>*\w\w\ta> + C \\w\\L~(D)

(5.11)

and \)j ^ (1, \w-\)j,D + CN'1 \\w\\L-{D).

(5.12)

Now (5.4) follows immediately from (5.12). To show (5.3), we use (5.7) and (5.10) to get J

This, together with (5.11), proves (5.3).

)

.

(5.13) •

Let u' be the interpolant from V to u. Using Lemma 5.1, we define Pu' e V such that Pu' = u' on A, (5.14) and i#|| + AT1 \Pu'\w]ia) + * C j v " '

50


Take x = u ~ Pu'in

C4-1)- N o t e

that

(Pu')(x*, t*) = u(x*, t*). We get

(U-u)(x*,t*) = B(U-Pul, G).

(5.16)

Set r,(x, t) = (u- Pu')(x, t). Using (2.21) and (1.1), we have B(U - Pu1, G) = (/, G + PGP) + (u0, G + ) o - B(Pu', G) , G) - e(Uxx, pG0).

(5.17)

Combining (5.16) and (5.17) and using (2.23), we obtain the following nodal error formula: (U - u)(x*, t*) = e(Vx, Gx) + (Vp, pGp) + (r,, G) + (r,, (p - 1)G^) M

- e(uxx, pGp) - 2 {T)~> [G])J ;=i

By RD(r), G) we mean that the integrations in (5.18) are only over D, for any domain D. Then (U - u)(x*, t*) = R^(V, G) + /? n ^(r,, G), (5.19) where (17", [G]); is split by ; = (v~, [ G J U + (r,-, [G])j,o^.

(5.20)

LEMMA 5.2

M l - G)|=£C/V-2lnMll/ll + l| Proo/ From (5.18)

yl We have

using e^N'1

and (5.15).

(5-21)


(5.18)

= /?(T?,G).


51

Since TJ(1, t) = 0 for all t e (0, T],

Hence II Vx II L»(i%) + e" 1 Ik I L ' ^ ) « C In yV || TJ, ||

O({h)

« C In N{ || u, || i l ( « 2 ) + || (Pu')x || L,(f22)} « C In JV{Vi ln^ N ||M,||fi2 + N \\u | | ^ ( f l b ) } ,

(5.23)

using (5.15) again. Finally, from (5.15)

(5.24)

7=1

From (1.1) it is easy to show that for each / E {1,..., M},

e ll«,ll|+ Il«lli, + H 2 * 11/11!,+ ll«olli%i)

(5-25)

where Sj = Uy/=1 Sj. Collecting (5.22)-(5.24) into (5.21), using (5.25) and Theorem 4.1 with v = 3, we obtain

7=1

3

T {(ln* /V + /V

The desired result follows immediately.

•

We are now ready for our main theorems. THEOREM

5.1 (Pointwise error estimate away from layers) Assume that II"oil W ) + IML>(fl) + ll««*rllz.W) * C,

(5.26)

and ll«Mflb)«C Then 1(1/ - M)(X*. /*)|

(5.27)


«Sl«ly + C7V||u|L.(A)).

52


where av is given in (4.14) and _{\, ll,

when (x*, t*) e ft otherwise.

Proof. Recalling (5.18), we have I*A.(^

G)|« Ve || T,X || t-t^Ve || Gx |

Using (5.27), Theorem 3.1 and (5.28)

where Aj(Q0) is as in (2.15), we get

VX IIGp

Hence, using e^N"1

and (3.4),

ill C||| (5.29) by Theorem 4.1. Applying (5.29) and Lemma 5.2 to (5.19), we are done.

•

We next give a pointwise convergence result for the case when (**, t*) lies in the boundary layer, under the assumption that the solution u exhibits typical boundary layer behaviour in the neighbourhood £20 of (x*, t*). 5.2 (Pointwise error estimate inside the boundary layer) Assume that (5.26) holds and that

THEOREM

[ for i+;=s2. Then \(U - u)(x*, t*)\

on Qo

(5.30)


meas (£2Q) + meas (A;(I20)) « Cav In A/,


53

where G)> w e bound each R,£r), G) (i = 1, 2, 3) separately. First, from (5.30) we have WuWc^^C. Thus by arguments similar to the proof of Theorem 5.1, we have

54


From the above estimates and the Cauchy-Schwarz inequality we get \Rh(r,, G)\ « CAT* |||G|||{[meas(/2)]i + AT* max [meas(yly(/2))]l}. (5.34) Now meas(/2) =£ C(A£ - \)av In N =£ C(e In (e/V)-1 + TV"1)0"')ln

N

^Co-^N"1 In TV,

(5.35)

using inteNyîeN)-*. Also (5.36)

Thus from (5.34)-(5.36) and Theorem 4.1, we obtain |7?/2(TJ,

(5.37)

G)\ ^ CcrJ^/V"' ln N.

Finally, for Rh(v, G), recall that p = 0 on /3) so that Rh(T], G) = e(r)x, Gx)h + (TJ, G),3 - (TJ, G0)h - 2 (v". [G1W 7=1

By Theorem 3.2(ii) and the Cauchy-Schwarz inequality, |/?/3(TJ, G)\ ^ C{N -1

In N

l|G;r[|/3+ N~2ln2 N ||G||/3 + N~2ln2 N \\G0\\h}[meas(l3)f

+ CN~2 In2 w2l[G]U[meas(A(/3))]*. Since meas(/3) =s CAo-^ In TV s (>„£ In2 N and meas(/l;(/3)) =s Ccrv In TV, inequality (3.4) and Theorem 4.1 now yield ' In2N\\\G\\\ .

(5.38)


meas(A;(/2)) « A£ - A « CAT1.


55

Combining (5.33), (5.37) and (5.38) gives |/?A,(t?, C)\ « C*\jri lnf N,

(5.39)

which together with Lemma 5.2 and (5.19) proves the desired result.

D

Recall (4.14). From Theorems 5.1 and 5.2 we reach the following conclusion for our streamline diffusion scheme (2.21). 5.1 Assume that (5.26) holds. Then in smooth regions, the scheme (2.21) is pointwise accurate of order almost O(eW~') when AT'ss E =£N~\ order almost O(e"W~ J ) when AT 2 s£e«AH and order almost O(AT5) when 0