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0
and
with the norm
For any
H� (n, ) { v I l v l r,w , a a
v H::, (n, ) and a
E
=
0 :::;
Jl
E
=
:::; r ,
of degree I is
n
= ( - 00 , oot
E
H�(n) I v tl xl H� (n) } Il v tJ + Xq ) t Ilr,w E
(1
where a is the largest integer such that Finally let
N
( 5.25 )
a
< r
( 5.26 ) + 1.
w(x) exp (- qt=l x;) . The Hermite polynomial HI(x) qII=l HI, (xq) .
and
=
=
n
Chapter
5.
Spectral Methods for Multidimensional and High Order Problems
The Hermite transformation of a function v
E
195
L! ({l) is as
v(x) = L v,H,(x) 1 1 1=0 00
with the Hermite coefficients
For any t(x) , 1#0
�
Obviously I ud ::; lid which implies that for any
(5.42)
1 ::; m ::; 3.
r � 0,
Hence by imbedding theory, (5.42) reads
+
I b(v, w , z) 1 ::; cl v ll l 1 w llL' ( lI z I l L' lI u Il L' ) cl v h ll w l l � lI z ll � ::; cl v h ll w lh ll z ll � ·
On the other hand, if w
E
Ht (n),
for v
then
Proof. It is easy to derive the above estimates.
E
Ht (n) .
For instance, the last one comes
immediately from (5.44). Theorem 5 . 1 .
(5.41)
Proof.
•
The representation (5. 31) is equivalent to (5.37) with (5. 34), or to
with (5. 34) ·
It is clear that a solution of (5.31) is also a solution of (5.37) with (5.34) . Now, suppose that is a solution of (5.37) with (5.34) . We have from Lemma 5 . 1 , Lemma 5.2 and (5.37) that for any E V(n),
(U, P)
v
E H- � (n) and so E H� (n) n Ht (n) . By Moreover by Green ' s formula, imbedding theory, E L12 (n) n W1,¥ (n) . By virtue of (5.37) and Lemma 5.4,
U
/).U
U
I fLa ( u, v) l :5 c ( ll U llw1'lf ll U l / o 2 + Il f ll ) Il v ll· By Green ' s formula again, /).U L2(n) and so U H� (O) n H2 (O) . The integration by parts in (5.37) leads to (5.31). A standard argument tells us that the normal derivative of U is also periodic and so U H; (O). Furthermore by (5.34), L: 1 ((U . V)U(j) - f(i), ¢l) 1 2 . L: 1 1 I� I (P, ¢l W :5 L: j=l Since (U . V)U - f L2(f!), the above series converges and so P H; (f!) . Using E
E
E
1
E
(5.34) again, we have
1
3
E
Chapter 5. Spectral Methods for Multidimensional and High Order Problems
203
and so P E, Finally it follows from (5.35) - (5.37) that On the other hand, by integrating by parts, we obtain that for 1 f 0,
l1-a (U, VI/JI ) = O.
H�(S1)nL�(S1).
a (U, V (I>t)
=
1 {)mU(j)(X){)m{)j �I(X) dx j,m=l 3
E
0
=-
3
E Im 1j
j,m=l
i l l l � E Ij Ulj l ,;" i l l l � (V · u, M . j=l (V . u, I/Jo) = 0 and so V . U = o . 3
1 {)mU(jl (X)�I(X) dx 0
(5.45)
In addition, The above statements lead to that P) is a solution of (5.3 1 ) . We now turn t o the equivalence of (5.37) with (5.34) and (5.41) with (5.34). If and so it U is a solution of (5.37) , then for any is also a solution of (5.41 ) . Conversely, if is a solution of (5.41) , then by taking VI/JI , 1 f 0 in (5.41) and arguing as in the previous paragraph, we can deduce that V · = o . Henc� V and for any E u, This means that is a solution of (5.37) . This completes the proof. _
(U,
v=
U
v E H�{S1 ), b(U, U, v) = b° (U, U, v) U v H�{S1), b(U, U, v) = b° (U, v).
UE
U
We are going to construct the Fourier-spectral scheme. For this purpose, we put (5.37 ) in an abstract framework. Let >. ..!:. and = Since
= E {S1 ) (HJ ( S1 ) ) ' . H! {S1 ) Hp! ( S1 ), E {S1 ) H- ! ( S1 ) and also E( S1 ) (H� ( S1 )) ' . We define an inner product on H� ( S1 ) by (5.46) (v, wh = a(v, w) + (v, w), V v , W E H;(S1), and a linear mapping A : (H� ( S1 ) ) ' -+ H� ( S1 ) by (5.47) (Av, wh = (v, w ) , V w E H;( S1 ). Moreover, by the theory of linear elliptic equations, A is a continuous mapping from W ( S1 ) with � - into Hr+2 ( S1) n H;( S1 ). Since E( S1 ) H-! ( S1 ), A is continuous from E( S1 ) into H�(S1)nH�(S1). But the latter space is comp&.ctly contained in H� ( S1 ), whence A is a compact mapping from E( S1 ) into H; ( S1 ) . Define G : rn.1 H; ( S1 ) -+ E( S1 ) by (5.48) (G( >' , v), w) = >'( b(v, v, w) - F(w)) - (v, w). From Lemmas 5.1 and 5.2, we know that for any >. E IR1 and v E H;( S1 ), I I G(>., v) II E :5 cl>'1 (II v ll � + Ilfll) . c
c
r
L
.
11-
C
C
x
Spectral Methods and Their Applications
204
Moreover G is a Coo mapping. Let D be the Frechet derivative of G. Then . for any positive integer m , DmG is bounded over any bounded subset of lR,l x According to (5 . 46) and (5.48) , (5.37) is equivalent to
(U, V ) l + ( G( .\, U) , v) = 0, \:I v E
Hi(f!).
H�(f!).
F(A, v) = v + AG(.\, v) . By (5.46)- (5.48) , we get that
an
(5.49)
equivalent form of (5 . 37) . It is to find U E
F('\, U) = O.
Hi(f!) such (5.50)
Let A* be a compact interval in IR\ and assume that (.\, U( .\ )) is a branch of isolated solutions of (5.50) . So there exists 8 > 0 such that for all .\ E A* and v E
11 (1 + ADG ( .\, U( .\ ))) vlll
H�(f!),
� 811vlll .
(5.51)
Now let VN be the same as in Section 5.1, and PN be the L 2 -orthogonal projection upon VN . It can be checked that for any v E
H�(n),
H;(f!) with
(5.4) asserts that PN is a continuous mapping in By (5.47) and (5.52) , AN E .c VN and
((H�(f!))' , )
(5.52) r
� o . Let AN = PNA. (5.53)
Maday and Quarteroni (19 82b) provided a Fourier-spectral approximation to (5.50) . It is to find UN E VN such that (5.54)
By (5.53) , (5.54) is equivalent to finding UN E VN such that By arguing as in the proof of Theorem 5.1 , it can be shown that \7 . UN = O.
Chapter 5. Spectral Methods for Multidimensional and High Order Problems
205
E
Let { (A, U(A» , A A" } be a branch of isolated solutions of (5.50). There exists a neighborhood @ of the origin in H;(n), and for N sufficiently large, a unique Coo mapping A A" -+ UN( A) VN such that for any A E A", FN( A, UN( A» = o and U(A) - UN(A) E @. Moreover, if the mapping A E A* U(A) E H; (n) is continuous for some r 2:: 1, then for any A, there is a positive constant b1 depending on I U(A) l I r such that Theorem .5 . 2 .
E
E
-+
Proof. We first use Lemma 4.26, with B = E(n), W = H;(n), nN = PN, A = A and g (A,v) = G(A,v ) . Clearly condition (i ) of Lemma 4.26 is satisfied. Further, by ( 5.4) ( 5.55 )
and a density argument,
Since
A is a c?mpact mapping from E(n) into H;(n), we derive that Nlim �oo II (PN I)A llc(E,Hl ) = O. -+oo II AN - A llc('E H') Nlim
Therefore both ( 4.132 ) and (4.133 ) are also fulfilled. So there exists a unique branch E A*}. The combination of ( 4. 134 ) , ( 5.4 ) and ( 5.50 ) yields that '
{ (A, UN(A» , A II U(A) - UN(A) l h
� � �
P
=
-
(5.56)
p
c ( I U(A) - PNU(A) l h + I ( A - AN)G(A, U(A» 1 1) c ( I U(A) - PNU(A) l h + 1 (1 - PN)AG(A, U(A » lh ) ( 5.57 ) c Il U(A) - PNU( A) 1I1 � cN1 -r Il U(A) lI r .
For the L 2 -error estimate, we put Lemmas 5.3 and 5.4 ensure the first part of cQndition ( ii ) in Lemma 4.27. By C condition ( iii ) of Lemma 4.27 is also satisfied. Since ( 5.4 ) ensures condition ( 4.135 ) . Finally by ( 5.3 ) , ( 5.4 ) and ( 5.57 ) ,
((H;)' ,H;) ,
(I - PN)A, I U(A) uN(A) II K < I U(A) - PNU(A) I K + I PNU(A) - uN(A) I K < I U(A) - PNU(A) I K + Nl ( I U(A) - PNU(A) l h + I U(A) - UN(A) l d < cN- lII U(A)1 I 2 . �
E
A A - AN =
Spectral Methods B{ld Their .�pplications
206
Hence condition (iv ) of Lemma 4.27 is also fulfilled. Now, using (4. 136) and ( 5.4 ) , ' we obtain that
II U(,x) - uN(,x) 1 I .
� =
cII FN (,x , U(,x)) ll = II(A - AN)G(,x , U(,x» II II (PN - I)AG(,x , U(,x» II = II( PN - I)U(,x) II � cN-r I l U(,x ) ll r
which completes the proof.
•
In actual computations, we can use the Newton procedure to evaluate UN (,x) . For instance, let uWl(,x) be the m-th iterated value of UN (,x). Then the iteration is as follows
(I + ANDG ( ,x , uW) (,x»)) (uW+Il(,x) - uWl(,x») = - (u W ) (,x ) + ANG ( ,x , uW ) (,x»)) , m � O. On the other hand, the approximate pressure PN (,x) = PN( X, ,x ) could be determined
by
with
PN (,x) =
PN,O(,x ) = 0, PN,I(,x ) = I I� ( 1 . « UN (,x) . V) UN (,x) - f) , 1/>1) ,
:
1 � O.
E
( 5.58 )
n
Assume that the hypotheses of Theorem 5.2 hold, P(,x) H;(n) and r � 1 . Then there is a positive constant b2 depending on IIU(,x) ll r such
Theorem 5 . 3 .
L5(n),
that
L PN,I(,x ) Ip, ( ,x )
I I ISN
Proof. Let
IIP(,x) - PN (,x) I + N- 1 IIP(,x) - PN (,x) lh � b2N -r . PN(,x) = PNP(,x) and p(,x) = PN (,x) - PN(,x). By (5.34 )
(P (,x) , rpo ) = 0
and for I � 0,
�
and ( 5.58 ) ,
, · I l � (l · [(UN ( .\) V) UN(.\) (U(,x ) · V )U(,x) ] M . I l l � ( I . [« UN (,x) - U(,x» V) UN (,x ) + (U(,x) V) ( UN(,x) - U(,x»] , rpl ) . By integrating by parts and using V · U(,x) = V . UN (,x ) = 0, we obtain that for 1 � 0,
(P(,x), rpl)
=
-
�
.
I (P (,x ), rpl ) I
=
.
Further, define
rn'!
Vk as (GN( A, V), W ) = A ( b� (v , v , w) - F(w » ) - ( V, W)N,
GN
:
x VN -+
(5.60)
and let
Let
AN
be the same as in (5.53) and set FN
:
]Rl VN x
-+
VN,
A Fourier-pseudospectral approximation of (5.41 ) is to find
UN E VN
such that (5.62)
By (5.53) , (5.60) and (5.61) , (5.62) is equivalent to finding
UN E VN
such that
By an argument as in the proof of Theorem 5 . 1 , it can be checked that We have the following result. E
'\7
.
UN = o.
Let { ( A, U( A», A A * } be a branch of isolated solutions of (5 ./.1), and rl > � , r2 > 2 . If f E H;' (O) and the mapping A E A* -+ U( A ) E H;2 ( O ) is continuous, then for N :?: No large enough, there exists a positive constant b3 depending on IIU( A ) II '2 and IIfll ." and a unique Coo mapping A E A* -+ UN ( A ) VN such that for any A E A *, FN ( A, UN( A» = 0 and
Theorem 5 .4.
.
E
N
In order to obtain an improved L 2 -error estimate, let M > be a suitably chosen integer. The corresponding set of Fourier interpolation points, the discrete inner product and the Fourier interpolation are denoted by OM, ( . , ·)M and 1M. If then M>
Nr':' ,
Define the mapping Giv( A,
v)
as
Chapter 5. Spectral Methods for Multidimensional and High Order Problems
where :FN
: YN -+ C(a) is given by :F;' (v) = ( f, V)M - L 1 /11 2 ( I · VI ) (1 . f, (Pt)M · III$N,I#O 2
209
Let F;' (A, V ) = v + ANGiv(A, v) . A new Fourier pseudospectr al approximation is to find uiv E VN such that (5.63) F;' (A, Uiv(A)) = O. We have the following result.
Let { (A, U (A)), A E A*} be a branch of isolated solutions of (5. ..{.1) and iv be the solution of (5.63). If f E Hr -1 ( n) with r > 2 and the mapping A E A* -+ U (A) E H: ( n) H;(n) is continuous, then there exists a positive constant b4 depending on II U ( A) lIr and IIf llr - 1 such that for any A E A*,
Theorem 5 . 5 . u
n
In actual computations, we can use the Newton procedure to solve (5.62) or (5.63) . For example,
(I + ANDG'N (A, U�m) ( A) ) ) (U�m+1 ) (A) - U�m) ( A) ) - (U�m) (A) + ANGiv (A, U�m) (A))) , m � O. =
On the other hand, the approximate pressure iN(A) is given by
with Piv,O ( A) = 0 and for
1 "# 0,
�
Piv,I ( A) = �I « UiV (A) · l ) (UiV (A ) . 1) + if · 1 , (Pt) M · If M > N
.:1 , then
(5.64)
where bs is a positive constant depending on I I U(A) l I r and I I f IlT- 1 . Theorem 5.4, Theorem 5.5 and (5.64) were proved in Maday and Quarteroni (1982b ) .
Spectral Methods and Their Applications
210
For the Fourier approximations of unsteady N avier-Stokes equation, we refer to the early work in Orszag 1971a, 1971b ) , Fox and Orszag 1973a, 1973b ) , Fronberg 1977 ) , and Moin and Kim 1980 ) . Hald 1981 ) , Kuo 1983 ) , Guo ( 1985 ) , Guo and Ma 1987 ) , Ma and Guo 1987 ) , Cao and Guo 1991 ) , and Rushid, Cao and Guo 1994 ) provided various schemes with numerical analysis for compuational fluid dynamics. The applications of Fourier approximations to other multi-dimensional equations can be found in numerous articles, such as Wengle and Seifeld 1978 ) , Feit, Flack and Steiger 1982 ) , Guo, Cao and Tahira 1993 ) , and Cao and Guo 1993b ) .
( (
(
(
5.3.
( (
(
(
(
(
(
(
(
(
Spectral Methods for Nonlinear High Order Equations
Many nonlinear partial differential equations of high order occur in science and en gineering, such as the Korteweg-de Vries equation possessing solitons, the Laudau Ginzburg equation in the dynamics of solid-state phase transitions in shape mem ory alloys, etc. . In this section, we consider the spectral methods for such kind of equations. We shall take the initial-boundary value problem of two-dimensional Navier-Stokes equation in stream function representation as an example to describe the Legendre spectral method for solving initial-boundary value problems of nonlin ear revolutionary equations of fourth order. The corresponding numerical solution converges to the weak solution of original problem. The spectral accuracy is obtained for smooth solution. The prediction-correction method is also used for raising the total accuracy of numerical solution. The numerical experiments coincide with the theoretical analysis. 1 1 2 and 80 be a non-slip wall. The initial-boundary value problem Let 0 of two-dimensional Navier-Stokes equation is of the form
{
=( , ) -
E
8tU + (U . V)U - p.!:iU + 'VP = /, x 0, 0 < t ::; T, 'V . U = 0, x 0, 0 ::; t ::; T, ( 5.65 ) U = O, x 80 , 0 < t ::; T, x n. U(x, O) = Uo(x) , Let V = { v I v V(O), 'V . v = OJ. The closures of V in L2(0) and H l ( O ) are E E
E
E
denoted by H and V respectively. Let
a(v, w) = 10 'Vv(x) · 'Vw(x) dx, b(v, w, z) = 1o (w(x) . 'V)v(x) · z(x) dx.
Chapter 5. Spectral Methods for Multidimensional and High Order Problems
211
The weak f�rm of (5.65) is as follows ·
(Ot U, v) + b(U, U, v) + p a(U, v) { U(O) Uo,
It is shown in Lions (1969) that if a unique solution in Tj V) n conservation =
L2 (O,
=
(f, v),
v
v E V, 0 < t :5 T, (5.66) x E n. and I E L 2 (O, Tj V'), then (5.66) has
Uo E H LOO(O, Tj H) . The solution of (5.66) possesses the
For numerical simulations, we can approximate (5.66) directly. But in this case, we should construct a trial function space whose elements satisfy the incompressibility. It is not convenient in actual computations. To avoid it, several alternative expressions are concerned. Firstly, we can take the divergences of both sides of (5.65) , and get
Ot eV' . U) + V' . ((U . V')U) - p l:l.(V' . U) + l:l.P = V' . f. By the incompressibility, we obtain
(U . V')U - pl:l.U {. Ol:l.Pt U ++ �(U) V' . I, =
+ 'V' =
P I, x E n, 0 < t :5 T, x n, 0 < t :5 T E
(5.68)
where
�(U) = 2 ( 02 U(1 )fhU(2) - fhU(1 )02 U(2 ») . Conversely if U and P satisfy (5.68) and . U(O) = 0, then for all t � 0, V' . U = 0. Thus (5.68) is equivalent to (5.65) provided that U and P are suitably smooth. The second formula of (5.68) is a Poisson equation for P at each time t. If we use it to 'V'
evaluate the pressure, then we need certain boundary conditions for the pressure. As pointed out in Gresho and Sani (1987) , we have that
O"P(x, t) = v . ( pl:l.U(x, t) + I(x, t)), x E On, 0 < t :5 T where O"P is the outward normal derivative of P . A simplified condition is O"P(x, t) = N(x, t), x E On, 0 < t :5 T. But the first condition is too complicated in actual calculations, while the simplified one is not physical.
Spectral Methods and Their Applications
212
The second way is to introduce the vorticity H(x, the stream function w(x, and J(v, w) = 82 w iAv - 81 w82 v, V V, w E Hl (n) .
t) ,
Then
{
(5. 6 5) i s equivalent to
atH + J(H, w) - p,/:).H = '\l -/:).w = H, H(x, O) = Ho (x) ,
X
x E n, x E n, x E n.
J,
0 < t ::5 T, 0 ::5 t ::5 T,
t)
(5. 69)
Since the incompressibility is already included, we do not need to construct the trial function space with free-divergence. Clearly w(x, = 0 on the boundary. But it is not easy to deal with the boundary values of the vorticity. In some literature, it is assumed approximately that H (x, = on an. But it is not physical. The third representation of Navier-Stokes equation is the stream function form. Let
t)
t) 0
{
Then the stream function form is as follows
8t /:).w + G(w, w) - p,/:).2 W = g , aw w = - = o, an w(x, O) = w o (x) ,
x E n, x E n, xEn
0 < t ::5 T, 0 < t T, ::5
(5. 70)
where 9 = - '\l x J. The main merits of this expression are remedying the problems mentioned in the above and keeping the physical boundary values naturally. But the appearance of the terms G(w, w) and /:). 2 W brings some difficulties in theoretical analysis. We shall derive the weak form of For any v, z E w 1 •4 (n) and w E H 2 (n), let
(5. 70).
Obviously
J(v, w, z) + J(z, w, v) = O.
(5. 7 1)
J(v, w, z) = -(G(v, w), z).
(5.72)
It can be verified that We can estimate IJ(v, w, z} 1 in several ways.
Chapter 5. Spectral Methods for Multidimensional and High Order Problems Lemma 5 . 7. For
' an y
213
v E HJ (fl),
Proof. By a density argument, it suffices to prove the conclusion for such a function, we observe that for x = (xt , X 2 ) E fl,
1:1 :
(8lv(X» 2 :5 2 18lv (Xl, x 2 ) 1 1 8�v (Xl, X 2 ) 1 dXl , (�v(X» 2 :5 2 1 ' 18lV (Xl! x 2 ) 1 1 �82 V (Xl! x 2 ) 1 dX 2 , Thus
v E CO" (fl) .
For
etc.
Similarly This completes the proof. Lemma 5 . 8 . For
any
•
v, w E HJ (fl), I J(v, v, w ) l :5 2 11'\7 v ll ll � v llll � w ll ·
Proof. By integrating by parts, we obtain that
J (v, v, w )
=
- 10 (8lV82 V8�w - (81 v? 8l82 w + (82 v) 2 �82 W - 8lV82 V8�w) dx + 10 (81 V82 W8�V - 81 v8l w8l 82 v + 82 v82 w81 82 v - 82 v8l w8�v) dx.
Integrating the last integral by parts again, we find that it vanishes. So by Lemma
5.7, I J (v, v, w ) 1 :5 118l v l b 1 182 v ll L4 " 8� w" + 1 1 8lV I l � . 11 8l82 W ll + 11 82 V Il�. 11 8l82 w ll + 118lV Ib 11 82 VI lL' " 8�w " :5 (2 11 8lV I l�. 1 I 82 V I l�. + 1 1 8lV lli. + 11 82 V lli. ) . 2 I 81 82 w II 2 + " 8�W ,, 2 + " 8�wl :5 2 11'\7 v ll ll� v llll� w ll ·
(
n
1
1
2"
2"
•
Spectral Metnod� and Their Applications
2 14
If v, z E HJ ( f! ) and w E H2 ( f! ) , then
Lemma 5 . 9 .
I J (v , w , z) l :5 2 11� w ll (IIVvll ll� v Il IlVzlI lI� zll) ! .
Proof. By Lemma 5.7, I J (v , w , z) 1
v'211� w Il IlV v IIL' IIV zII L' :5 2 11� w ll ( IIV v ll ll� v Il IIV z ll lI�zll) ! . :5
•
Lemma 5 . 1 0 .
For any v , w , z E HJ ( f! ) ,
Proof. By integrating by parts, we have that J (v , w , z) = 10 (OlW02Z0:V ..... �W�ZOl02V + 02W02Z01 02V - 02WOIZO�V) dx + 10 (OlVOIW��Z - 02VOIWO:Z + OlV02WO�Z - 02V02WOI02Z) dx. (5.73) Then the desired result follows from an argument as before. Lemma 5 . 1 1 .
Let v , w , z E HJ ( f! ) . If in addition w E Hr ( f! ) with r > 2, then
I J (v, w, z)1 :5 cll� vlll l w ll r IlV zlI · Proof· We first let 2 r 25 ' Set r = I-' + 1 , 2 _2 I-' Hl ( f! ) '-+ LP ( f! ) and Hr - 2 ( f! ) '-+ La ( f! ) Thus
<
2, then
I J(v, w , z) 1 � c ll v llrll.6. wI I IIV z ll , I J(v, w , z) 1 � c ll v llr IIV w ll ll.6. z ll .
Proof. Since W- 1 (!1)
'-+
I J(v, w , z) 1
L<Xl ( !1), we get that �
:S
IlolV IlLoo II.6. w ll llo2Z I1 + II 02V llLoo II.6. w ll llolZ II c ll v llrll.6. w I I IIVzll·
On the other hand, we have that
\In 02VOlwoiz dX \ � I I 02V IILoo IIVw ll l l.6. z ll ·
Moreover for
and for
2 < r < '25 '
r 2: % ,
The remaining terms i n (5.73) can b e estimated similarly. Finally, the second con clusion follows from imbedding theory as in the proof of the previous lemma. _ The weak solution of ( 5 . 7 0) is a function that
{
111
E L 2 ( 0, Tj Ha) n L<Xl ( 0, Tj H1 )
such
(OtV1l1 (t), V v) + J (1l1(t), 1l1(t), v ) +p( .6. 1l1(t), .6. v) + (g(t ) , V )C(H-2(n),H�(n» = 0, v v E Ha ( !1), ° < t � T, x E n. 1l1 (X, 0) = 1l1o(x),
(5.74) It is shown in Guo, He and Mao (1997) that if 1l1o E HJ ( !1) and 9 E L 2 ( 0, Tj H- 2 ), then (5.74) has a unique solution. The solution satisfies the conservation
Spectral MetJ; ogs .and Their Applications
216
We can adopt Legendre approxi � ations or Chebyshev approximations to solve (5.74) numerically. For the sake of simplicity, we focus on the Legendre spectral method. Let lP�f = lPN n H� (O) and
(v, wh = (Vv, Vw), (v, wh = ( b. v, b. w),
v
v, W E HJ (O) , V v, w E H� (O) .
It can be checked that in the space H� (O), the inner product (v , wh here is equal to the inner product a 2 (v, w) in Section 5 . 1 , and I v l 2 = lI b.v l l . The orthogonal projections P;:;'o : HO' (O) --+ lP�o , m = are two such mappings that for any v
E
1 , 2,
HO' (O),
(v - p;:;,o v , ¢>L = 0,
V ¢> E lP�o , m = 1 , 2. It is not difficult to sh�w that for any v E HJ (O) , I l v - Pk o v l 1 1
--+ ° as N 00 . Clearly the projection p 2 ,0 is the same as the projection p;/ in Section 5 . 1 . Now let tPN be the approximation to W. A Legendre spectral scheme for (5.74) is to find tPN(X, t ) E lP�o for all t E f4 (T) such that
{
--+
(Dr VtPN(t), V¢» + J (tPN(t) + orDr tPN ( t ) , tPN(t) , ¢» +/1- (b. (tPN(t) + O' rDr tPN(t)) , b.¢» + (g(t), ¢» = 0,
tP(O ) = tPN,O(X),
where 0' , (j are parameters, ° ::; 0', 0 ::;
(5.76)
1 and
I I W o - tPN, o l l l -+ 0, as N --+ 00 . (5.77) 0 1 For W o E HMO), we take tPN,O = PJII WO. While for W o E H� (O) , we take tPN, O = Pi/W o o Both of them satisfy (5.77). Thus the approximations on the initial level are well defined. Now, assume that tPN(t - r ) is known. Let r be suitably small and A(v, w) = (Vv, Vw) + /1-O'r ( b. v, b. w) + or J (v, tPN(t - r) , w) , V v, w E lP�o . By Lemma 5.9, A(v, w) is a continuous bilinear elliptic form on lP�o x lP�o . Hence by Lemma the numerical solution at the time t is determined uniquely. So (5. 76) has a unique solution as long as W o E HJ (O) and 9 E L2 (0, T; L2(0) ) . Especially, if 0' = 0 = then we deduce from ( 5 . 7 ) and ( 5 7 ) that
2.1 ,
�,
�
1
.6
IIVtPN(t ) 11 2 + /1-r L I Ib.V'N(s) + b.tPN(S + r ) 1 I 2 +r
seRr(t - r )
L (g(S), tPN(S) + tPN(S + r) ) ds = IIVtPN,0 11 2
seRr(t - r )
Chapter 5. Spectral Methods for Multidimensional and High Order Problems
217
which is a reasonable simulation of (5.75). In practical computations, a good choice of basis functions is essentially important. Shen proposed a nice set of basis functions. Let Lm (Y) be the Legendre polynomial of degree m. Set
(1994)
Then
lP�P = {(Pi(x) 1 0 :S It , 12 :S N - 4}.
B y the orthogonality of Legendre polynomials, both the matrix with the elements ((Pz, � ' 7 = (N-'\ ) , A � max (�, 7 - 2ro ) , (b) 0' = � ' 7 (N-4 ) , 2 I ) 0' < "21 ' 7 N � ' being a positive constant such that for any J.! c�(1 _ 2 0' ) C(O, Tj L2)
=
0
=
( C
4
CJ
Cp
v E IP�o , II� v ll � cpN2 11V' v ll .
J.!,
Then there exists a positive constant bI depending only on n and the norms of and g in the mentioned spaces, such that for all t E RT (T), If 0' > � and 6 is suitably large, then the above result is vaild for any 7 --+ O .
I}i
Chapter 5. Spectral Methods for Multidimensional and High Order Problems
2 19
Proof. LeOliN = P't;° ifl . We derive from (5.74) that
{
(D,. V'WN(t), V'ifJ) + J(WN(t) + OrD,. W N(t), WN(t), ifJ) + j.£(�(WN(t) + urD,.wN(t)), �ifJ) o = -(g (t), v) + B(w(t), WN(t), ifJ), V ifJ E IP� , 0 < t ::; T, W N( O ) = p;/ Wo, x E fl
(5.79 )
where
(V (D,.iflN - 8tifl ) , V ¢1) + J(iflN + 8r D,. iflN, iflN, ¢1) - J(iJ!, ifl, ¢1) + par (D,. � ifl, � ¢1) . Let W = .,pN - iJ!N. We obtain from (5.76) and ( 5 7 9 ) that (D,. VW (t) , V ¢1) + J (W (t) + 8r D,. W (t) , iflN(t) + W et), ¢1) + J (iflN (t) + 8r D,.iflN (t ), W et), ¢1) +o p (� (W (t) + ar D,. W(t») , � ¢1) = - B (ifl(t) , ifl N(t), ¢1) , 'V ¢1 E IP� . By taking ¢1 = 2 W (t) + e r D,. W (t ) and using an argument as before, we obtain from (5.71) that D,.I I VW (t)1 1 2 + r (e - 1) li D,. VW( t) 1 2 + 2pl l �W(t)1 I 2 +pr (a + n D,.II �W (t)1 I 2 (5.80 ) +pr2 (ea - a - �) I D,. � W(t)1 I 2 = � Gj (t) .
where
G1 (t) = -r(e - 28)J (W(t), W(t),D,.W(t») , G2 (t ) = -r(e - 28)J (W (t), iJ!N (t ), D,. W (t») , G3(t ) = -2J (iflN(t) + 8r D,. iflN(t), W et), W (t») , G4 (t ) = -fod (iflN (t) + 8rD,.iflN(t), W(t) ,D,. W(t») , Gs(t ) = 2 ( D,.iflN(t) - 8tifl (t), �W (t») - for (V ( iflN (t) - 8tifl (t» ,D,. VW (t») , G6 (t) = -2par (D,.� ifl(t ) , � W(t») - fopur 2 (D,.� ifl (t) , D,. � W(t») , G7 (t) = -2J (iflN (t) + 8rD,.iJ!N (t) , iflN(t), W et») + 2J (ifl(t), ifl(t ), W(t») , Gs (t) = -erJ (ifl N (t) + 8r D,. ifl N(t ) , iflN(t), D,. W (t») + er J (ifl (t), ifl(t), D,.W(t») .
220
Spectral Methods aDd Their Applications
I Gj (t) l .
c
D
Let > ° and be an undetermined positive We are going to bound Lemmas and 5.8-5.10, we have that constant. By (5.71)
I G1(t)1 S 2TI � - 281 \ VW (t)\J I �W(t) I I DT� W(t ) 1I S coT I DTVW(t)1I 2 + CITN4 I1 VW(t)1I 2 1 � W(t) 1I 2 , I G2 (t)1 S 2TI � - 2ol l � wN(t)1 1 ( I VW (t) I I �W (t) I I DT VW (t) I I DT�W(t)\1 ) � 2 (Vrl � - 20II l � WN (t)I I I \ � W(t)l \ t I DT VW (t) l I t) (T I VW(t)I I I D T�W (t) l I ) t S T I V W (t) I I DT�W(t) 1I + T(� - 20)21 1 � WN (t)1 1 2 1 � W(t) I I DT VW (t) 1I S cO IlT2 I1 DT�W(t)r + 4coll1 I V�(t)1I 2 + cO T I DT VW(t)\r + C2 T I �W(t)\r , IG3(t )1 S 2 \J (W (t) , W (t) , WN(t) + 8TDT WN(t)) \ S 2D I V W (t) I I � W (t)ll l I � (WN(t) + OTDT WN(t ) )11 S D I � W(t)11 2 + c3 1I VW (t ) 1I 2 , 1
2
_
I G4 (t)1 S 2�T I VW (t) l I ! I � W(t)lI ! (I I DTVW(t) lI ! I DT�W(t) lI t ' I � ( WN(t) + OTDT WN (t))1 1 + I DT�W(t)II I V (WN(t) + OTDTWN(t))l l t I � (WN(t) + OTDT WN(t)) I ! ) S 2�TI I WI I C(o,T;H2) (II DT VW (t) lI t I DT�W(t) lI t I V � (t) I ! I � � (t) lI ! + I DT�W (t) I I VW(t) lI ! I �W (t) l I t) S 2�TI I Wl l c(o,T;H2) I DT VW (t) 1I 2 I 1 DT�W (t) 1 I 2 I1 VW (t) 1 I 2 1 � W(t) 1I 2 + cOIlT2 I1 DT�W(t) 1I 2 + c:: I l wl l �(O.T;H2) I VW (t) I I �W (t) 1I S 2�TI I WI I C(o.T;H2) I DT V � (t) l I ! I DT � � (t) l I ! I V � (t) l I t 1I � � (t) lI ! + cOIlT2 I1 DT�W(t) 1I 2 + D I �W(t) 1 I 2 + c4 1I VW (t) 1 I 2 S 2coIlT2 I1 DT�� (t) 1I 2 + 4C�Il Il VW (t) 1I 2 + cO T I DT V � (t) 1I 2 + D I � W(t)1I 2 + c4 I1 VW (t) 1I 2 + C4T I �W(t ) 1I 2 1
1
1
1
Chapter 5. Spectral Methods for Multidimensional and High Order Problems
where
221
Cl = c1o ; ( e - 2D) 2 , C3 = IIwll � (o,T ;If2 ) ' -c
�
I Fs (t) 1 and I F6(t) l . It can be verified as before that IID1'WN(t) - otw(t) 1I 2 ::; IID1'WN ( t) - D1'W (t) 11 2 + IID1'w (t) - otw(t) 1I 2 +1' 21' r+1' w ( s) 2 ds . (5.81) ::; -;;:2 itr 11 0. (w (s) - wN(s))1I 2 ds + "3 it I o� 11 A similar estimate is valid for 11 '\7 (D1'wN(t) - Otw (t» 1I 2 . Thus 1' I Gs ( t) 1 ::; D 11L).� ( t) 11 2 + cOT IID1' '\7� ( t) 11 2 + � + 110. ( W (S) - wN(s» 11 2 ds + CS * 110. (W(S) - wN ( s» l I� ds + CsT * II O�W ( s) 11 2 ds 1' + CST 2 + II O� W ( s) l l : ds
We next estimate
[
1
where
Cs =
[ max ( e, � ) . Clearly 2�0
1
1
t I G6( t) 1 ::; D 1 1 L).� ( t) 11 2 + CO/1- T 2 1ID1'L).� ( t) 1 1 2 + l1;T +1' lIo.w(s) lI� ds where Ca = /1- (1' 2 of Lemma 5.10,
(� + 4�0 e T 2 ) . Finally, we estimate I G7(t) 1 and I G8 (t) l . By virtue
2 1 J (WN(t) + DTD1'WN(t) , W (t) - WN (t), � ( t») 1 ::; 8cllwl l c(0,T ;H2 ) IIW (t) - WN(t) lI f IIW (t) - WN(t) lIl llL).� (t) 11 ::; D 11L).�(t) 11 2 + c7 ,1 Ilw (t) - wN(t)ll l ll w(t) - wN (t) 1I 2 16 ' . t Lemma where C7, 1 = c 11 W 11 2C T 2 . Accordmg 5.9, D
(0, ;H )
( 5 . 82 )
�
2 \ J ( wN(t) + DT D1' W N ( t) - W (t), W (t), � (t») I ::; 4c llwll c(0,T ;H2) 11L).� (t) 11 IIW (t) - WN(t) ll f IIW (t) - WN(t) ll l + DT IID1'WN (t) 1I 2 ::; D 1IL).� ( t) 1 1 2 + C7,1 1IW (t) - WN (t) III IIW (t) - wN(t) 11 2 t + C7,2 T itf +1' II 0.W ( s) 11 22 ds (5.83 )
(
)
Spectral Methods ahd Their Applications
222
� 52cI I 1l1 11�(o,T;H2)' Let C7
where C7, = 2 to (5.83) leads
=
2C7,1 + C7, . 2
The combination of (5.82) and
t
r IG7 (t) 1 ::; 2D I tJ.W (t) 11 2 + c7 11 1l1(t) - 1l1N(t)1I1 1l1l1(t) - 1l1N(t) 11 + C7T + 118.w(s)ll� ds. 2
I Gs (t) 1
2 ' then the values of { and all do not depend on 7 N4 • By Lemma 3.2, we only require A 2 (t) � 0, i.e., Cj
(5.88)
Spectral Methoids and Their Applications
224
being a positive constant independent of N and T. Since � 2, a sufficient is that T = o(N->' ) , ).. � max 7 - 2 . Finally, the above statements together with. the triangle inequality lead to the first part of this theorem. then we can take e = 28 and so C = = O. If in addition 0' and 8 � Therefore the condition ( a) can be weakened to any T -> O. •
b
2 condition for (5. 8 8)
(�,
�,
>�
ro)
ro
l C2
Theorem does not provide an optimal estimate. But if \[f is a little smoother, then the optimal estimate exists, stated as follows.
5.7
Theorem 5 . 8 . Assume that (i) condition (i) of Theorem 5.7 holds; (ii) ro � rl > 2, or ro > rl = 2 and E LOO (O, T; Hr) for some r E ( 2 , ro); (iii) one of the following conditions is fulfilled (a) 0' > �, T = (N->') , ).. � max (�, 6 - 2ro) , (b) 0' � 'T 0 (N-4 ) , (c) 0' "21 ' TN4 :::; Jl C� (12 20' ) Then for all t E k,.(T), \[f
0
=
=
then E LOO(O, Tj for E We have
r1 2,
w
Hr ) r (2, r1 ) ' 4 G7 (t) =. jL=l Gr,j (t)
225
I R2 (t)1 more
where
G7,1 (t) = 2J (WN(t) - W (t), W(t) - WN(t), q,(t)) , G7,2 (t) = 2J (w(t) + t5-r DTw(t), W(t) - WN(t), q,(t)) , Gr,3 (t) = 2J (w(t) - WN (t) - t5-r DT WN(t ), W(t), q, (t)) , G7At) = 2t5-rJ (DT WN(t) - DTW(t) , W(t) - WN(t), q,(t)) . By Lemmas 5.8 and 5.12, we obtain that
I G7,1 (t)1 ::; 4 1 � q,(t) l l I V' (W(t) - wN(t)) l l � (W(t) - WN(t))1 I ::; D 1 � q, (t) 1 2 + C7,3 1 I W (t) - WN(t)l I � and
I G7,2 (t)1 ::; 2c Il w (t) + t5-rDT W(t)l l r I W (t) - WN(t)1 1 1 1 � q,(t) 1 ::; D 1 � q,(t) r + C7,4 I 1 W (t) - WN (t)l I � where C7, 3 = � I W llioo (O , T ; H2 ) and C7,4 = � I W l i oo (O , T ; H r) ' Moreover by ( 5.71 ) and Lemma 5 . 1 1 ,
Since
I DT WN(t)1 I 1 ::; -r- � (1* I O.WN(S)I I � dS) � ::; -r- � ([+T I O.WN(S)I I � dS) � and I l otWN(t)1 I ::; I l otw( t)1 I ' we see that 2 2 I G7,3 (t) 1 ::; D I � W- (t) 1 2 + C'i,4 I 1 W (t) - WN (t)1 I 21 + Cr,s-r itr+T l I o.w(s)1 1 22 ds
226 where
Spectral Methods
and'
Their Applications
C7,S = 282C7,4 ' Using Lemma 5.9, we deduce that
. l i D,. (llT(t) - IlTN(t)) ll i � D 1 � q, (t)1 1 2 + C7, S T 2 1 1 � (llT(t) - IlTN(t))1 I 2 1 I D,. (llT(t) - IlTN(t))l l l . l i D,. (llT (t) - IlTN(t))1 I 2 +,. � D I � q, (t) r + C7, S T I � (llT(t) - IlT N(t)) 1 1 2 ([ 118. (llT(s) - IlT N(S )) I � dS ) � . (1* 118. (llT(s) - IlTN(s)) I � dS) < D I � W (t) 1 2 + C7,S [+,. 11 8. (llT(s) - IlTN(s)) ll ; ds + 41 c7,sT 2 1 � (llT(t) - IlTN(t)) 1 I 4 i(t +,. 118. (llT(t) - IlTN(t)) ll � ds 82 where C7 ,6 = c ' The above statements lead to D I G7 (t)1 � 4D 1 � q, (t) 11 2 + c7 111lT(t) - IlTN(t) l I � + C7 It t+,. 118. (llT(s) - IlTN(s)) I � ds + C7T It t+,. 1 8. IlT(s) 1 I 22 ds + C7T2 11� (llT(t) - IlTN(t))1 1 4 1t t+,. 118. (llT(s) - IlTN(s))1 1 22 ds where C7 = max (C7 ,3 + 2 C7 , 4 , C7, S , C7 ,S ) ' Similarly, 1
1
2
where
Cs
=
(C7, 7 + 2C7,S , C7,9 , C7,10) and 4e ce 2 T',Hr) , C7,7 = -l ' coil l llT llioo(O'T,'H2) , C7,S = -4coil-I l IlT I Loo(o
max
Now, by taking = obtain from ( 5.89 ) that
D i6
C7,1 0 = c482oe . ll
and using an argument as in the derivation of ( 5.84 ) , we c
D,. 1 \7q, (t) 1 2 + T(e - 1 - 4co) I D,. \7q, (t) 1 2 + � 1 � q, (t) 1 2 + IlT (0- + �) D,. I � q, (t) I 2 + Il T 2 (e o- - 0- - � - 8co) I D,. tlq, (t) I 2 � Al I \7q, (t) I 2 + A2 (t) I � W (t) I 2 + A3 (t), ( 5.90 )
Chapter 5. Spectral Methods for Multidimensional and High Order Problems
227
where
Let
qo ;::: 0
and choose
�
as follows. For
(1' > 21 ' we take
)
� ;::: 6 = max 1 + 4co + qo , 2(1'2(1'+-16co 1 . Then
�(1' - (1' - 21 � - 8co ;::: 0
(
and so
(
�
7"(1-�-4co ) li D.,. V � ( t)1 1 2 +1'7" 2 �(1' - (1' - � - 8co 7" = 0 ( N-4 ) , then we take
If (1' =
�
and so
(5.91) also holds.
and
) I D.,.6�(t)11 2 ;::: qo7" li D.,. V �(t)11 2
.
(5.91)
Finally, for (1' < � and
2
2 4 . 7" N < Jt cr>2 ( 1 - 2(1' ) ' we put
and then (5.91) also follows. Now, by the same procedure as in the proof of Theorem 5.7, we obtain from (5.90) an estimate similar to (5.86). In particular, we have
p (t ) = 7"
L: A3(S) sERr(t-.,.)
=
0
(7" 2 + N2 - 2ro + N- 2rl + 7" N2 - 2rl )
•
and
Spectral Methods Their Applications
228
T = ( N-4 ) , then for all t E R... (T) , p(t) ::; � 2�TN . On the other hand, for > �, we require that A2(t) ::; 0 , i.e. , ) ::; b4 , T ::; b4 , T N4 (T + +T b4 being a positive constant independent of and T. The proof is complete. We now turn to numerical experiments. For describing the errors, let x �jq ) and w{jq) be the Legendre-Gauss-Lobatto interpolation points and weights. The error If
0"
::;
�2
and
0
N 2 - 2 To + N- 2 T,
2
0"
N2 - 2 T,
N
_
E( tPN, t) is defined as
1 eN (1 + cos ) (1 + cos 2) (5. 92) 10 The numerical results of scheme (5.76) are presented in Tables 5.1-5.3. The first table shows the high accuracy of (5. 7 6) even for small and the good stability in long time calculation even for large T. Table 5. 2 indicates that when -+ and 0 proportionally, the numerical solution converges to the exact solution. Table 5. 3 shows that the computations are also stable for very small even if = O. Table 5.1. The errors E(tPN, t) , fl = 0. 5 , 0" = 0. 5 ,b = O , = 10. T = 0. 5 T = 0.1 T = 0. 0 5 T = 0. 0 1 T = 0.005 T = 0. 0 01 t = 10 2. 497E - 2 4. 9 95E - 3 2. 4 97E - 3 4. 9 95E - 4 2. 4 97E - 4 5.000E - 5 t = 20 2 . 4 85E - 2 4.969E - 3 2. 4 84E - 3 4. 9 70E - 4 2 . 4 85E - 4 4. 9 83E - 5 t = 30 2. 438E - 2 4.880E - 3 2. 440E - 3 4. 8 79E - 4 2. 4 39E - 4 4. 8 76E - 5 In the previous part, we considered the two-level spectral scheme (5.76). Since we I
We take the test function
W(x, t)
t
=
7r X l
7rX
•
N,
T
N
-+
fl,
00
fl
N
adopt partially implicit approximation for the nonlinear term, the convergence rate in time t is of first order. It limits the advantages of spectral approximations in the space. A modified version is the following
{
(D ... \11/;N(t) , \1 0 for 1 $ k $ Po and ),,. inner product of Vk with (5.94), we obtain
(5.95)
'I , I + ¢ >'I, I) TI ( Y (x)) . 1=0
Since IT1 (y) 1 ::; 1 for IY I ::; 1, we know from the above two equalities and the last formula of (5.120) that for x E [-1 , a] ,
In a similar way, we see from the proof of Lemma 5.16 that for
x E [a, 1),
The above estimations lead to the conclusion.
•
Theorem 5 . 9 . The iteration applied to (5. 1 1 6}- (5. 1 1 8) converges, as m
---t 00 .
Proof. The result follows from Lemmas 5 . 1 7 and 5.18. Actually, since E i s nonsingular, the convergence of ej, m implies that of U,,/, N, m - U,,/,N , 'Y = 1 , 2. •
In the final part of this section, we present some numerical results. The time discretization considered is the second-order Crank-Nicolson scheme with the mesh size T = 0.01. The test function is
U ( 1 ) (X, t)
=
et arctg
(ax + i) ,
U ( 2 ) (X, t)
=
e t arctg
(ax i) . -
We use the Chebyshev pseudospectral domain decomposition method to solve this problem. For comparison, we also use single domain method for this problem. In Tables 5.8 and 5.9, we list the maximum norms of the errors between the exact solution and the corresponding numerical ones at t = 1 , -with different values of a and
.
Chapter 5. Spectral Methods for Multidimensional and High Order Problems
247
N. It can be seen that t �e domain decomposition procedure provides better numerical results, in particular, for the solutions possessing oscillations. Furthermore, if the matrices associated with the pseudospectral method are full, then the use of several sub domains allows the reduction of the overall complexity. It is noted that similar results are also valid for multi-sub domain method and the Legendre pseudospectral domain decomposition method.
Table 5.8. The errors of single domain method. a = 10 a=1 a=4 N = 4 4.767E - 4 1 . 288E - 2 2.812E - 2 N = 8 1 . 094E - 4 5.079E - 3 2.345E - 2 N = 20 2.572E - 7 2.860E - 4 1 . 005E - 2
Table 5.9. The errors a=1 N = 4 1.937E - 4 N = 8 7.377E - 7 N = 20 2.577E - 7
of two domains method. a = 10 a=4 6.1 19E - 3 3.668E - 2 1 .024E - 3 1.893E - 2 1.943E - 4 9.231E - 3
Another early work of spectral domain decomposition methods for hyperbolic sys tems was due to Kopriva (1986, 1987). We also refer to Quarteroni (1987b), Macaraeg and Streett (1988) , and Macaraeg, Streett and Hussaini (1989) for this technique and its applications. Carlenzoli, Quarteroni and Valli (1991) also used it for com pressible fluid flows. Recently Quarteroni, Pasquarelli and Valli ( 1991) developed heterogeneous domain decomposition method arising in the approximation of certain physical phenomena governed by different kinds of equations in several disjoint subre gions. Various applications of domain decomposition methods can be found in Chan, GlowinBki, Periaux and Widlund ( 1988) , and Canuto, HUBsaini, Quarteroni and Zang (1988). For the applications of Chebyshev pseudospectral methods to other nonlinear problems, we also refer to Hitidvogel, Robinson and Schulman (1980) , and Maday and Metivet ( 1986).
248 5.5.
Spectral Methods and Their Applications Spectral Multigrid Methods
Multigrid methods have been used for spectral approximations since the beginning of the last decade. Zang, Wong and Hussaini (1982, 1984 ) described this approach and its applications to periodic problems and non-periodic problems. It makes the spectral methods more efficient. In this section, we present some basic results in this topic. We first let A = (0, 211') and consider the model problem
{
x < 00 , - 00 < x < 00 . 2 and /j = J (xU» ) . Let N be an even positive integer. Set xU ) = approximation to the value U (xU» ) , 0 :5 i :5 N - 1 . Then -o�U = J, U (x) = U (x + 2 11') ,
Uj =
if - I
E
/ l12
-00
E VM.N and 2 � Proof. Set
4>1 CX I )
Then 4>1 E
114>II Loo
.•
.•
dI L oo(At} cMN i ll 4> ll . �
��
�
00,
= -2111" lA. 4> (Xl , X2 ) e-i1"2 dX2 .
lPM•1 and by Theorem 2.7, �
p
N. VM.N = lPM. I VN.2
f
� cM �
��
C6 . 1)
II4>dI L2 (At} � cMN i ( � 1 4>l I i'(At} )
i
��
Consequently, which leads to the conclusion.
•
Chapter 6. Mixed Spectral Methods
259
As a consequence of Theorem 6. 1, we have that for any , 'IjJ E
VM.N, (6.2)
Theorem 6.2. For any E VM•N , Proof. Let
that
, (X I)
1101 11 2
be the same as in
(6.1). We know from the proof of Theorem 2.8
= 27l' IILI $N II Ol d/;" (Al) � 1 !7l' M4 IILI$N lI d/i,'(A,) = � M4 I11I 2 .
On the other hand, 1102 11 2
=
27l' IILI $N 12 II 11 i,. (A,) � N2 11 1I 2 .
Then the desired result follows.
•
v v(x) mL= L vm .I Lm (xd ei1x, O 1 ' 1 =0
The mixed Fourier-Legendre expansion of a function E L 2 (!1) 00
IS
00
=
where
Vm. 1 is the Fourier-Legendre coefficient, Vm.l = 217l' (m + �) 10 v(x)Lm (X I) e -i1x• dx.
The L 2 (!1)-orthogonal projection PM.N : L; (!1) VM.N is such a mapping that for any v E L; (!1), - PM.N V, N :::; cM- 2r 2::: II V d11r(At l cN- 2 • 2::: 1W' l I v d l i'(Al ) 111> N 111$ N :::; cM- 2r l l v l l i'(A2 ,w( Al » CN- 2 B l v I 1' (A2 P (At l ) · 27r
+ +
It is noted that for any v
E
�
H�:;(!l) and r, 8
•
0, (6.4)
When we use the mixed Fourier-Legendre approximations for semi-periodic prob lems of partial differential equations, we need different kinds of orthogonal projections to obtain the optimal error estimates. For instance, the H l (!l)-orthogonal projection Pk,N : H� (!l) -+ VM, N is such a mapping that for any v E H� (!l), (V ( v - PiI,N V) , V¢»
+ ( v, ¢»
=
While the HJ (!l)-orthogonal projection plf?N (!l) any v E HJ,p (!l) ,
Theorem 6.4. If v E M;'B (!l) with r, I I v - piI,N v l l " :::; c (Ml - r
8
0, V ¢> E VM, N .
-+ V,,'},N is such a mapping that for
� 1 , then for J.L = 0, 1 ,
+ Nl-B) (M,,-l + N,,-l ) II V I l Mr,
• .
If, in addition, for certain positive constants Cl and C2 ,
(6.5) then
Chapter 6. Mixed Spectral Methods
261
Proof. Let VI ( Xl ) be as before, and Pk be the Hl (l\.t )-orthogonal projection. Set V. (X) 1 L1 :5N (PiIVI (Xl )) ei1x, . =
By Theorem 2. 10,
I l v -w l � :::; I v - v. l I � < 211' I!LI $N ( l i vi - PiIvdl k1 (A') + [2 1 vl - PiIvdl i' (A, » ) + 211' L ( I vdl k1 (A' ) + 12 I 1 vdl i2 (A, » ) :::; cM2-2IlrI>lLN N ( I vdl kr (A' ) + 12 I 1 vdl kr-1 (A1» ) I :5 + cN2 - 28 L 128 - 2 ( I vdl k1 (A, ) + 12 1 I vdl i2 (A' » ) :::; c (M1 -r +I!NI>Nl -S) 2 1 I v l �r" . inf
wE VM.N
Then the first conclusion with J.L = 1 follows. By means of the duality, we deduce that
I v - PiI,NVI l :::; c (M1 -r + N1 -S) (M-1 + N-1) I V I l Mr" .
The last result comes from the above inequality and (6.5) .
v
Theorem 6 . 5 . If E HJ,p(n)
n
M;,8 (n)
with r, s 2: 1, then for J.L = 0, 1,
If, i n addition, (6. 5) holds, then
pj,l
Proof. Let Vl (Xl ) be as before, and be the HJ (Al )-orthogonal projection. Put v. (X ) 1!LI:5N (pltvl(xl )) ei1x2• =
By virtue of Theorem 2 . 1 1 ,
•
262
Spectral Methods and Their Applications �
+
+
�
�
+
c 1LN (i vi - pl/ vd1-'(A') [2 1 v - pli�NvdI 12 (A, ) ) 1 I$ c I ILI >N (I vd1-'(A' ) [2 1 I vl lli2(A, ) ) cM2-2r ILN ( I vdl 1-r(A' ) [2 I1 vdl 1-r-l (A, ) ) I I$ CN28-2 ILN [28-2 (Ivd1-'(A, ) 12 1 I vdl i'(Al) ) I I> c (M-r N-8 ) 2 I 1 VIl �r, +
+
+
+
• .
The remaining results come from an argument as i n the last part of the proof of Theorem 6.4. • In numerial analysis of mixed Fourier-Legendre spectral schemes for nonlinear problems, we also need to bound the norm One of them is stated in the following theorem.
I Pli�NVllwr,p .
Theorem 6.6. Let (6. 5) hold and 8
n
> !. IIv E HJ.p ( O ) H; ( A2 , H1 ( A1 )) , then
I Pli�NVI I Loo � c ll v I H'(A2 .H1 (AtllIf, i n addition, v E X;· 8 ( O ) with r � 1 , then Proof. We first let VI ( Xl) be as before, and let Obviously
VI E HJ(A1 ) and vi E lP�. l ' Define
We assert from the orthogonality of the Fourier system and the property of that Obviously
PlioN
Chapter
6. Mixed Spectral Methods
263
By Schwarz inequality and Poincare inequality, we deduce that
I VI -v71 I iT' (A, ) + [2 1 I vl - v7I1 i' (A, ) :::; c ( l i vi -
!,
l L V LOO(A, I I I ::;N 1181 iI l ) :::; cM I ILI ::;N I VI - v7I H (Al) + c IlLi ::;N I vdl w+1(A, ) < (M1 -r + 1 ) II2:N I vdl w+1(A, ) I ::; < c (II2:N (1 + [2 fS ) ( I2:N (1 + [2) " I VdI W+1(A, ») I I ::; I ::; C
1
"2
Spectral Methods and Their Applications
264 Similarly, we have that
L I l v7 1 I LOO (A, ) s: cli v Ii H'(A2 ,H' (A, ) ,
III� N
L I l i li vi I i L OO (A, ) s: cli v I IH'+'(A2 ,W(A, ) '
III� N
The proof is complete.
•
Finally we consider the mixed Fourier-Legendre interpolation. Let 1M be the same as in the previous paragraphs and IN be the Fourier interpolation on the interval respectively. Set IM,N = 1MIN = IN IM.
A2
Theorem 6.7. If v E H: (A2 , W(A 1 )) n H; (A2 , H"'(A1 )) n H; (A2 , H>'(Ad) , O < (
( )
O
a s: min r, A ) , S: j3 S: mi n s , , and r, s , A, , > � , then I I v - IM,Nv I I H,B(A2 ,H"(A, )
s: cM2 ",-r + t Ii V I i H,B(A2 ,W(A, ) + cN,B-s l l v I I H'(A2 ,H" (AI » + cq ( j3) M2 01 - >. + t N,B --y Il v Il m ( A2 ,H� (AI » )
where q (j3) = ° for j3 > � and q(j3) = 1 for j3 s: � . Proof.
Let I be the identity operator. We have that
where
Dl
=
I I v - IM v II H,B(A2 ,H"(Ad )
+ I l v - INv Il H,B(A2 ,H"(A, ) ,
D2 = II (IM - I) ( IN - I) v II H,B(A2 ,H"(A, )
.
From Theorems 2.4 and 2.12,
If j3 > � , then by Theorems 2.4 and 2. 12,
If j3 s: � , then
The proof is complete.
•
Chapter
6. Mixed Spectral Methods
265
It is noted that if 0 a :::; 1 and a < min(2r - 1 , 2>' - 1 ) , then we can improve Theorem 6.7 by using (2.63) . In this case, we have that
Il v -
:::;
IM,NVl l wJ(A2 ,HQ(A, )) :::; cM"- r l v I l H�(A2 ,W(A, » + cNfj- sll vI I H'(A2 ,HQ(Al » + cq ( j3 )M"->'Nfj-,,! I l v II H� ( A2 , H A (A, ») '
q( j3 ) being the same as in Theorem 6.7. Theorems 6.3 and 6.5 can be found in Bernardi, Maday and Metivet (1987) .
6.2.
Mixed Fourier- Chebyshev Approximations
This section is devoted to the mixed Fourier-Chebyshev approximations. Let A I , A 2 1 and n be the same as in the previous section. Set w(x) = w(x I ) = (1 - XD - 2 , and
(v, w) w = Define
10
1
v(x )w(x )w(x) dx, I l v ll w = (v, v) � .
{v I v is measurable, I l v ll w < We define the spaces L�(n), H� (n), W� ,p (n) and their norms in a similar way. We L� (n)
oo}.
=
also introduce some non-isotropic spaces as follows
H�, S(n) = L 2 (A 2 ' H� (A I )) n HS (A 2 ' L�(A I ) ) ' r, S 2: 0, l �, 8(n) = H�, 8(n) n H I ( A 2 , H:- I (A I ) ) n W - ( A 2 , H�(Ad ) , r, s X�, S(n) = H8 ( A 2 ' H:+ I (Ad ) n W+ 1 (A 2 , H:(A I ) ) , r , S 2: 0
M
2:
1,
equipped with the n9rms
The meaning of Vp(n) is the same as in the previous section. The closures of Vp (S1) in H�, s(n), M�, S (n), X�, 8(n) are denoted by H�:; ,w(n) , M�:; ,w(n), X�:; , w(n) respectively. Their norms are denoted also by I . IIH:" , I . for simplicity. The corre I . The space H;:�(n) is the sponding semi-norms are denoted by I . I ':; and I .
1 M:' " Il x.:;' · 1 M':;' " H "
Spectral Metbods and Tbeir Applications
266
closure in H:;;" ( n) of the set containing all infinitely differentiable functions with the period 211" in the x 2 -direction, etc. Let M and N be any positive integers. The subspaces lP M, ! , lP1t, l ' have the same meanings as in Section 6 . 1 . We first give some inverse in and equalities.
VN, 2 , VN,2 , VM,N
VA},N
Theorem 6.8. For any 1> E
VM,N and 2 ::; p ::;
00 ,
Proof. Let 1>1 ( Xl) be the same as in (6. 1 ) . According to Theorem 2.13, we deduce
that
1 1> I Loo whence
::; IILI �N I 1>dI LOO(A, ) (IILI�N ::; c ( MN) !
::; cMt
1 I ILI �N l 1>dI L�(A, ) l 1 1>d l h(A ) = c (MN ) ! II 1> ilw ')
1
2'
As a consequence of Theorem 6.8, we have that for any 1> , 'IjJ E
This completes the proof.
Theorem 6.9. For any 1> E
11 1> 'ljJ I I � ::; cMNII 1> II � II'ljJII � ·
•
VM,N ,
(6.12)
VM,N ,
Proof. Let 1>1 be the same as in (6. 1 ) . We see from the proof of Theorem 2.14 that
The mixed Fourier-Chebyshev expansion of a function v E L�(n) is
The rest of the proof is clear.
v(x)
= mL=O ILII=O vm,ITm( Xl)eilx2 00
00
•
Chapter
6. Mixed Spectral Methods
267
with the Fourier-Chebyshev coefficients
(
1 v(X)Tm(x l )e -ilx2 dx . Vm , l = 1r 2 Cl in The £ 2 (O)-orthogonal projection PM, N : L; ,,,, ( O) -+ VM, N is such a mapping that for any v E L; ,,,, ( O ) ,
or equivalently,
PM,NV ( X )
=
M E E vm ,l Tm ( Xl ) eilx• . m =O I I I:5 N
Theorem 6 . 1 0 . For any v E H;:�(O) and r, II v
�
s
;:::
0,
PM, NV Il", � C (M-r -+- N-' ) II v I l H':;'· .
Proof. Let vl (xd b e the same as in (6.3) , and PM : L�(A l )
orthogonal projection. Then
PM,N V (X)
=
-+ IPM,l b e the L�(Al )
E PMvl (x l )ei lx2 . Il l:5 N
Using Lemma 2.5, we find that
IIv - PM,NV I I
=
2 1r E I I Vl - PMvdl ii!,(At l
Il l :5 N
+ 21r EN II vdlii!, (A, ) l' I >
+ cN-2• EN W· ll vdli� (A, ) I l l> I l l:5 N cM-2r l l v l l p(A. ,H':;(A,)) + C N- 2 0 l l v llh'(A2 ,Li!,(A, ) ) .
� cM - 2 r E II vdlh':; (A' ) �
It is noted that for any v E H�:; ,,,, ( O) and r, s
;:::
•
0,
(6 . 1 3) When we apply the mixed Fourier-Chebyshev approximations to different semi periodic problems of partial differential equations, we adopt different orthogonal pro jections to obtain the optimal error estimations. For instance, the H� (O)-orthogonal projection Plt,N : H!,,,, ( O) VM,N is such a mapping that for any v E H!,,,, ( O) ,
-+
Spectral Methods and Their Applications
268
While the HJ.w e!1)-orthogonal projection PJ.,;.�N that for any v E HJ.p .j!1),
Theorem 6 . 1 1 . If v E M;:� (n) with r,
8
:
HJ.p .we!1)
--+
Vi} . N is such a mapping
� 1 , then for f1- = 0 , 1 ,
If in addition (6. 5) holds, then
Proof. Let VI (X l ) be as before, and Pit be the H!(A l )-orthogonal projection. Put
v. (X) = L: (PIt vl (X l ) ) ei lx2 . I I I� N
By Theorem 2.16,
Il v - pk. NVII i.w �
N
N cM2-2T. ILN ( I vdl �':;(A, ) + 12 1 I vd l�':;-1 (A, » ) I I$ + cN2-2 8 L: 12 8 -.2 ( I vd��(A' ) + 12 1 I vdl h(AJ ) ) II I >N l (M -T + Nl - ·r I v l t-.:;' · · c
The rest of the proof i s clear.
•
In numerical analysis of mixed Fourier-Chebyshev spectral schemes for nonlinear One of them is stated in the problems, we have to estimate the norm following theorem.
IIpif�NVllw.:;'p .
Theorem 6 . 1 3 . Let
If in addition
S
>
!.
v E x;:�(n), then
Proof. Firstly let
Clearly
(6.5) hold and r,
VI ( X l) be as before, and set
VI E HJ,w (A1 ) and vi E lP�,l . Define
v
If E HJ,p,w (n) n H; (A 2 ' H:(A l ) ) , then
270
Spectral Methods and Xheir Applications
w E HJ,w (A t }, then by Lemma 2.8, al,w (w, w) ;::: � l w l ��(A, ) + [2 I1 w ll h (A, ) . Moreover, for any w, z E HJ,w(A 1 ), If
al ,w (vi - vi, ¢i) ° for all ¢i E lP�,I ' we see that � l I v, - vill ��(A, ) + 12 11 vI - vil l i�(A, ) ::; al,w (VI - V; , VI - vi ) al,w (lil - vi , VI - ¢i) ,
Since
=
=
V
¢i E lP�,I '
Consequently,
� l I v, - vill ��(A, ) + [2 1 vI - vil l h (A, ) ::; ..j� G ll v, - ¢i 1 l ��(A, ) + [2 1 vI - ¢i ll h (A' » ) .
Now, let pJ/ be the same as in the proof of Theorem 6.12. By taking and using Theorem 2. 18, we deduce that If'
M,l
¢i pllv, =
� l I v, - vill ��(A, ) + 12 1 vI - vil l i�(A, ) ::; cM2-2r ll vdl �':;(A, ) .
(6.14) II vl - viIlH�(A' ) ::; cM!'-r llvdIH.:;(A') , J.I 0, 1 . We now estimate IIpli�NV I w.. . oo , J.I = 0, 1 . Obviously, I l Pli�NVllw.., oo ::; IILI $N ( 11 8r viIlLOO (A, ) + ( 1 + I J.l I !' ) I vi i Loo (A' » ) , J.I = 0, 1 . (6.15) We can bound the terms on the right-hand side of (6.15) . Let 1M be the Chebyshev
By means of the duality,
=
Gauss-Lobatto interpolation on the interval A I . Then
Using Theorems 2.13 and 2.19, we obtain that for r >
I � vi - IM8I VdI LOO(A, )
�,
cMtl l 8lvi - IM8I VdI L�(A, ) ::; cMt ( l vi - VdHMA, ) + 1 8lVI - IM8I VdIL�(A' » ) < cMt I vi - vdH�(A, ) + cMt -r l l vdI H':;+ ' (A') ' (6. 1 7)
!, then Il v - IM,NV IlHl'(A2 ,H�(A, )) ::; cM2or -r ll v Il Hf3(A. ,H':;(A, )) + cNi3- s llvI I H '(A2 ,H�(A, )) + cq( j3 )Mzor - .\Ni3 -'Y llvI I H� ( A2 ,H; (A, )) where q(j3) = 0 for j3 > ! and q(j3) = 1 for j3 ::; ! ,
n
n
.
Spectral Methods and Their Applications
272
Proof. We have with
D1 = D2 =
I l v - IMV I H!l(A2 ,H�(Al)) + I l v - INV I Hil(A2 ,H�(Al)) ' II (IM - I)(IN - I)V Il H!l(A2 ,H�(Al)) '
From Theorems 2.4 and 2.19,
For (3 >
!, we have
while for (3
:::;
!,
The proof is complete.
•
Theorems 6.10, 6.12 and 6 . 14 can be found in Guo, Ma, Cao and Huang (1992) , and Guo and Li (1995a) . Quarteroni (1987a) also considered this kind of mixed approximation. The filterings are also available for the mixed spectral methods. For example, let
¢(x )
=
M N
L: L: �m" Tm (X1)ei'x2 . m=O 1 ' 1=0
Following the same line as in the proofs of Theorems 2.26 and 2.27, we can prove that if 0 :::; r :::; 01 and 0 :::; S :::; 02, then In order to keep the spectral accuracy, we can take OJ = oj ( N ) and oj ( N ) N � oo, j = 1 , 2, see Guo and Li (1995b).
� 00
as
Chapter
6.3.
6. Mixed Spectral Methods
273
Applications
In the final section of this chapter, we take two-dimensional unsteady Navier-Stokes equation as an example to describe the mixed spectral methods. For simplicity, we focus on the mixed Fourier-Legendre spectral schemes. Let and be the same as in the previous sections, and - 1 , 1 , ° :s; :s; 21l'} . I and > ° are the speed, the pressure and the kinetic viscosity, respectively, and and t) are given functions with the period 21l' for the variable We look for the solution with the period 21l' for the variable such that
At , A 2' X2
00 = {x X l = Il U = ( U( l ), U(2 ) ) . Uo(x), Po(x) f(x, X2 . (U, P) X2 , Ot U (U . V)U - Ill::. U V P = f, E 0, ° t :s; T, E 0, ° :s; t :s; T, V · U = O, E 00, ° t :s; T, U = O, x E n, U(x, O) = Uo(x), X E n. P(x, O) = Po(x), +
+
X
X
X
0 U(x, t), P(x, t)
!, r � bI ll in for some tt E R.,. (T) , then for all t E RT ( tI ) ,
Theorem 6 . 1 b35 . Let 0 p(tI) e b2 t, �
2 rM N
(6.26),
= a =
(6.33) hold.
If
where bI , b2 and b3 are some positive constants depending on Il and II UM,N II C(0,T;W1 , OO )
•
We next deal with the convergence of (6.26) with 0 = a = 0 and {} > !. Let PM1 0,N U, U = UM,N - UM,N and P = PM,N - PM,N P , We get from (6.20) and (6.26) that
[
UM,N =
( DT U(t) +_ d(U(t) , UM,N (t) + U(t)) + d( UM,N(t), U(t)) , v) + J1,a(U(t), v) - b(v, P(t) +(hDT P(t))
=
( Gt (t) + G2 (t) + G3(t), v) - b(v, G4 (t)),
(3(DT P(t) , w ) + b(U(t) + (hDT U(t), w ) U( O )
=
p( O )
=
0
=
't v E VM,N , t E RT(T) ,
- ,8 ( DT P(t) , w) + (Gs(t) , w ) , 't
w
E WM ,N , t E RT (T) ,
Spectral Methods and Their Applications
278
where
Gt(t) = 8tU(t) - DTUM,N (t), G2 (t) = (U(t) . V) U(t) - (UM,N (t) . V) UM,N (t), G3 (t) = "21 (V · UM,N(t)) UM,N(t), G4 (t) = p et) - PM, N P(t) - ()rDTPM,N P(t), Gs(t) = V (U(t) - UM,N (t) - ()rDTUM,N (t)) . -
•
By Theorem 6.5 and (4.24),
I Gt (t) 1 I 2 � cr l U ll k' (t,t+T;L' ) + � (M-2r + N- 28 ) I U l k1 (t,t+T;Mr, , ) .
By Theorems 6.5 and 6.6, we deduce that for
I G2 (t) 1 I 2
r, s > 1 ,
1 (U(t) . V) U(t) - (UM,N (t) . V) U(t) 1 I 2 + I (UM, N (t) . V) U(t) - (UM, N (t) . V) UM,N (t) 1 I 2 c ( II U(t) lI � l,oo Il U(t) - UM,N(t) 11 2 + I UM,N (t) 1I100I l U(t) - UM,N (t)II� ) ( M - 2r + N-2. ) ( I U(t) lI �r+ l" + l + I U(t) I tvl ,00 )
� 2
� �
C
U = 0, we derive in a similar manner that I G3 (t) 11 2 = I G3 (t) - � (V . U)U f � (M-2r + N-2. ) ( l 1 U(t) ll �r+ l" + l + I U(t) l I tv" OO ) .
Since V .
Obviously,
•
C
C
I G4 (t) 1 I 2 � (M-2r + N -2B ) I P(t) lI kr" + c()2 r ll P l k1(t,t+T;£2 ) , I Gs(t) 11 2 � (M -2r + N -2S ) II U(t) lI�r+ l , ' +l + c()2 r l U l k'(t,t+T;M" ' ) . C
Finally, by an argument as in the proof of Theorem 6.15, we get the following result.
Let 8 = q = O, () > �, r � b4J.t in (6.26), and (6.33) hold. If for
Theorem 6.16.
r, s � 1 ,
n
n
n
n
U E C (0 " T · H0t,1' Mr+1,s+ 1 Wt,oo) Ht (0 T · Mr,s) H2 (0 T · L2 ) P E e (0 T · Hr," ) n Ht (0 T· L2 ) then for all t E Rr (T) , E ( U M,N UM,N , PM,N - PM,NP, t) � bs (� ( r 2 + M-2r + N- 28 ) +(3) where b4 and bs are some positive constants depending on J.t and the norms of U and ,
,
p
P
-
P
in the spaces mentioned above.
,
,
p ,
"
p
"
p
'
Chapter
6.
279
Mixed Spectral Methods
We find from Theorem 6.16 that the best choice is f3 = OCr) = 0 (M- 2 r + N-28 ) , and the corresponding accuracy is of the order f3! . It seems that the artificial com pression brings the convenience in calculations, but lowers the accuracy. Indeed, if we compare the numerical solution to other projection of the exact solution, called the Stokes projection, we can remove this problem, and get a better error estimation for any f3 2': O. We shall introduce this technique in the next chapter. Obviously, since we use the approximation (6.24) for the nonlinear term ( U · V)U, . a reasonable simulation of conservation (6.27) follows, even the trial functions do not satisfy the incompressibility. Also because of this reason, the effects of the main nonlinear error terms are cancelled, i.e. ,
(d (u(t), u(t)) , u(t)) = (d ([r(t), [r (t) ) , [r(t)) = O. Therefore the nice error estimates follow. · Conversely, if the trial function space VM, N fulfills the incompressibility, then it is better to approximate the convective term by
d( UM,N, UM,N) ,
d( v , w)
=
81 (w(I)V) + 82 (W( 2 )V) .
In this case, the conservation can also be simulated properly. For the error analysis, we have
�
(J (u(t), u(t)) , u(t)) = (lu(tW, V · u(t)) .
As V . u(t) =I- 0 usually, this term cannot be cancelled. This fact shows that a good approximation to the nonlinear convective term plays an important role. Bernardi, Maday and Metivet (1987) applied the mixed Fourier-Legendre spectral methods first to semi-periodic problems of steady Navier-Stokes equation. The mixed Fourier-Chebyshev spectral methods have also been used for semi periodic problems of various nonlinear differential equations, such as the vorticity equation in Guo, Ma, Cao and Huang (1992) , and the Navier-Stokes equation in Guo and Li ( 1996). Recently, the Fourier-Chebyshev pseudospectral methods have also been developed, e.g. , see Guo and Li (1993, 1995a) .
CHAPTER 7 COMBINE D SP E CTRAL METHOD S
In many practical problems, the domains are not rectangular. We might use some mappings to change them to rectangular and then use pure spectral methods to solve the resulting problems numerically. But it is not easy to do so. On the other hand, such procedures usually bring in the complexity and errors. Thus it is better to adopt the finite element methods in those cases. However, sometimes the intersections of the domains may be rectangular, such as a cylindrical container. So it is reasonable to use the combined spectral-finite element methods or the combined pseudospectral-finite element methods. Since finite element methods are based on Galerkin method with interpolations of functions, the latter seem more attractive. In this chapter, we present the combined Fourier spectral-finite element methods for semi-periodic problems, and the combined Legendre spectral-finite element methods and the combined Chebyshev spectral-finite element methods for non-periodic problems. Finally, we take Navier Stoke� equation as an example to show the applications of these combined methods.
7. 1 .
Some Basic Results in Finite Element Methods
In this section, we recall some basic results of finite element methods in one dimen sion, as a preparation of the forthcoming discussions. Let A = ( - 1 , 1 ) and x E A.. Decompose A into 2 M subintervals D..j , -M $ j $ M - 1 . hj represents the length of D..j . Let h = -M<max j <M-l hj . Assume that there exists a positive constant p inde- -
pendent of the partition such that IIl,ax h $ p. Let k be non-negative integer -M$J$M-l j and nh , k be the Lagrange interpolation of degree k over each D..j . Define h
Sh , k = {v I v is a polynomial of degree $ k on D..j , -M $ j $
280
M - I}.
7.
281
Chapter Combined Spectral Methods
n
Moreover Sh,k = th,k n Hl (A) and S�,k inverse inequalities exist.
= Sh,k
Lemma ·7. 1 . For any E Sh,k and 1 ::5
p ::5 q ::5
HJ (A) . In the space Sh,k , several
00,
l1j
Proof. is a polynomial of degree at most k on each with the left extreme point = xU) . Let y x - xU») - 1, A = {y I l y l < I } and �(y) = (x). By virtue of
:( .
Theorem 2.7,
J
On the other hand,
Consequently, 1 I I I Lq(llj) ::5 c
from which the desired result follows.
Lemma 7.2. For any E Sh,k and 0
(��)
::5 r ::5
1
1
. - r;
1 I I I LP(llj ) •
1' ,
( h ) r -/I-
1 1 1 1 /1- ::5 c k2
I I l I r .
Proof. By Theorem 2.8 and an argument as in the proof of the previous theorem, we
deduce that Hence
1 1 1 1 /1- ::5
( h ) r-/I-
C k2
1 1 1 1 . ,
I'
=
0, 1.
The general result comes from induction and space interpolation. The L 2 (A)-orthogonal projection Ph ,k : L 2 (A) any v E L 2 (A),
--+
•
Sh,k is such a mapping that for
(7. 1 )
282
Spectral Methods and Their Applications
For simplicity, let c(k) be a generic positive constant depending on k. We have from Ciarlet ( 1 978 ) that for any v E � < r :5 k + 1 and 0 :5 I-' :5 min(r, 1 ) ,
W(A),
IJ v - n h,k V I I :5 c (k)hr-I-' Iv l r , I-' Il v - nh,kv lJwl',oo :5 c( k) hr-I-' Iv l wr,oo . Lemma 7.3. For
any v E Hr ( A ) , �
I I : , p, = 0, 1 . . I I I� N _ Finally the desired result comes from induction and space interpolation. The L 2 (51)-orthogonal projection Ph,k,N : L; (51) Vh,k,N is such a mapping that for any v E L; (51) , (v - Ph,k,NV, ¢» = 0, V ¢> E Vh,k,N. Set f = mi n ( r , k + 1 ) . For simplicity, Ph,k denotes the L 2 (A 1 )-orthogonal projection on Sh,k, 1 , and PN denotes the L2 (A 2 )-orthogonal projection on VN, 2 ' Similarly, Pl.,k stands for the H 1 (A 1 )-orthogonal projection on 8h,k, 1 and P�:� stands for the HJ (A 1 ) orthogonal projection 'on 8£,10, 1 ' Theorem 7.3. If v E H; , S (51), r > ! and s � 0, th en 1 1 &� ¢> 1 I 2 :::; C
-+
IIv
-
Ph,k,NVIl :::; c(k) ( hr + N - ' ) I l v l l w,
• .
Proof. Let
Then Ph,k,NV(X) = Using Lemma 7.3, we obtain that I l v - Ph , k,NV I 1 2
E
I I I� N
E Ph, vI ( 1 ) e il", x
k
I'I� N
I l v l - Ph,k vt l l i' (A ' )
< c( k ) h 2 r
E
I I I� N
+E
•.
I I I> N
I l v t l l i' (Al )
I l v t l l �' (A, ) + CN-2B
E
I'I> N
I Ws l l v t l l i' (A, )
:::; c( k ) h 2 r l l v l l i' ( A. ,w (AI l ) + CN-2B l v l �' (A2 , L2 (A, ) ) '
In particular, for any v E H�:;(51) , r >
!
-
and s � 0,
The H 1 (n)-orthogonal projection Pl.,k,N : H; (n) for any v E H; (n) ,
( V (v - Pl,k,N V ) , V ¢» + (v , ¢»
-+
= 0,
Vh,k,N is such a mapping that V ¢> E Vh,k,N '
286
Spectral Methods and Their Applications
The H6 (n)-orthogonal projection for any E H6,p (n) ,
v
P�:�,N
:
( v (v - P�:�,N V) , V¢»
H6 ,p (n)
-+
V�k,N is such a mapping that v,.�k,N '
If v E M;, O ( n ) and r, s � 1, then Il v - Pl,k,N vt :s; c( k ) (hr- I + NI - . ) (hl -1' + NI'-I ) I v Il M" " If, in addition, for certain positive constants Cl and C2 , = 0,
V ¢>
E
Theorem 1.4.
p. =
0, 1 .
clN :s; h1 :s; C2 N,
then
(7. 10)
Proof. Let VI (Xl) be as before, and
v. (x) = LN (pl, k VI (Xl)) eil"' I) I �
By Lemma 7.4,
Il v - Pl, k ,N V I :
:s;
:s;
:s;
:s;
•.
I v - v. l I � v -P V L v -P V I I I �N ( 1 l l,k II I :'(A ) + 12 1 I l,k III :' (A1 ) ) + LN (I I vdlk1(A, ) + 12 1 I vdl i' (A, ) ) II I > c( k) h 2r- 2 ILN (i l vdl k' (A') + 12 1 I vd l k.- 1 (A, ) ) I I� + cN2 - 2 • LN 12 0- 2 ( 1 I vdl k1(A') + 12 1 I vdli2 (A1 ) ) III> 1 k) c( (hr + NI -O r Il v l �.. . . 1
S o the first conclusion i s true for
If v E M;" (O)
n
p. =
1 . The rest of the proof i s clear.
and r, s � 1 , then I v - P�:�,N vt :s; c( k ) (hr- 1 + NI -o ) (h l -1' + NI'-I ) I v I M" " If, in addition, (7.10) holds, then
Theorem 1 . 5 .
H6,p (n)
p.
(7. 1 1 ) _
= 0, 1 .
Chapter
7.
Combined Spectral Methods
287
Proof. Let VI(Xl) be as before and
v. (x) 1 L:1:5N (p�:�vl (xd) eil:r2 . Il v - p�:�,NV l l � c in! I v -wi t � cl v - v.l l · =
Then
W EVh,k,N
By an argument as in the der�va.tion of (7. 1 1 ) , we prove the first result for Finally by means of the duality and (7.10), we complete the proof.
Il p�:�,NV l wr,p . Let (7.10) hold, k and r, > ! . For any v H; (A 2 ' Hr (A 1 )), Il p�:�,NV I LOO � c( k) l I v I l H' (A2 ,W(Ad) · If, in addition, v then We can also bound the norm
�
Theorem 7.6.
E
1
s
E
fJ, =
1. _
HJ,,,(fl)
n
X;" (fl) ,
VI (Xl ) be as before and pl,k,NV (X) 1 L:1:5N vi (xl) eil:r2 . Obviously vi S� , k , l . By an argument as in the proof of Theorem we know that for any S� , k , l ' I VI - vil l �' (Ad 12 vI -vil l 12 (Ad � c( l I vl - l k1 (Ad 12 vI - 1 I 12 (A') ) · By taking P�:� v" we obtain from Lemma that Proof Let
E
=
E
=
6.6,
+ 11
+ 11
7.5
By means of the duality, It is clear that for fJ,
=
( 7.12) 0, 1 ,
Spectral Methods and Their Applications
288
We estimate the terms on the right-hand side of (7. 13). Firstly, we have
1 81 ViI I LOO(A,J :::; 1 81 Vi - nh,k 81 VdI LOO (A, ) + 1 81 v/ - nh,k 81 VdI Loo (A, ) + 1 81 VdI LOO (A, ) ' (7. 14) From Lemma 7. 1 , ( 7. 2 ) and (7.12), 1 81 vi - nh,k Bt vdI LOO (A, ) :::; c(k) h - t ( l i vi - vdI Hl (A, ) + 11 81 v/ - nh, k 81 vdI L2 (Atl ) < C(k) ( l I v dI H!(Al) + hr- t l vdlw+ l (Atl ) . By (7.3),
11 81 v/ - nh,k 81 VdI Loo (A1) :::; c(k) Il vdl w+l (A1) '
By imbedding theory,
11 81 VdILOO (A1 ) :::; I vdlw+ l (Al) '
The above statements yield that
:::; c(k) /EN Il vdl w+ l (Atl I I� 1 " " :::; c(k) ( I /EI �N ( 1 + 12 r ' ) ( I /EI �N ( 1 + [2 ) " I Vdl kr+ l (Atl ) (7.15) :::; c(k) l I v I l H'(A2 ,W+ l (Al)) ' 1
Similarly, it can be shown that
v , I /EI �N I iIl LOO (A, J :::; c(k)llvI l H'(A2 ,W (At l ) 1 E v I / I �N III I i I Loo (Atl :::; c(k) l I v I l H·+ (A2,W(At l ) ·
The combination of (7.13) and (7.15) -(7.17) completes the proof.
(7. 16) (7. 1 7) •
Finally we consider the combined Fourier pseudospectral-finite element approxi be the Fourier interpolation on the interval mation. Let associated with the set of interpolation points. Define = = Set 2� [ = = 2N 1 E
IN
A2 , h,k,N Ph,k IN INPh,k ' 1· dX N v(x)W(X) l (v, v v) ' II l i k i + "'2 EA2,N A1
(V, W)N Then for any v E e (A 2 ' L2(Al )), (v - Ih,k,Nv, ( ' , X 2 ) E lPN, 2 ' By Lemma 7. 1 and Theorem 2.7,
ck 11 ¢> II Loo :::; sup I I ¢> I I LOO (AI ) :::; . fL sup 11 ¢> lI v ( A I ) xEA2 x2 EA2
yh
� ( r sup ¢>2 (X) dX I ) t :::; ck: II ¢> I I . yh y h lA, x2 EA. c
E li,k, N and 0
Proof. Let
:::; r :::; 11 ,
N
1=0
¢>(x) = :L ¢>I(X I )L,(X 2 ) ' Then ¢>,(X I ) E Sh,k,l ' As in the proof of Theorem 7.2, we have (7.18)
On the other hand, 02 ¢>( X) =
N
:L ¢>1 (X I ) 02 L,(X 2 ) ' 1=0
By (2.49) and the same argument as in the proof of Theorem 2.8,
II02 ¢>(X I , · ) II L' (A. ) Therefore
N
:::; :L fi{l+l) I ¢>,(X I ) I :::; cN2 11 ¢>(XI , · ) ll v (A. ) · 1=0
r
II ¢> (X I ' · ) l I i' (A. ) dX I = cN4 1 1 ¢> 1 1 2 I I 02 ¢> 11 2 :::; cN4 lAl
which together with (7.18) imply the result for 11 = 1 and r = O. The proof is complete by induction and space interpolation. • The L 2 (n)-orthogonal projection Ph,k, N : L 2 (n) Vh,k,N is such a mapping that for any v E L 2 (n) , (v Ph,k,NV , ¢» = 0, 'if ¢> E Vh,k, N .
-
-+
Chapter
7.
Combined Spectrill Methods
Theorem 7.10.
If v E HT"' (!1) , r > !
291
and s :2: 0, then
Proof. Using Theorem 2.9 and Lemma 7.3, we find that Ilv
- Ph,k,NVi l
In particular, for
::; II·v - PN Ph , k Vl l ::; IIPN (v - Ph,k ) II + IIv - PNvi l ::; �( k ) ( hr llvllu(A" H'(A' ) + N- s llvllH' (A,p (A1» ) < c( k) ( hr + N -' ) IIvll w,"
v E H�"
•
(!1) ,
The HI (!1)-orthogonal projection for any v E H I (!1) ,
Pl,k,N
:
HI (!1)
-+ Vh,k,N is such a mapping that
(\7 (v -PI, k,N v) , \7 9'» + (v, 9'» 0,
The HJ (!1)-orthogonal projection any v E HJ (!1) ,
P�:�,N
=
:
HJ (!1)
V 9'> E
Vh,k,N .
-+ Vh�k,N is such a mapping that for
If v E MT, S (!1)
and r, s :2: 1, then fo r J.L = 0, 1, Il v - Pl,k,N v t ::; c( k) ( hr-I + h-I N- s + NI- S ) ( hl - 1L + NIL-I ) IIvIIM" " If in addition (7.10) holds, then
Theorem 7 . 1 1 .
Proof. Let v.
=
Ph,k P�V Vh,k,N . It is easy to see that E
Using Lemma 7.3, we obtain
IIv
-
Ph,kVl l �
0,
I!PN V I l LOO(A.)
� �
c I!PNV - vIILOO(A.) + IIv Il LOO(A.) c (l + In N)N-" IIv l lw&. OO (A.) + IIv I l L OO(A.) '
Chapter
7.
Combined Spectral Methods
Taking Jl
O.
In the same manner, we verify that
B1 � B2 � B3 �
c(k ) ll vI I H!(A2 ,Hl (Al» ' c(k)ll v I H�(A2 ,L2 (A, ) ' c(k)ll v I Hi + 6 (A2 ,H � + ' (A, ) '
6'
> o.
The above six estimations together with (7.33) provide the bound of •
I l p�:2,N V l wl 'OO '
Finally we consider the combined Chebyshev pseudospectral-finite element ap proximation. Let IN be the Chebyshev-Gauss-Lobatto interpolation on the interval 11. 2 , associated with the interpolation points x� ) and the weights w (j ) , O � j � N. Define = Ph,kIN = INPh,k . Set N 1 (
h,k,N (V, W)N = � JfAI V (X t , X 2(j») W (X t , X 2j» ) (j) dx t , I vll N = (v, v) 1 · Then for any v E e (11. 2 , L2 (A 1 )), (v h,k,NV , ¢» N = 0, V ¢> E Vh,k,N ' Theorem 7 . 2 1 . If v E H�,S(f!) and r, s > �, then W
-
Theorems 7.17, 7.19 and 7.20 can be found in Guo, He and Ma ( 1995) .
Proof. The result follows from Lemma 7.3 and Theorem 2.19. 7.5.
•
Applications
In this section, we again take the two-dimensional unsteady Navier-Stokes equation as an example to describe the combined spectral-finite element methods. We focus on the combined Fourier pseudospectral-finite element schemes. But the main idea and the technique used can be generalized to non-periodic problems defined on a
Chapter
7.
Combined Spectral Methods
301
three-dimensional complex domain, and to other kinds .of combined spectral-finite element methods and combined pseudospectral-finite element methods presented in the previous sections. Let if, P and I' > 0 be the speed vector, the pressure and the kinetic viscosity. We consider the semi-periodic pr.oblem and require P ( x , t) to be in the space and A ( P ( x , t)) = for 0 :::; t :::; T. The L� ,in) , i.e. , P has the period for bilinear forms a ( v , w) and b ( v, w) are given by We adopt the artificial com pression ( .2 3 ) with the parameter (:J � O. When (:J = it is reduced to the original problem and so requires the incompressibility automatically. As explained in Section 6.3, a reasonable approximation of non-linear convective term plays an important role in preserving the global property of the genuine solution and in increasing the com putational stability and the total accuracy of n�merical solution. But it is not easy to do so for pseudospectral methods, since the interpolation and the differentiation do not commute. Besides, for the improvement of stability, a filtering will be used. Now let � 0 and N > 0 be integers. We use all the notations of Section 7.2. For simplicity, let be the set of scalar be the set of vector functions and functions, defined by
(6. 22) 271" .X 2 (6. 21 ).
6
0
0,
k
Vh,N
Wh,N
Wh,N = {w I W E Sh,lc,l VN,2 ,A(w) = O } . ®
The interpolation n h , lc+ 1 is denoted by nlc+ l for simplicity. Furthermore, for v (x)
N
=
L v/ (xl ) e i/",. , /=0
the filtering with single parameter a � 1 is given by
( ) v = � ( 1 - 1 � 1 " ) v/ ( xl ) ei/"'2 . 1 1 D(v , w , z) = 2 (( w . V') v , z) - 2 ( ( W . V')z, v) .
RN V = R
N a
Let
If V' . w = 0, then D( v , w , z) = (( w . V' ) v , z) . Thus we define the following trilinear form d(v, w , z) = 2 . V')z), v) . V') v ) , z) 2
1 (IN((w ·
1 (IN((w
We shall approximate the nonlinear term by d(v , v, z) . Clearly -
d (v , w , z) + d( z, w , v) = o.
(7.34)
302
Spectral Methods and Their Applications
Guo and Ma (1993) proposed a scheme for (6.22) . Let 7 be the mesh size in time. Uh,N and Ph,N are the approximations to U and P, respectively. 8, and 0 are parameters, 0 ::; 8, 0 ::; 1 . The combined Fourier pseudospectral-finite element scheme for (6.22) is to find Uh,N(X, t) E Vh,N, Ph,N (X, t) E Wh,N for all t E Rr(T ) such that (DrUh,N(t) , v ) (RNUh,N(t) 87 RNDrUh,N(t), RNUh,N(t), RNV) (Uh,N(t) DrUh,N(t), v ) - b (V, Ph,N(t) 0 7 DrPh,N(t » v E Vh,N, t E J4 (T), = (IN nk+1 J(t) , v ) , 0 [1 (DrPh,N(t), w) b (Uh,N(t) 7 DrUh,N(t), w) = 0, w E Wh,N, t E J4(T),
a-
a-,
+d + a-7 +
+/La
+
+
+
Uh,N(O) = IN nk+1 Uo , Ph,N(O) = IN nk+1 Po
V
(7.35)
where Po is determined by �Po = 'V . (f - (Uo ' 'V) Uo) .
a-
If 8 = = 0 = 0, then (7.35) is an explicit scheme. Otherwise a linear iteration is needed to evaluate the values of unknown functions at the interpolation points. In particular, if 8 = = 0 = �, then we get from (7.34) and (7.35) that 2 1 2 2
a-
II Uh,N(t) 1I =
+ f3 IIPh,N(t) 1 I + 2/L 7 Bellr(t-r) 1: IUh,N(S) + Uh,N(S + 7) 1 1
l I u h,N( 0 ) 11 2 + f3 I 1 Ph,N( 0 ) 1 I 2 + 7 1: ( IN nk+ 1 J(S) , Uh,N (S) + Uh,N (S + 7 » . BeRr(t - r)
Clearly it simulates the conservation (5._6 7) properly. We analyze the generalized stability of scheme (7.35). Let constant such that for any v E Vh,N ,
CI = cI ( k ) be a positive (7.36)
Assume that
a- > -21 '
or
A = 7 ( h - 2 + N2 ) < /LCI ( 12- 2a-)
Let c be a suitably small positive constant and qo > two cases, ( i ) 20a- ;::: 0 ( ii ) 20a- ::; 0
O.
(7.37) '
We consider the following
+ a- + 2c and 20 ;::: 1 + 2c + qo, + a- + 2c and -1 20 ;::: (1 + 2c + qo + /LACI( a- + 2c » ( 1 - /LACI G - a-))
(7.38) (7.39)
Chapter
7.
303
Combined Spectral Methods
Uh,N(O),Ph,N (O), !N nk+1 j and the right-hand side of the second uo , p, j and g , respectively, which induce the errors ( 7.35 ) , Ph, N u p for simplicity. Then from ( 7.35), it follows that Uh,N (DTU(t), v) + d (RNU(t) + 87" RNDTu(t), RNUh,N (t) + RNU(t), RNV) + d (RNUh,N (t) + 87" RNDTUh,N (�)' RNU(t), RNV) + J-Ia (u(t) +0"7" DTu(t), v) - b (v,p(t) + 97" DTP(t)) = (j(t) , v) , v E Vh,N , t E R, (T), ,8 (DTP(t), w ) + b (u(t) + 97" DTu(t), w) (g(t), w ) , V w E Wh,N , t E �( T). ( 7.40 ) Take v = u + 97" DTu in the first formula of ( 7 . 40 ). Since 1 297" (DTu(t), j (t))1 � e J-l7" 2 I D.Tu(t) 1 21 + c9e 2 1 Jet) 1 2- 1 ' J-l we get from (1.22) and ( 7 .34) that DT I l u(t) 11 2 + 7"( 29 - 1) I DTu(t) 1 I 2 + J-I(2 - e) l u (t) l � + J-I7"(0" + 9)DT l u (t) l � +J-I7" 2 (290" - - 9 - e) I DTU(t) l � - 2b (u(t) + 97" DTu(t),p + 97" DTP(t)) ( 7 .41) + t Fj (t) � C (le: 92 ) I l j(t) l � l
Suppoee that have the errors formula of denoted by and and of
{
=
0"
where
2d (RNUh,N (t) + 87"RNDTUh,N (t), RNU(t), RNU(t)) , 297" d (RNUh,N (t) + 87" RNDTUh,N (t), RNU(t), RNDTu(t)) + 27"( 9 - 8)d (RNU(t), RNUh:N (t), RNDTu(t)) , = 2 7"(9 - 8)d(RNU(t), RNU(t), RNDTu(t)) . Next take w p + 97" DTP in the second formula of ( 7. 4 0). We obtain that ,8DT I p (t)1 1 2 + ,87"(29 - 1 - e) I DTP(t)1 I 2 +2b(u(t) + 97"DTu(t),p(t) + 97"DTP(t)) � ,8 I p(t) 11 2 + (� + :;) I g (t) 1 2 . ( 7.42 ) Putting ( 7 .41) and ( 7. 4 2) together, we derive that F1 (t) F2 (t) F3 (t)
=
( 7 . 43)
Spectral Methods
304 We are going to estimate
an d
Their Applications
I Fj (t) l · Let v, w, z E Vh,N and
By virtue of (2.23), Theorems 2.1, and Theorem 2.4,
Bq(v, w , z) � l (wOqV, Z)NI + l (wOq Z, V)NI � c(k) lIwll (IIIN(zoqv) II + Il v llLoo lzh) � c(k)lI w ll Olzoq VII + II v IlLoo lzh ) · Moreover, for any 0
< 'Y
O.
We know from Theorem 7.22 that if (j > � , then the restriction o n A disappears and so we can use larger T . If (J = Ii > 2""- 1 for (j > then the restriction on also disappears and so the index of generalized stability, This improves the stability essentially. Indeed, it is the best result. On the other hand, since we adopt the operator the effect of the main nonlinear term is cancelled, i.e.,
�,
p e t)
s = -00.
d(v, w, z),
d (u(t), u(t) , u( t)) = o. This fact again shows the importance of a reasonable treatment with the leading nonlinear terms in the approximation of nonlinear problems. We are going to analyze the convergence of scheme (7.35) . In order to obtain the uniform optimal error estimation for (3 � we first consider the corresponding Stokes problem, which is itself very important in fluid dynamics. Let be the dual space of (HJ, p( O ) and = L�,p(O) . ( ' , - stands for the duality between and Let "I and = The Stokes problem is to find and such that
0,
V. pEW
V=
E V'
r
W E W e ' W. ,
{ b(au(,uw)v)=- b(v,p) w),
)
V'
uEV
V'
( Tf, v ) , 'v' v E V, (7.48) (e, 'v' w E W Let Vh,N be the trial function space of the speed vector u. We approximate u and p by u h,N E Vh,N and ph, N E Wh,N satisfying { l-'ab (u(h,uNh,N, w, )V)=-(e,b (w),v ,ph,N) = (Tf, v) , 'v''v' wv EE Vh,Wh,N'N (7.49) ' I-'
=
(7.50) is a combined Fourier pseudospectral-finite element approximation to (7.49) . Following the same line as in Girault and Raviart (1979), it can be shown that 7 . 5 has a unique solution provided that the following BB condition ( the inf-sup condition ) is fulfilled, sup 7.5 ) �
( 0)
vE V':.N
b(v, w) cllwll , 'v' w E Wh,N ' 11 V 11 1
---
( 0
Chapter
7.
Combined Spectral Methods
3 07
This condition is fulfilled for Vh,N = ( S�,k + ,l 181 VN, 2 , see Canuto, Fujii and Quar 2 teroni (1983) . If the components of trial functions belong to the different spaces Ah,N and Bh ,N , then it is denoted by Vh, N = Ah,N X Bh ,N. It was pointed out that ( 7.5 1 ) holds for Vh, N = S�, k + 2,l 181 VN, 2 X S�, k + l,l 181 VN, 2 . G o and Ma (1993) improved this result essentially, stated as in the following lemma.
)2
(
Lem m a 7.6.
If Vh,N
(7.51) is fulfilled.
) (
= S� , k + 1,l 181 VN, 2
(
) �
) X ( S�,l,l
181
VN, 2 )
and (7.10) holds, then
We define the linear operator P : V x W Vh,N X Wh ,N by P. ( u , p ) = ( P.u , Pop) = h (u ,N , ph ,N) where (uh,N , ph,N) is *the solution of (7.50 ) . Let f = min(r, k + 2). Using
-+
Lemma 7.6 and an abstract approximation result in Girault and Raviart (1979) , we obtain the following result. =
M; , 8 (f2) n HJ,p ( f2 ) , p E H;- l, 8- l (f2) n L5 ,p ( f2 )
Lemma 7.7.
If U J
E
Let Vh,N
Vh ,N '
(7.50) has a unique solution. Moreover if u and r, 2:: 1, then s
l, s- l (f2) n L2O (f2) Hrp , s(f2) n HQl,P (f2) ' p E Hr,P P
and r, s > 1 then -
By (7.52) and Theorem 2.26, it can be shown that if u H�" - l (f2) n L5 ,p ( f2 ) and 1 ::; ::; lX, then
s
We now deal with the convergence. Let (6.22) that
U*
E
=
P.U, p .
E
=
J
M; , S(f2) n HJ (f2) , p E
PoP. We deduce from
(D.,. U*(t), v) + d (RNU*(t) + 8r RND.,. U*(t), RN U*(t), RNV) v) - b (v, P*(t) + (hD.,. P *(t)) +/l a (U*(t) + O'rD.,.U*(t), 7 = (IN nk+ 1 f(t), v) + L Gj (v, t), V v E Vh ,N , t E R.,. (T) , j=l f3 (D.,. P *(t), w) + b (U·(t) + Or D.,.U*(t), w) = ( Gs(t), w) , V w E Wh ,N , t E R.,. (T), (7.54)
Spectral Methods and Their Applications
308 where
G1(v, t) = (D'TU(t) - 8tU(t), v) , G2 ( v, t) = (D'TU* (t) - D'TU(t), v) , G3 (V, t) = d (RN U*(t), RN U*(t), RNV ) - D (U(t), U(t), v) , G4(V, t ) = 07d ( RN D'TU* (t), RNU* (t), RNV ) , G5(V, t) = O'wra (D'TU*(t), v) , G6 (V, t) = -8rb (v, D'T P* (t)) , G7(V, t) = (f(t) - IN nk+ l f(t), v) , Gs( v, t) = (JD'T P* (t) . Let
U = Uh ,N - U* and F = Ph,N - P*. Then by (7.35) and (7.55) , we obtain that
(D'T U (t), v ) + d (RNU(t) + Or RN D'T U (t), RNU* (t) + RNU (t), Rv ) + d (RN U* (t) + Or RND'T U* (t), RNU (t), RNV) + p,a (U (t) + 0'7 DT U (t), v) -b (v, F(t) + 8rDTF(t) ) = - � Gj (v, t), V v E ltJ. ,N , t E R,. (T) , j= l (J (DTF(t), w) + b ( U (t) + 8r DT U (t), w ) = - (Gs(t), w) , V w E Wh,N , t E R,. ( T) , U ( O ) = IN nk+ 1 Uo - U*(O), F(O) = IN nk+ l Po - P *( O ) . 7
(7.55) We now estimate the terms on the right-hand side of (7.56) . Firstly by (4.24) and (7.53 ) ,
7 � II DTU(s) - OtU(s) 11:1 ::::; c(k)r 2 IJ UI I�(o,t;H- I)' seRr(t-T) r � IIDTU*(s) - DTU(s) 1I 2 aeAr (t-T) ::::; c(k) ( h 2f + N - 2 a ) ( IJ U II�(o,t;H" ') + II P II11(o,t;H.-I -I) ) . To estimate the nonlinear terms, we first let v, w, Z E Vh,N , ;::: I + � and I > O. Set .•
Aq = I «RNIN - I) ( RNW8qRNV ) , z ) l , Bq = I(IN ( RNW8qRNZ ) - RNW8qz , RNV ) I ,
S
q = 1 , 2, q = 1, 2.
By Theorem 2.26 and an argument as in the proof of Lemma 10 of Guo and Ma ( 1991 ) , we know that if 0: ;::: max ( s - p" 1 ) , P, = 0, ":"' 1 and s > !, then for any
Cbapter
7.
309
Combined Spectral Metbods
II RNIN (RN V RNW) - RNvRNwII HI' (A, ) :::; c(k) NI' -· (I v I H '(A, ) l w Il H"Y+ ! (A, + I w I H '(A, ) l I vI I H"Y + ! (AJ . )
(7.56)
By virtue of imbedding theory and (7.57) , we deduce t hat
Aq :::; I (RNIN - 1) (RNWOq RNV) 1I -1 1 I zl h < c( k) N - S (i, (I Oq vl � ' -l (A2) I w ll �"Y+ !. (A ') + l w I 1'-' (A2) I Oq v l �"Y+ ! (A2» ) dX l ) t 1I ;l h :::; c(k) N -S (IIOqvIl H ' -' (A2 P(A' ) Il w Il H"Y+! (A" H"Y + ! (A ') + I Oq v Il H"Y+ ! (A, P (A, )) I w I l H' -l (A" H"Y+! (Al)) ) I l zlh · Similarly,
Bq :::; I (RNIN (RN v RNw) - RN V RNW, Oq z ) 1 " :::; c(k) N - · (i , ( I vl � '(A') II w ll �"Y+ ! (A + I w I 1 '(A, ) I H �"Y+! (A ) dX l ) Ilzl h 2» 2) :::; c( k)N-S (1 I vII H '(A,P(A l)) ll w I H"Y + ! (A , H"Y + ! (A )) 2 ' +l I w Il H' (A2 P(Al)) I I v ll H"Y+ � (A2 ,H"Y+ ! (A' ) ) Il z l k 1
The above arguments imply .that
from which and (7.54), it follows that
(RNU* (t) , RNU*(t), RNV) - D (RNU* (t), RNU*(t), v)1 (7.57) :::; c(k)N - S ( 1 I U(t) I Ml " + I P(t) I H' -' (A2 P (A, )) f I l vlll. Similar to (7.44) and (7.45), we have from Theorem 2.26 and ( 7 . 53 ) that for r � 1 , I D (RNU*(t), RNU*(t), v) - D(U(t), U(t), v)1 :::; c(k) ( II U(t) I M',"Y + l + II U*(t)II Mlo"Y+ l ) II RNU*(t) - U(t) ll ll vl h (7.58) :::; c(k) ( hi' + N - s ) ( I U(t) I Mr" + I P(t)I I Hr - " , -, ) 2 1 I vl k Id
310
Spectral Methods and Their Applications
The estimates (7.58) and (7.59) lead to the bound for have from Theorem 2.26 and (7.45) that
I G3 ( v , t ) I I ' Furthermore we _
Moreover, by (7.53) and (7.54) ,
11 U* (s) l I �l"'+l II DrU* (s) 1I 2 � c(k) r2 ( 1 U 1 I &(o,t ;Ml ,..,+ l ) + I P I &(o,t ;H-r (A2 P (Ad » ) ( 1 I U Il h1 (o,t;Hl ) + I P l h1 (o,t ;L2 » ) ' We can estimate I Gs(v, t) 1 and I G6(v,t)1 in the same manner. Also by Theorem 2.4 and (7.2) , I G1(v , t ) 1 I � c( k ) ( hi' + N-' ) Il f(t) l w, . l v l . r3 L
.eR. (t-r)
In addition,
1 0 ( 0 ) 1 � c( k ) ( hi' + N -B ) ( ll UoII Hf + IIPoII Hf-1 , . -d , i P (O) i � c( k) ( 1 Uo lll + I PoI I H' + 6 (A. ,H ' + 6 (Al » ) ' 6 > O. .•
Finally we apply Theorem 7.22 to (7.56) , and the convergence follows.
Assume that (i) (7.10) and condition (i) of Theorem 7.22 are fulfilled; (ii) for certain constants r � 1 and s > �, U E C (0 , T', Mpr,. Jio,l p) HI ( 0 , T ,· Hr,B ) H2 (0 ' T · H - I ) P E e (0 , T', L02 ,1') HI (0 , T' Hr - l ,.- l ) f E e (O , T H; ) (iii) for certain t l E R,. ( T ) and certain positive constant b2 , ) eb2 t1 (r2 h2i' + N-2• + (3) - r Np.hc(k 6 ( 9)2 . Then for all t E Rr(tl) , I l uh,N (t) - U* (t) 1 I 2 � ceb2 (r2 h 2i' + N- 2• + (3) where b2 is a certain positive constant depending on k, f3, and the norms of U, P and f in the mentioned spaces.
Theorem 7.23.
n
i
n
i
"
n
n
,
,
P
. w dS = - is zw dS, V w E 1)( S ) ,
,
A,
then we say that z is the derivative of v with respect to denoted by 0>. v etc. Furthermore, we can define the gradient, the Jacobi operator and the Laplacian in the sense of distributions. For example, if
is v 6.w dS = is zw dS, V w E 1)( S ) , then we say that
z = 6. v, etc. Now, let (v , w) = is vw dS, I lvl l = (v , v)� ,
and
L2( S ) = {v E 1)'( S ) I lI v ll
< oo } .
Furthermore, define
equipped with the semi-norm and the norm as
For any positive integer r, we can define the space Hr (s) with the norm I . Il r by induction. In particular, the norm I . 11 is equivalent to (11v112 II 6. v I 12)! . For any 2 real r 2: the space Hr (s) is defined by the interpolation between the spaces H[r]
0,
+
Chapter 8. Spectral Methods on the Spherical Surface
315
the space H: (S) is the duality of H - r (s) . Particularly, and H[r] + l . For r < 2 O H (S) = L (S) and Il v l l o = Il v ll . It can be verified that for any v , w H 2 (S) ,
0,
E
(8 . 1)
- (�v , w ) = (\7v , \7w ) .
. The space C(!1) i s the set of all continuous functions o n S, with the norm I I · 1 1 0 . Lemma 8 . 1 . For
C(S) .
any
0 � J.L � r , HT (S)
'-t
HI-'(S) . For
any
r >
0, W+1 (S)
'-t
Proof. The first assertion follows immediately from their definitf�ns. We now prove
the second one. Let B be the unit ball in the three-dimensional Euclidean space, and w be a function defined on B. We denote by Tw the restriction to S of w . We can take HT + 1 (S) to be the trace space of HT + � (B) with the norm II v IIW+l (S) =
By imbedding theory, W + � (B)
On the other hand, for any v and
'-t
inf
weHr + � (B) Tw=v
Il w li H "+ � (B l "
C(B) , and so for any w E Hr + � (B),
E HT+1 (S), there exists w E HT+� (B) such that Tw = v
Consequently, Il v l i o(s) = sup Iv(x) 1 = sup Iw(x) 1 xES xES
su� Iw(x) 1 � c Il w(x) I i H "+ � (B) � 2cll v ll w+I (S) . � xEB
This implies the assertion. Let
L� (S) = Lemma 8 . 2 . For
any
•
{ v E L2 (S) I A (v) = O } ,
vE
Hl (S) n L6(S),
A(v) =
is v dS.
Spectral Methods and Their Applications
316
Co being a positive constant independent of v .
Proof. By the Poincare inequality, we claim that and the desired result follows. Lemma 8.3. If v
E
L 2 (S), w
•
E
Hr+ 1 (8)
and
r
> 0,
then
I l v w ll :s c ll v ll ll w ll r+ 1 '
Proof. By Lemma 8 . 1 , •
L I (x)
We next turn to the L 2 (8)-orthogonal system. Let be the Legendre poly nomial of degree I. The normalized associated Legendre function is given by
{
+ l ) (m - I)! (1 X 2 ) t fi Lm ( X ) , . I,m ( X ) = (2m2(m + I) ! L I,m (X) = L _I,m (x), Moreover, the spherical harmonic function Yi, m (..\ , 0) is
L
_
x
for 1 2 0, m 2 for I < 0,
m
III,
2 I ll .
(8.2)
(8.3)
The set of these functions is the L2 (S)-orthogonal system on 8, i .e. , 21r 1r = 0) cos 0
la l� Yi,m (>', O) "f/',m'(>" 2
dOd>' 81 ,118m,ml•
The spherical harmonic expansion of a function v E L 2 (8) is 00 v= L L
v (>', O) 1= - oo m� 1 1 1 I,mYi,m
where the coefficient
VI,m is given by VI,m = l" l: v ( >. , O) "f/,m (>', 0) cos 0 dOd>'. 2
Chapter 8. Spectral Methods on the Spherical Surface
It can be checked that
- sYl,m ( \ 0) = m ( m
317
+ 1 )1I,m (A, 0) .
(8.4)
So 1I ,m ( A , O ) is the eigenfunction of the operator - � on S, corresponding to the . eigenvalue m ( m 1 ) . Thus in the space Hr ( s ) , the normJ v ll r is equivalent to
+
mr ( m
( f :E
1= - 00 m?: 111
VN = span {1I ,m 1 1 1 1
+ 1 )" I VI,m I 2 )
1
2
(8.5)
Now let N be any positive integer, and define the finite dimensional space VN as
:::; N, I I I :::; m :::; M ( l)} . Clearly, VN depends on the choice of M ( l ) . Usually we take M ( l ) N or N + I l l . For fixing the discussion, we take M ( l ) = N . Let VN b e the subset of VN containing �
all real-valued functions. In this space, we have the following inverse inequality. Theorem 8 . 1 . For any 1>
E
VN and
j.t ,
0 :::; r :::;
:E :E JI,m1l,m (A, 0) . 1> = 1=-N m=111
Proof. Let
N
N
Then by (8.5) ,
<
I I ; · •
The L 2 (S)-orthogonal projection PN : L 2 (S) any v E L 2 (S), or
equivalently,
PNV(A, 0) =
N
N
L L
1= - N m=111
-+
VN is such a mapping that for
vI,m1l,m (A, O ) .
318
Spectral Methods and Thei� Applications
Theorem 8 . 2 . For any v E Hr (s) and 0 :::;
p.
:::; r,
(8. 5), L mJ.l ( m + 1)J.l l vl ,m I 2 + L L mJ.l ( m + 1)J.l l vl ,m I 2 l i v - PNvl l ! I=LN-N m=N+ NL L 1 mJ.l(m + 1)J.l l vl,m I 2 + I LI>N m=L1 1 mJ.l(m + 1 )J.l l vl,m I 2 1=-Nm=N+l 1 1>Nm=N+l 1=-oom=N+l mr (m + 1)' I VI,m I 2 The proof of the above two theorems is due to Guo and Zheng (1994) .
Proof. By
00
:::; c
c
c
00
:::; c
00
00
:::; cN 2 J.1 - 2r :L
8.2 .
:L
00
00
:::; cN2 J.1 - 2 r l l v l l � ·
•
Pseudospectral Approximation on the Spherical Surface
(j ) w A ( - 1,1). SN {(..x ( ") 0 (; » ) l ..x ( " ) - } + 0(;) arcsin x (;) - The discrete inner point ( . , · ) N and the discrete norm I . l i N are defined by (V ,W)N + 1 E2N �N (.�(k) ,O(i») (..x(k) ,O(i» ) w(i) , I v i N The interpolation IN C(S) VN is such a mapping that for any C(S) , INv 1=N-N m=N1 1 VI,mY/,m (..x , O), (8. 6 ) where 2N N (..x(k) , OW) ,m (..x(") , O(i») w(i) . VI,m + 1 k=03=0
We now present another approximation on the spherical surface, based on the in terpolation. Let x U ) and be the Legendre-Gauss interpolation points and the weights on the interval = Define SN as a set of grid points, =�
,
=
=
2N
271"
, 0 < k < 2N, 0 < -
vE
--t
2
_71" _
2N
""
L..J �
v
J.
' " m e'., � 8",L " m ( sin B ) cos B. L.. m=O lll$m
8.
321
Chapter Spectral Methods on the Spherical Surface
I:;I,m ' denote the sum in the above formulae. Then
Let
I� 12 cos B OA
0,
(J(v, w), z) + (J(z, w), v) = O. Indeed, the integration by parts yields
(8. 13)
{" L: Z (O>.V09W - O>.W09V) d9dA " - fo 2 L: (O>.Z09W - 09Z0>.W) d9dA - l" Z (A, i) v (A, i) O>.W (A, i) dA + l" Z (A, -i) v (A, -i) O>.W (A, -i) dA. From Courant and Hilbert (1953) , the regularity of w implies that w approaches the limits independently of A, as 9 -+ ± � . This means that O>.W = 0 at 9 = ± � , and so (J(v, w), z ) =
2
2
v
Chapter 8. Spectral Methods on the Spherical Surface
323
H
(8.13) follows. Next let v = in (8.12) and integrate the resulting equation from a to t . We obtain from (8.13) that II H(t) II 2 + 2 p.
l I H (s ) l� ds
= IIHo ll 2 + 2
1o\j(s ) , H(s)) ds.
(8. 14)
Now let V� = VN n L�(S) . and lIT are approxiiQated by TiN and ""N . The spectral scheme for (8. 12) is to find TiN( ). , 0, t) E VN , ""N( ).. , 0, t) E V� for all t E R,. ( T ) such that
{
H
(DT TiN (t) + J (TiN(t) + CrDTTiN(t) , ""N(t) ) , 4» 4> E VN , t E RT ( T ) , - I' (t. (TiN (t) + UrDTTiN(t)) , 4» = (J(t), 4» , 4> E V� , t E RT ( T ) , - ( t.""N(t) , 4» = ( TiN (t) , 4» , ( ).. , O) E S TiN ( a ) = PNHo ,
V V
(8.15)
where c and u are the parameters, a � c, u � 1 . If both of them are zero, then (8.15) is an explicit scheme. Otherwise a linear iteration is needed at each time t E R.,. ( T ) . We now consider the existence and uniqueness of the solution of (8.15) with c i- a or u =F a. Let F (t) = TiN(t - r) - r ( l - c ) J (TiN(t - r ) , ""N (t - r )) +p.r(l - u )t.TiN(t - r) + rJ(t - r ) .
Then for any t E R.,. ( T ) ,
Clearly i t i s a linear algebraic system for the coefficients o f the spherical harmonic expansion of TiN (t) . Hence we only have to verify that TiN(t) is a trival solution when F (t) == a. In fact, by taking 4> = TiN (t) and using (8. 13), It implies TiN (t) == a , whence TiN (t) is determined uniquely at each time t E R,. ( T ) . In particular, if c = u = � , then we have from (8.13) and (8.15) that p.r ll TiN(t) II 2 + 2
L..J
"
= l l TiN( a ) 11 2 + r
ITiN (S) + TiN(S + r ) ll2
BERr(t-T)
�
BE Rr(t-T)
(J( ) , TiN (S) + TiN (S +. r )) .
s
(8.16)
324
Spectral Methods and Their Applications
This is a reasonable ana!ogy of (8. 14) . f We now analyze the generalized stability of scheme (8. 15). Suppose that and the right-hand side of (8.15) have the errors j and respectively. They and denoted by induce the errors of and Then
17N(O), �o , g, �N ¢N. 17N 1/;N, (DT�(t) + J (� (t) + 8TDT�(t) , 1/;N(t) + ¢(t)) + J (17N(t) + 8TDT17N(t), ¢ (t)) , rp) - It (� ( �(t) + (7TDT�(t)) , rp) = (l(t), rp) , V rp E VN, t E RT(T), � - ( ¢ (t), rp) = ( �(t) + g(t), rp) , V rp E V� , t E RT(T), A (¢ (t)) = 0 , t E RT(T), � ( O ) = �o , (A, 8) E S. ( 8. 1 7 ) Let e be an undetermined positive constant. By taking rp = 2 � + eT DT fJ in the first formula of (8. 1 7 ) , we get from ( 1 . 22 ) and (8.13) that DT 11�(t)1 1 2 + T(e - 1 ) I DT�(t) 1 2 + 21t 1�(t) l ; + itT (0" + �) I DT�(t) l; + ItT 2 (eO" - 0" - n I DT�(t) l ; + � Fj (t) = (l(t) , 2�(t) + eTDT�(t)) (8.18) where
Fl(t) 2 ( J (17N(t) + 8TDT17N(t), ¢(t)) , �(t)) , F2 (t) eT (J (17N(t) + 8TDT 17N(t), ¢ (t)) , DT � (t)) , F3 (t) T(e - 28) (J ( � (t), 1/;N(t)) , DT�(t)) , F4(t) T(e - 28) (J (� (t), ¢ (t)) , DT �(t)) . Further, we put rp = ¢ in the second formula of (8. 17), and then obtain that =
=
=
=
By Lemma 8.2, we find that
from which and Lemma
8 . 2,
(8 . 1 9)
Chapter
8.
325
Spectral Methods on the Spherical Surface
e
We are going to estimate I Fj (t) l . Let > O. By find that for , >
0,
I FI (t) 1
$ $ $
( 8.13 ) , ( 8.19) and Lemma 8.1, we
2 1(J (7} (t) , t,b(t)) , 71N(t) + 6TDT71iv (i)) I ep, 1 7} (t)l � + e: II 71N I I �(O,T;LOO ) 1 t,b (t) l : ep, 1 7} (t)l � + e: I171N I I �(O,T;H"Y+') ( U 7}(t) 112 + II g (t) 11 2) .
Similarly,
8.3 yields T (e - 26) 2 I I .,pN I I C(o,T;H"Y+2) 2 171 (t) II2 . 1 F3(t) 1 $ e T I I DT71 (t) II 2 + C e
Furthermore, the use of Lemma
_
_
Since
8.1 that _ T 2 (e - 26)2 II 1/N I 1 C(O,T;L2) 2 1 1/- (t ) 1 I2 · IF3(t) 1 $ e 'T I I DT71 (t) I I 2 + c N "1 e Also, by Lemma 8.3, Theorem 8.1 and ( 8. 2 0 ) , T(e 2 ) 2 1F4 (t) 1 $ e T IIDT 7}N I I 2 + C � 6 1 t,b (t) I :+ I 7} (t)l � 2 $ eT I IDT7}(t) 11 2 + C'T (e � 26) 2 ( 1 I 7}(t) II; + I Ig (t) II;) 17} (t) l � $ eT I I DT 7}(t) 11 2 + cTN2"1 (ee - 26) 2 (1 I 7} (t )1 I 2 + I l g (t)1 1 2 ) 1 7} (t) I � . Clearly, I (i(t) , 7}(t) + e� DT7}(t)) 1 $ eT II DT7} (t) 112 + C 1 7} (t)1 1 2 + C ( 1 + T;2 ) I j(t) 1 2 . By substituting the above estimates into ( 8.18 ) , we claim that DT 1 I �(t) 11 2 + T( e - 1 - 3e ) I I DT�(t) 112 + p, 1 � (t)l � + p,T (0" + � ) DT 1 � ( t) l � +p,T2 (eO" - 0" - � - e) I D �(t )l � $ Al 1 I � (t) 11 2 + A2 (t) 1 �(t )l � + A3(t) ( 8. 22)
we deduce from Theorem
T
Spectral Methods and Their Applications
326
l + e I1 77N ll 2c O,T;H'Y+ 1 ) , Al c ( 1 + � ( ) CTN2-y C - ( e 2 6 ) 2 ( I1 77N l c2 (O , T ; L2 ) + 1 I � (t) 1 I 2 + I l g ( t )1I 2 ) , A2 (t ) . = - p, + p, + Te ) 1 J-(t ) 1 2 + C ( 1 + e2 ) I1 77N ll c2 O,T;H"I+ 1 II g- (t )1I 2 . A3( t) = C ( 1 + e ( ) p, Let qo > 0 and c be suitably small. We choose e in three different cases. (i) u > !. We take 2 u + 2c ) e � 6 = max 1 + qo + 3c, 2 u ( 1 .
where
=
c
-
c
_
T(e - 1 3c ) IIDT�(t )1I 2 + p,T 2 (eu - u - � - ) IDT�(t) l � � qoT IIDT�(t )1I 2 . (8.23) (ii) u = !. According to Theorem 8 . 1 , there exists a positive constant c] such that I v ll � � c]N2 11 v1l 2 for all v VN. For this reason, we take e � 6 = 1 + qo + 3c + P,C] T N2 G + ) Then
c
-
E
c
and so (8.23) also holds.
( 111. . ) U .
< "2
1 and T N2
c
.
whence (8.23) is still valid. Now let
E(v , t) = II v(t )11 2 + T z: T ( l v( s ) l � + qoT IIDTv( s )1I 2 ) , ( ) p e t ) = 11� (O) 11 2 + p,T (u + �) I�( O )I� + T z: A3(S). p,
seRr t-
seRr (t-s)
.R,. ( t ) yields E (� , t ) � p (t ) + T z: (AIE ( i7 , s ) + A2 (S)E2 (� , S)) .
Then the summation (8.22) over
seRr(t-T)
Applying Lemma 3.2 to (8.24), we claim the result as follows.
(8.24)
Chapter 8. Spectral Methods on the Spherical Surface
Assume that (i) N2,,! is suitably small, being an arbitrary small positive constant; r.") > -1 or N2 CII-' ( 1 2- 2U ) ,. U 2 ·· (iii) I l g (t) 1 2 ::; T�b 2"! and p (tl) eb2 t' ::; TNbs "! for some tl E Rr (T). Then for all t E R,. (tl)
Theorem 8.4. T
{ n
T
327
I
l , ( 8.26 ) for u = l , for (j < C l-' . It leads t o the following result.
�.
If (8. 26) and condition (ii) of Theorem 8.4 are fulfilled, then for all p (t) and t E R,. (T) , the estimate (8. 25) is valid.
Theorem 8 . 5 .
We know again from Theorem 8.5 that a suitable implicit approximation of non linear term can improve the stability essentially. Indeed in the case of Theorem 8.5, scheme ( 8. 15) has the generalized stability index = -'- 00 . In the final part of this section, we deal with the convergence of ( 8.15 ) . First of all, we derive from ( 8.4) that for v
s
PN fl ( >' , fJ) v
E H2 ( S),
m ( m l)vl , m Y/, m (>', fJ) 111:SN m �111 2: 2: VI ,m flY/,m ()., fJ) = flPN V(>', fJ). ( 8.27) I I I:SN m �111 Then we have from ( 8.12) and ( 8.27) that
- 2: 2:
+
HN = PNH and WN = PNW. (DrHN(t) + J (HN(t) + 8T DrHN(t), WN(t)) , r/J) + I-' (V (HN(t) + UT DrHN(t)) , V r/J) = (� Gj (t) + f(t), r/J) + (VG4 (t), Vr/J ) , 'V r/J E VN, t E Rr(T), (VWN(t) ; V r/J) = (HN(t), r/J ) , 'V r/J E V�, t E R,. (T ) , A (W N(t)) = 0, t E R,.(T), = HN(O) PNHo , ()., fJ) E S
Let
( 8.28 )
Spectral Methods and Their Applications
328 where
Gl(t) = DrHN(t) - OtHN(t), G2 (t) = J (HN(t), WN(t » - J (H(t), w (t » , G3 (t) = DTJ (DrHN(t), WN(t » , G4 (t) = J.LO'TDrHN(t). Next, set fI = 1IN - HN and q, = 'ifJN - WN. We get from (8.15) and (8.28) that (DrfI(t) + J (fI(t) + DTDrH(t), WN(t) + q, (t) ) +J (HN(t) + DTDrHN(t) , q, (t) ) , t/» - J.L ( (fI(t) + O'T DrfI(t) ) t/» = - (� Gj(t), t/» - ( V'G4 (t), V't/» , V t/> VN, t Rr( T ) - (Aq, (t), t/» = (fI(t) , t/» , V t/> V�, t Rr(T), A ( q, (t) ) = 0 , t Rr(T), fIC O) = 0, (A, 9) S. By an argument as in the proof of Theorem 8.4, we can deduce an estimate similar to (8.24) . But 1IN and ii are replaced by HN and H respectively, while p et) is replaced A
E
E
E
E
E
,
,
E
by
As before, we have
I: I Gl(s) 1 I 2 � CT 2 I 1 H I �2 (O .t ;£2 ) ' T ERT(t 8 -r)
By Lemma 8 . 1 , Theorem 8.2 and a similar estimate as (8.21) , we verify that for any 'Y > 0,
I G2 (t) 1 I 2 � c I WN I �+2 I 1 HN(t) - H(t) lI� + c Il H(t) lI� I WN(t) - w (t) I �+2 � cN-2r ( I W(t) I ;+2 I 1 H(t) I �+1 + I w (t) II�+'Y+2 I1 H(t) l I n � cN-2r ( I H(t) II ; II H(t) I �+1 + II H(t) II�+'YI I H(t) I �) .
By virtue of Lemma 8.3 and Theorem 8.2, it is easy to see that T
I: I G (s)1 I 2 8ERT(t-r) 3
�
<
O , H E C (O , Tj W+l ) Hl (O , Tj Hl ) H2 (O , Tj L2 ) . Then for all t E RT (T),
Theorem 8.6.
n
n
J-t
b4 being a positive constant depending on and the norms of H in the mentioned spaces. The above numerical analysis comes from Guo (1995) . The spectral methods on the spherical surface have also been used for the barotropic vorticity equation in Guo and Zheng (1994), and for a system of nonlinear partial dif ferential equations governing the fluid flows with low Mach number in Guo and Cao (1995) . The applications of the pseudospectral methods on the surface can be found in Cao and Guo (1997) .
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