0, such that
o< ao :::; a(u) :::; al, u E R. We also assume that the functions a(u) and feu) are smooth and bounded together with their derivatives. The global nature of our assumptions ,with respect to u(x, t) does not constitute any serious restriction. In fact, the approximation solutions to be considered will be shown to be uniformly close to the exact solution u(x, t) and thus depend only on the nature of a(u) and f(u) in a neighborhood of the range of u(x, t). Let v == a(u)(Vu+ J~Vu(x,T)dT),b(u)== a(u)l. We note that
The weak form of (1.1) appropriate form mixed methods follows that, find a 111ap
(u, v) : [0, T] + H x Z, such that (a) (Ut,
+
Wh
X
Vh such that
(a) (Ut,
v« +
it
'ilu(x, T)dT
~ b(U)V = g(t),
321
S1ll1: MIXED FINITE ELEMENTS FOR PARABOLIC EQUATIONS
No.3
=
and obtain (Vu)t + Vu gt. Therefore the approximation of Vu can be obtained by working out this ordinary differential equation. Before defining the fulldiscrete approximation, we introduce some notations. Let N be a positive integer.
g(x, ti),
gi+1 /
2 = (gj+l
Let ~t == T/N,ti = j~t,j = O,l,···,N. We also define gi ==
+ gi)j2, Otgi+ 1 / 2 =
(gi+ 1
gj)j(b.t).

The fulldiscrete approximation is defined by the sequence ({Ui}f=o' {Vi}f=o) : [0,T] W h X Vh such that (a) (8 t Uj+l / 2, tp )  (V· Vj+l/2,tp) == (!(Uj+l/2),tp),tp E Wh,
(b) (b(Uj+l )Vj+l,,,p)
j
+ (Uj+l, V . 7/)) + (b.t E
r=O
U r+ 1 / 2, \1 . "p) = O,,,p E Vh,
+
(2.3)
(c) UO = W(O), V O = Y(O), where j = 0,1,"" N  1, W(O) and V(O) are defined by (3.1). U sing arguments similar to those used to derive the existence and uniqueness of (2.2)
and Brown's fixed point theorem, we can easily obtain that (2.3) has a unique solution. Let X be a Banach space with norm
1I,B11l,2(x) =
r 11,B(t)llidt, Jo
11·llx and f3 : [0, T] =>
T
and
II,BIIL=(X) = ess
X. Define
sup
O~t~T
11,B(t)llx.
We will make extensive use of the ztype inequality: ab ~ Ea2 + (b2 j4E), E > O. And in what follows, G will be used to denote various positive constants that are independent of h.
3 VolterraType Mixed Projection We define the Volterratype Mixed Projection of the solution u(x, t) to be a map (W, Y) : [0, T] + W h X Vil such that
(a) (\1. (Y  v),
(b) (b(u) Y, 7/,) + (VV, \1 . 7/,) + (J~ W (T)dT, V . "p) == O,,,p E Vh, ( 3.1 ) In order to prove the existence and uniqueness of (W, Y), we introduce two families of projections[lO][13]: ITh : Z + Vil and Ph : H + Wh such that
(a) (V· (q  llhq),cp) = O,q E Z,cp E W h , (b) (\1·"p,w  Phw) = O,W E H,,,p E Vh,
=
=
(c) (\1·)IT h Ph(V,) (or divIlj, Phdiv): Z + Who Then ITh and Ph have the following approximation properties:
(a) I/q  IThql/o :S Gllqllrhr,q E H T(0 )2, 1 :S r:S k + 1. (b) 11\1· (q  IT hq)lIs :S GII\1 . qllrhT+ s, q E H r + 1 (0 )2, 0 :S r, s :S k + 1. (c) Ilw  PhW"s~ Gllwllrhr+s,w E Hr(o),O ~ r,» ~ k + 1. Lemma 3.1 If (~Y) is defined by (3.1), then there exists a constant G
(3.3)
>
0 such
that
IIWllo :S GIIYllo. Proof We introduce an auxiliary problem, for ~a
(3.4)
f3 E L 2(0), let a be the solution of:
= j3 in O,a == 0 on
ao.
322
ACTA MATHEMATICA SCIENTIA
Then we have a E H 2(O)
(W,f3)
n HJ(O),
II al12:S Gllf3llo.
and
Vol.17
It follows that
= (W, V . (Va)) = (W, V· (II h Va)) = (b(u)Y, II h Va)  (J~ W(r)dr, V· (II h Va))
~ G(II Yllo + J~ IIW(r)llodr)llf3llo.
IIWllo::; C(llYllo +
Then
it
IIW(r)lIo dr).
Thus, by applying Gronwall's inequality, we have (3.4). . Theorem 3.1 Volterratype mixed projection (3.1) has exactly one soluiton (W, Y) E W h X Vh • Proof In order to show existence it is clearly sufficient to prove uniqueness. Thus let v = 0 and set ep = W,,,p = Y in (3.1), we obtain
(b(u)Y, Y)
+
(it
W(r)dr, V · Y) = O.
J;
Then by setting ep = W(r)dr in (3.1a), we have IIY115/al ~ (b(u)Y, Y) = 0, so that Y = O. By Lemma 3.1 we conclude at once that W = 0 which completes the proof. In the error analysis of (W, Y) to (u, v), we introduce the following notations: dh = v  Y, eh = II h v  Y, Zh = PhU  w: We first substract (3.1b) from (2.1b) and use (3.2b), it follows with (3.1a) that (a) (V· d h , ep) = O,ep E W h , (b) (b(u) dh , "p) + (Zh' V . "p) Lemma 3.2 For s ~ 0,
+ (J~ Z h ( r )dr, V · "p) = O,,,p
E Vh.
( 3.5)
(3.6) Proof We introduce an auxiliary problem, for
V· (a(u)Va) =
I'
Then we have a E H s+2(O) n HJ(O), and It follows from (3.3a) and (3.3c) that
I'
E H*(O), let a be the solution of
in O,a = 0 on
Ilalls+2
ao.
(3.7)
~ Gllf3l1s.
(Zh,{3) = (Zh' V· (IIha(u)Va)) = (b(u)dh,a(u)Va  IIha(u)Va) +(V . dh, a  Pha)  (J~ zh(r)dr, V . (IIha(u)V'(a))
~
+ IIV· dhllohlnin(s+2,k+l)}llf3/1s + J~ /Izh (r)lls drllf3lls. G{lldhllohmin(s+l,k+l)
Then by using Gronwall's inequality, we have (3.6). Lemma 3.3 For s 2: 0,
(3.8)
323
Sun: MIXED FINITE ELEMENTS FOR PARABOLIC EQUATIONS
No.3
Proof Let w E H 2(O). Then, by (3.5a),
and (3.8) follows from (3.3c). Lemma 3.4 For s
Ildhll s
::;
2: 0,
C{(lldhll o+
it
+(IIV· dhll o +
Ildh(r)llodr)hmin(s,k+l)
it
IIV· dh(r)lIodr)hmin(s+1,k+1)}.
(3.9)
Proof Let a E H s (O)2, it follows from (3.5a) and (3.2a) that
(b(u)d h, a) == (b(u)d h, IIha)
+ (b(u)d h, a 
~ (Zh' V · Iha) I(b( u)d,,, a)11
Then
(it
::; (lI zh lIs+1 +
IIha)
zh(r)dr, v . Iha) + (b(u)d,.. a  IIha).
it
Ilzh (r) lIs+1 dr )llali s
+Clld h Iioliali s hmin(s,k+l). Thus we have (3.9) from (3.6). For s 2: O. Assume that u E Hs+2 (O), and let then we have error estimates:
Theorem 3.2
J~
Ilu(r)llpdr,
Ilulip
==
Clllulllrhr+s,O:::; s:::; k 1,2:::; r:::; k + 1,
Ilu 
WIIs :::;
{
Cillulllr_lhr+k,s == k, 1:::; r:::; k + 1,
Clllulllr+2hr+k+l, s == k + 1,0 :::; r
:::; k + 1;
Ilv _ YIIs :::; { Clllulllr+lhr+s,O:::; s :::; k, 1 :::; r :::; k + 1, CII'1Llllr+2 hr+k+ 1 , S == k + 1,0:::; r:::; k + 1;
IIV· (v 
Y)IIs
< Clllulllr+2hr+s, 0 ~ s, r
::; k
(3.10a)
+ 1.
(3.10b) (3.10c)
Proof The methods of proof are similar to those of [12],[13] except the analysis of the integral term. We take 'ljJ == eh in (3.5h) and use (3.3a) to have
(b(u)eh,eh) == (V ·dh,Zh)  (b(u)(v  IIhv),eh)  (V .dh,J~ zh(r)dr) == (b(u)(v  IIhv),eh). Then Ilehllo ~ Cllv  IIhvllo. The following arguments are similar to those of [12],[13], we can easily have
324
ACTA MATHEMATICA SCIENTIA
Vol.17
It follows that
lIu  WI/s :::; II z hll  s + Ilu  Phulls :::; C{/Ivll ph min(p,k+l)+min(s+l,k+l)
+ IIV' . vllrhnlin(r,k+l)+min(s+2,k+l)
+ Iluli r hmin( r,k+ l)+min( s,k+1)},
where p
and
~
1.
Ilv  YIIs :::; C{llvllphmin(p,k+l)+min(s,k+l) +I/V' . vllrh min(r,k+l)+min(s+l,k+l)},
IIV'· (v  Y)IIs :::; CIIV' · v/lr hr+s. So we have
(3.10).
Theorem 3.3 For s ~ O. Assume that u, Ut E Hs+ 2(0), then we have error estimates:
lI(u  W)tlls :::;
C(I/Iulllr + Ilutllr)hr+s,O:::; s:::; k 1,2:::; r:::; k + 1, C(lllulll r+l + IIUtllr+l)hr+k, s == k, 1 :::; r :::; k + 1, { C(IIlulllr+2 + Il utllr+2)h k+ 1 , s == k + 1,0 ::s r ::s k + 1;
(3.11a)
T+
II (v
_ Y) t 1/_ s
C(III u IIIr+1 + II Ut II r +1 ) hr+s , 0 < s < k, 1 < r ::s k + 1, C(llI u lllr+2 + Il u t llr+2)h r+k+1 , s == k + 1,0 :S r:S k + 1; s Y)t IIs :::; C(lllulll r+2 + II u t llr+2)h r+ , 0 ::s S, r ::s k + 1.
:::; {
IIV' . (v 
(3.11b)
(3.11e)
Proof Let dd h == (v  Y)t, eei, == (Ilh v  Y)t, ZZh == (PhU W)t. We differentiate (3.1) to obtain (a)
(b)
(V' · dd h , VJ) == 0, VJ E W h , (b(u)ddh, 'l/J) + (bt(u)d h, 'l/J) + (ZZh' V' · 'l/J) + (Zh' V' . 'l/J) == 0, 'l/J EVil.
(3.12)
We still introduce the auxiliary problem (3.7). Then we have
(ZZh,{3) == (ZZh' V'. (Ilha(u)V'a))
== (b(u)ddh, a(u)V'a  Ilha(u)V'a) + (V' . dd h, Q

Pha)
(bt(u)dh, Ilha(u)V'a)  (Zh, V'. (IIha(u)V'a)) < C{lIddhllohmin(s+l,k+l) + IIV'. ddhllohmin(s+2,k+l) +lId hll s 1
+ Il z hll  s }II,6lls.
It follows from (3.6) and (3.9) that
/Izzhlls
::s C{(llddhll o + Ildhll o + J~ /Idh(r)llodr)hmill(s+l,k+l) +(IIV· ddhllo + IIV· dhllo + J~ IIV'· dh(r)llodr)hluin(s+2,k+l)}.
(3.13)
Let a E HS (0)2, then
+ (b(u)dd h, a  Ilha) == (bt(u)dh, Ilha)  (ZZh' V'. Ilha)  (Zh' V'. IIha) + (b(u)dd h , a :S C{lIdh/l s + II zhlls+l + IIzzhlls+l + Ilddhllohmin(s,k+l)}llalls.
(b(u)dd h, a) == (b(u)dd h, Ilha)

IIha)
Sun: MIXED FINITE ELEMENTS FOR PARABOLIC EQUATIONS
No.3
325
It follows from (3.6),(3.9) and (3.13) that
Ilddhlls
Ildhll o+ J~ IIdh(r)lIodr)hmin(s,k+l) +(IIV· ddhll o+ IIV· dhll o+ J~ IIV· dh(r)llodr)hmin(s+l,k+l)}.
~ C{(ddhll o +
(3.14)
If we take 'ljJ == eeh in (3.12b) and use (3.3a), then we have
(b(u)eeh,eeh) == (b(u)(v  IIhv)t,eeh)  (bt(u)dh,eeh)
(V· dd h, ZZh)  (V . dd h, Zh) == (b(u)(v  IIhv)t,eeh)  (bt(u)dh,eeh). Then, f1eehflo ~ C{II(v  IThv)tllo + Ildhllo}. So we can obtain II ddhllo ~ Ileehllo + II(v ~
IThv)tllo
C(llvllr + IIVtllr)hr, 1 ~ r
~ k
+
1.
Similarly as (3.8) was derived, we can easily.obtain from (3.12a) that
Next, note that
(V· eeh,
v· eeh == 0. r, ~ C(IIV· vll r + IIV· vtllr)h 1 ~ r
where we have used (3.2a) and (3.12a), so that
~ k + 1. Hence, \\V· ddhllo == IIV· (v  IIhv)tllo Using areuments similar to those used to derive Theorem 3.2, we have (3.11). Corollary 3.1 Assume that u, Ut E Hk+1(0) n Loo(O), k > 0, then there exists
constant C
== C( u) > 0, independent of h such that
IIWllo,oo + IIWtllo,oo < c. And if u,
Ut
E H
k+ 2(0)
(3.15)
n W1,ooCO), k > 0, then we have again
II Yllo,oo + IIYillo,oo
~ C.
(3.16)
Proof It is easy to obtain from the inverse inequality, triangle inequlity, (3.3), (3.10) and (3.11). We omit this proof here.
4 SemiDiscrete Mixed Finite Element Approximation In this seciton we will employ the new projection (W, Y) and derive the optimal L 2 error estimates for the semidiscrete mixed finite element approximation. Theorem 4.1 Let {U, V} be the solution of (2.2). If the solution u of (1.1) is such that u E L oo(H k +2 ) n LOO(W1,00), Ut E L 2(H k +2 ) n Loo(W1,00), where k is positive integer, then there exists a constant C > 0, independent of h such that
326
ACTA MATHEMATICA SCIENTIA
Vol.17
~ Ch + { llu IILOO (Hk+2) + Ilu t IlL2(Hk+l )} . k
Proof Let
~
1
=U
W,1]
=W
 u,(
=V 
Y. We substract (2.1a) from (2.2a) and
(2.1b) from (2.2b), it follows from (3.1) that (a) (et, ep)  (V· (, ep) == (1]t, ep) + (f(U)  feu), ep), ep E W h , (b) (b(U) ( , 'ljJ) + (e, V . 'ljJ) + (J~ r )dr, V . 'ljJ) == ((b(u)  b(U) )Y, 'ljJ ), 1/, E
e(
Vit,
(4.2 )
We differentiate (4.2b) and take 1/J = ( to obtain (b(U)(t, ()
+ (bt(U)(, () + (~t, V
= (( b(u)  b(U))yt , ()
+ (e, V
. ()
+ ((b(u)  b(U) )t Y, o.
e and ep == et in (4.2a), respectively.
We take ep = have
. ()
(4.3)
Adding both of them to (4.3), then we
~ ~ {(b(U)(, o + 11~116} + II~tIl6
=
~(bt(U)(,()  (17t,~) + (f(U)  f(u),~) + ((b(u) +((b( u)  b(U))t Y, ()  (7]t, et)
+ (f(U)
b(U))Yf,() 7
 I( u), et) ==
LG
j .
(4.4)
j=1
If we integrate (4.4) with respect to t, it follows from Theorems 3.23.3, Corollary 3.1 and ztype inequality that
it where ~(o)
~ ~ {(b(U)(,() + 1I~1I6}dt ~ Co{II~1I6: 1I(116},
= ((0) = O.
r
1
ft
i o c,« = 2 io (bs(U)(~t + wt)(, ()dt ::; C(lI(IILOO(LOO) + 1) it IICII6dt + e it II~tll6dt, where b, denotes one order derivative of b.
i t Gsdt = it {(bs(u)(Ut  ut}Y,() + ((bs(u)  bs(U))UtY,C)}dt
::; C(1I(IILOO(Loo) + 1){ h Zk+ Z i
t
(1IulI~+l + IIUtll~+l)dt
+ it (1I~1I6 + 1i(1I6)dt} + [; it lI~tIl6dt. Using arguments similar to those of G 1 and G 5 , we have
Sun: MIXED FINITE ELEMENTS FOR PARABOLIC EQUATIONS
No.3
327
We now make an inductive hypothesis: (4.5) where K
< 1 is a positive constant. They by taking
E
sufficiently sm all and applying
Gronwall's inequality, we have
To complete our argument, we must show that (4.5) is right for h sufficiently small. We use the inverse inequality and (4.6) to see that
It is well known that we can easily have (4.5) naturally for k > 0 and h sufficiently small through inductive reasoning. We see that Theorem 4.1 follows from (4.6),(3.10) and
(3.11).
5 FullDiscrete Mixed Finite Element Approximation In this section we will give the optimal £2 error estimates for the fulldiscrete mixed finite element approximation. Theorem 5.1 Let ({uj }f=o, {vj }~o) be the solution of (2.3). If the solution u of (1.1) is such that u, Ut E L OO(H k+2 ) n Loo(W1,OO), Utt, Uttt E L 2(L2 ), where k is positive
integer, and the following condition is satisfied: there exists a positive constant R such that (5.1) then we have sup {II(U  u)jllo O~j~N
+ II(V 
v)jllo + l::,.t
j
L 118 (U t
u)'l/21Io}
1=1
(5.2) Proof We first take values at times of tj+1 and tj in (2.1a) , add both of them and
devide it by 2. In the second place we take value at time of tj+1 in (2.1b). Then we obtain (a) (u{ +1/ 2,
(b) (b(uj+ 1 ) , v j + 1 , 'l/;) + (ui+ 1 , V· 'l/;) + (J~j+l u(r)dr, V· 'l/;) = 0, 'l/; E Z. We subs tract (5.3a) from (2.3a) and (5.3b) from (2.3b) to obtain (a) (at~j+1/2,
(5.3)
(5.4)
328
ACTA MATHEMATICA SCIENTIA
Let "p
Vol.17
= (j+l/2, it follows from (5.4b) that
+ (Otei+ 1 / 2 , \1. (i+1/ 2 ) + (e i+ 1/ 2 , \1. (i+ 1/ 2 ) = (b(Ui);tUi+1)(i, (i+1/ 2 ) + ((b(ui+ 1 ) _ b(Ui+ 1 ))Oty j+1/ 2 , (i+ 1 / 2 ) (b(U i+ 1)Ot(i+ 1 / 2 , (i+1 / 2 )
+(
(b(ui+l)b(ui»)(b(Ui+l)b(Ui» yi ij+1/2) ~t
+(It
.r: W(r)dr  LltWi+ J
,~
1/ 2 } ,
(5.5)
\1. (i+ 11.2 ) .
j+ We take tp = e 1 / 2 , tp = Otej+ 1 / 2 and tp = Q = It{J/~+l W(r)dr  LltWj+1/2} in J (5.40a), respectively. If we add these equations to (5.5), then we have (b(U j+ 1 )Ot(i+ 1/ 2 , (j+l/2)
+ (Otej+ 1/ 2 , ej +1/ 2) + (Otej+ 1/2 , Otei+ 1/ 2 )
+( b(Uj);~Ui+l)(j, (i+1/2) + ((b(ui+1)
_ b(Ui+l n8tyi+1/2, (i+1/2)
+((b(ui+l)b(U~))~~b(Ui+l)b(Ui))yj, (j+1/2)
_ (Otej+1/2 , Q) ( Otrf+ 1 / 2 , Otej+ 1 / 2 + ej+ 1 / 2+ Q) + (u1+ 1/ 2 _ Ot uj+ 1 / 2, Otej+ 1/2 +e j+ 1/ 2 + Q) + (f(U j+ 1 / 2)  fj+1/2( u), Otej+ 1 / 2 + ej+1/ 2 + Q)
(5.6)
7
= L: K j . i=l
Now we multiply (5.6) by 2Llt and sum on j from 0 to M  1, where M is positive integer such that 1 :::; M
< N. We often use stype inequality, Taylor's formula, Theorems
3.23.3 and Corollary 3.1 in the following error estimates, where the methods of [15] are often used.
2Llt
M1
L:
i=O
{(b(Ui+ 1)Ot(i+ 1 / 2, (i+ 1 / 2 )
2 Coll(MlI6+ IIe MI16 + 2Llt
+ (Otei+ 1/2, ej + 1 / 2 ) + (Otej+ 1/2, Otej+ 1 / 2 ) }
L: II Otej+1 / 2116
Ml
i=O
M1
L: {K 1 + K 2 + K 3 + K 4 + K 6 } i=O M 2k 2 jI16)} :::; C{h + + (Llt)4 + Llt L: (II(jI16 + IIe j=O 2Llt
Ml
2Llt
L:
j=O
{K s + K 7 }
:::;
C(
sup Ilejllo,oo 05:j 5: Al 1
+Llt
M1 + €Llt L: II 0tej +1/2116. j=O
+ 1){(h 2k+2 + (Llt)4
L: (lI(j116 + Ile j 115)} + ELlt L: IIOtej+1/2116. M
M1
i=O
i=O
SlID: MIXED FINITE ELEMENTS FOR PARABOLIC EQUATIONS
No.3
If we take
E
329
sufficiently small, apply discrete Gronwall's inequality and make an induc
tive hypothesis: sup O~j~Ml
II(i 110,00 < K < 1,
it follows that
(5.7)
Ml
sup O~i5:M k
:::; C{h +
{11(illo+ Ileilio + ~tL Il ote'+1/ 2 110}
1
1=0
(llu IIL OO (H k+ l ) + Il u t IIL oo(H k+l ) ) + (~t)2}.
(5.8)
Applying inductive reasoning[15], we can easily see that (5.7) is still right for M == N under the condition (5.1). So that (5.8) is right for M == N also. Thus, by combining (3.10),(3.11) with (5.8), we have proved Theorem 5.1. Acknowledgement The author would like to express his sincere thanks to Professor Yuan Yirang and Professor Liang Guoping for their elaborate instruction. References 1 London S, Staffans O. Volterra Equations. Lecture Notes in Math. Spinger Verlag, Berlin, New York, 1979. 2 Cannon J R, Lin Y. A prior £2 error estimates for finiteelement methods for nonlinear diffusion equations with memory. SIAM J. Numer. Anal., 1990,27:595602. 3 Cannon J R, Lin Y. Nonclassical HI projection and Galerkin methods for nonlinear parabolic integrodifferential equations. Calcolo, 1988 , 25: 187201. 4 Lin Y. Galerkin methods for nonlinear parabolic integradifferential equations with nonlinear boudary conditions. SIAM J.Nurner. Anal., 1990,27:608621. 5 Thomee V, Zhang NY. Error estimates for semidiscrete finite element methods for parabolic integrodifferential equations. Math. Comp., 1989,53: 121139 6 Sun Pengtao. The finite element methods for nonlinear hyperbolic integradifferential equations J.Eng.Math. (in Chinese), 1994,11(2):7682 7 Sun Pengtao. The interpolated postprocessing technique of the F .E.M. for nonlinear parabolic integrodifferential equations. J.Syst.Sci.& Math. (in Chinese),1996,16(2):159171 8 Sun Pengtao. Elliptic H2_ Volterra projection and the H1Galerkin methods for the iritegrodiffereritial equations of evolution. J.Appl.l\1ath. of Chinese Univ.,1995,10(1):1124 9 Sun Pengtao. The finite element method with moving grid for the nonlinear parabolic integrodifferential equations. J.Shandong Univ., 1995,30(1):4351 10 Raviart P A, Thomas J M. A mixed finite element methods for 2nd order elliptic prolems. Mathematical Aspects of Finite Element Methods, Lecture Notes In Math., Vol.606, SpringerVerlag. 1977.292315 11 Douglas Jr J, Roberts J E. Mixed finite element methods for second order elliptic problems. Mathernatica Applicada e Computacional, 1982,1: 91103 12 Douglas 11' J. Global estimates for mixed methods for second order elliptic equations. Math. Comp, 1985, 44: 3952 13 Brezzi F, Douglas Jr J, Marini L D. Two families of mixed finite elements for second order elliptic problems. Numer. Math., 1985,47: 217 235 14 Johnson C, Thomee V. Error estimates for some mixed finite element methods for parabolic type problems. RAIRO Model. Math.Anal.Numer., 1981,15:4178 15 Wang Hong. The estimates of stability and convergence of fully discrete finite element methods for nonlinear hyperbolic equations. J.Cmpu. Math., 1987,2: 164170(in Chinese) 16 Lin Y, Thomee V, Wahlbiu L. SIAM J. Numer. Anal., 1991,28: 10471070