super convergence of a chebyshev spectral collocation method

8/3/2019 Super Convergence of a Chebyshev Spectral Collocation Method

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of a Chebyshev Spectral Collocation

Zhang

IS February 2007 1Revised: IS September 2007 1Accepted: 21 September 2007 1r 2007Business Media, LLC 2007

We reveal the relationship between a Petrov-Galerkin method and a spectral colof the second kind (l and zeros of Uk) for the

as theof the first kind (Zeros of h) . Super-geometric convergent rate is estab

of solutions.Chebyshev polynomials Collocation Spectral method Superconvergence .

of the spectral method [2-5, 8-10. 12. 15, 16} is its

to some higher-order numerical integration errors. Naturally,to know if a similar result can be proved for collocation at the Chebyshev

, this generalization is not straightforward.cation at the Legendre-Gauss-Lobatto points can be naturally linked to a symmetpoints can only be linked to a non-symmetric (or weighted)

It turns out that thisa Petrov-Galerkin method, i.e., different trail and test spaces are used.

rk was supported in part by the US National Science Foundation grants DMS-0311807 and(181)

of Mathematics and Computer Science, Hunan Normal University. 410081 Changsha, [email protected]

of Mathematics. Wayne State University. 48202 Detroit. MI, USASpringer
mailto:[email protected]:[email protected]


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238 J Sci Comput (2008) 34: 237- 246We start from an observation to the Chebyshev polynomials: if h E Pp interpolates I at

the following set of the Chebyshev points8 _ lfjXi = cos8;, ,-- . 0, t .. . ,p, (Ll). p ' J

then the remainder can be expressed asI (x ) hex) l [xo,xl, . . . ,xp,x] I ) T ~ ( x )

I 2= f[xo, XI, . . , Xp, X] 2p-j(x I)Up _ 1(X)I= I[xo, XI , ' " Xp, x] 2p (Tp_1 (x ) - Tp+1 (x, 0.2)

where n and Uk are the Chebyshev polynomials of the first and second kinds, respectively.It is well known that

l ( p + I ) ( ~ ) f[xo,Xj, . .. ,xp,x] (p+ 1)1 'for some E ( - I , I) . Now we introduce Condition M for the derivatives of I:

max l /k)(x)l:::cM k, k=0.1,2, . . . .- 1 ~ x : : s : J

By the Stirling's formula(1.3)n! ( ~ ) " J 2 l f ( n + ~ ) .

we haveI / (x) - h(x)1 ::: (eM)P+1 (1.4).fP 2p

This is a better convergent rate than O(e- 0, since the right hand sideof (1.4) is equivalent to .

eP(r-1np) y =In eM2In the following sections, we shall realize the same convergence rate when the Chebyshevcollocation method is used to solve two-point boundary value problems.

2 A Petrov-GaJerkin MethodConsider the boundary value problem

_u " f. u( - I )=O, u'( l) I. (2.1)Its weak form is to find u E HM- I , I] {v E HI [-1. 1]: v( -I ) = 0) such that

(u',v') = ( f ,v )+v( l ) . " I v E H ( ~ [ - I , I ] . (2.2)i l Springer


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of auxiliary functions in H ( [- I , I] :

= rr: - arccos x,

1/fm (x ) = fX Tm I(t)dt, m l.-I4>1(-1) 0= 1/f(-1),4>m = 0 1/f(l), for m 2,

JI=X2 Urn - 2 (x )m -I

1/f(l) = 2,

j=l.ical approximation is the following.

Find u{J E Pp[-I , 1], up(-1) = 0, such that

239

m ~ 2 , (2.3)(2.4)

(2.5)on up(x) I:LI bk- l1/fk(X). It isrd to obtain

1bo = - (1,4>1) + 1. (2.6)rr:

(2.7)of Theorem 2.1, is the truncation of the first p terms in theu'. Using the differential equation

an expression for 1/fj:I 2--2-[(1 - x )mUm - 1(x ) + xTm(x) + Tm(-I)] .m -1 (2.8)

The method is a Petrov-Galerkin type since we use different trial and test spaces.Tk and Ub see (2.3)

we impose Condition M on u:max lu(k)(x)l:s:cM k , k=0 , I ,2 , . . . .

xEI- I , l l

of Condition M was discussed in [18].Springer


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240 J Sci Comput (2008) 34: 237-246We need the following notation.I I u(x)v(x)dx(u , v)w IIvllw (v, v ) ~ 2 . -I ~ 1 - x 2 '

Theorem 2.1 Let u an d up be solutions of (2.2) and (2.5), respectively. Assume that usatisfies Condition M and the non-degenerate condition: IbpI > Ibp+ll. Then when p + I >M /.J2, we have

(u u p ) ( x ) ~ b p 1 / l p + 1 ( x ) , ( u - u p ) ' ( x ) ~ b p T p ( x ) , (2.9)I (eM)P+IJ l I u ' - u ~ l I w ~ O [ _ 1..[ii (eM)PJl I u - u p l l w ~ O [ ...[ii 2p , 2p . (2.10)

Proof Actually, bms are the Chebyshev coefficients of u'. To see this, we start from theChebyshev expansion

00u'(x) = LCnTn(X),

n=Oand use the orthogonal property of to verify, for n # 0,

C = 11 u'(x)Tn(x) 2 ,! 2n - (u ,tPn+l) = - ( f , Pn+1) bn,rr -I rr rrand for n 0,

1 11 u'(x)To(x) dx I ! I 1 ,- (U, tPI)=-( f , tP l)+U(l) booCo = -; v'f=X2 rr rrIHence, the solution of (2.5) satisfies

p -Iu ~ ( x ) = LcnTn(X). (u up)'(x) = LCnTn(X).

n=O n?.pBy Theorem 2.2.3 in [13, p. 71], we have

u ( n ) ( ~ n )bn =Cn (2.11)for a suitable E I, I). Therefore, by condition M,

n, , rr 2 rr ( cM )2lIu -up '2 LCn:S '2 L 2n- 1n!n?:p n?.pMP )2( M2 M4 )2rr c - I + + + ...(2Pp! 24(p+l)2(p+2)2

( MP)2( 1 1 )< 4rrc2 - I + - + + ...- 2P p! 2 'Springer


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241

0

p + I > M /./2. The last estimate in (2.10) follows by using the Stirling's formula.st estimate in (2.10) can be obtained similarly from

00u(x) I),,1ftn+l(X)

n=O

bp1ftpH and bpTp are dominate terms in u - up andu ~ , respectively.

We refer the reader to [14] for another usage of the Petrov-Galerkin method in the

hev Spectral Collocationp + 1 Chebyshev points defined by (1.1). Here we seek wp E

HM-I, I] in the form(3.1)

ao = bo, such thatjr rx i =coS - . j = I, .... p 1. (3.2)p

p. 67] or [7, p. 104]) that when usingrr rra>o=wp= , Wj= j = 1, 2, . . . . p - I,2p P

pLg(Xj )Wjj=O

of (3.2) by Um(xj)(l - xJ)Wj and sum up over j 1,2, . . . ,1. Note that (1) it makes no difference with summing up over j = 0, L . . . , p here;W ~ ( X ) U m ( x ) ( 1 x2) E P2p- l when m :s p I and consequently the integral on the

m 1,2, . . . , p,p - l

- L W ~ ( X j ) U m - l ( X j ) ( l - x ; ) W j j= 1


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242 J Sci Comput (2008) 34: 237-246

I I dx= -m W ~ ( x ) T m ( X ) _I v I -x2-mil W ~ ( X ) ~ + I (x)dx = - m ( w ~ , ~ ' + I ) ' (3.3)

-I

On the other hand, the right-hand side is, for m = I, .. . , p - 2,p-IL ! (xJUm- I (X j ) ( l x;)Wjj= 1

p - IL!p(X j )Um-I(X j ) ( l -x ; )Wjj= 1

= II / J ( X ) U m _ l ( x ) ~ d x = -m(/J.m+t> (3.4)-I

andp-I L !(Xj)Up-2(Xj)(1 - X;)Wj - (p - I ) ( f l . p)*, (3.5) j= 1

where (' , .)* is the (p + I)-point Chebyshev-Lobatto quadrature rule defined above.We see that the collocation method, is equivalent to the following variational fonnulation:(V 1) Find wp E Pp[ -1,1], w p(- I ) 0, in the fonn (3.1) with ao boo such that

( W ~ , ~ ) (fI,k)*, k 2, . . . ,p. (3.6)In light of (3.1),

W ~ ( X ) = ao + a l TI (x ) + ... + lp_1 Tp - 1(x), (3.7)W ~ ( X ) aIUo(x) + 2a2UI(X) + ... + (p -1)a p- I Up- 2 (X). (3.8)

Substituting (3.7) into (3.6), we find, by (2.7), that2. ap-l - ( /J ,p) , (3.9)rr

2ak- I=- { /J ,k)=bk- l . k=2,3 , .. . ,p l. (3.10)rrConsider an auxiliary problem(V2) Find vp E PP[-I, 1], vp(-I) 0, in the fonn (3.1) withao = bo, such that

( V ~ , ~ ) ( /J ,d, k=2 , .. . ,p . (3.1 \)

We can see that( u ~ v ~ , D = ( f /J.k), k 2, . . . ,p. (3.12)

Springer


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up and wp are solutions of (VO) and (V 1), respectively. Therefore,Cp-l - ap-l

p. 104]

1g(x)-=d=x=-I v'I=X2[f/(X)Up_ 2(X) ( l -X 2)](2p)

C:)f?)[UP-2(l -X 2 )](PJ

E 1, 1). Here we have used the formula [11, p. 38]2(1-x 2)Up_2 (x) Tp(x) Tp_2 (x),

243

(3.13)

(3.14)

(3.15)

(3.16)

Tp(x) =2P- 1x P + .. '. Substituting (3.17) into (3.16) and (3.16) into (3.14), we2Pp!(p 2)' (3.18)

(3.19)

I I v ~ w ~ l I w ~ I c p - l - a p - d + h . o . t . ~ ~ ( ~ ~ r + l (3.20)of Pk in (3.12), we have

(u 1) ) '(x) /1 fX (u 1) )'(t)vp)'(x) P p dx = (f II )(x) P P dtdx. J l - x2 - I -1

v'p ( u ~

j (u 11 )'(t) II= 'Jrb ( f - 1I)(x)dxdt. (3.21)-1 1 - t2 tfd(s)ds.

Springer


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244 J Sci Comput (2008) 34: 237-246

Recall the interpolation error (1.4), we then have

I l u ~ v ~ I I 1 1 ! : S IIEpllw C (eM)P+l (3.22)..jP 2pIn light of Theorem 2.1, (3.20), and (3.22), we see that the difference u' is the dominantpart for the error u' - in the sense that

C( u - W p ) ' ( X ) ~ ( u - u p ) ' ( x ) = b p T p ( x ) + b p + 1 T p + l ( X ) + " ' ~ . . j P (eM)P .pHere we have used (2.11), Condition M, and the Stirling's formula to estimate bp Let Yj(2j - 1)n/(2p), j = 1, . .. , p, i.e., zeros of Tp, then

(u - wp)'(y) (u - up)'(Yj) bp+1Tp+1(Y ) + ... ~ ( e M ) P + l..jP 2pHence, The Chebyshev points y/s are derivative superconvergent points. Results in thissection can be summarized into the following.

Theorem 3.1 Let u and wp be solutions of (2.2) and (3.2), respectively. Assume that usatisfies Condition M and the non-degenerate condition: bp =I- 0. Then when p +1 > M/,,[2,we have(3.23)u - wp)'(x) ~ b p T p ( x ) , Ilu' - w ~ l I w o[)P ( ~ ; r l

Furthermore, the zeros of Tp are superconvergent points for u' - in the senseM)P+IJ j = 1,2, .. . , p. (3.24)u - o[Jp ( ~ p ,p)'(Yj) Yj =

4 Further Extension

Consider the two-point boundary value problem2_U " K U =0, u( - l ) =0, u'(1) = 1. (4.1)

The solution satisfies Condition M with M =K. We consider the case f =0 for simplicity.For collocation methods, the only difference with f =I- 0 is in numerical integration, whoseimpact is of higher order as we have shown.

The weak form is to find U E HM- I , I Jsuch that(U',V') K2(U, v) v(I), YVEHM-I,I].

The Chebyshev spectral collocation method of (4.1) is to find wp E Pp n HM-I, I] inthe form of (3.1) such that

- W ~ ( X j ) K2Wp(Xj) 0, x j =cos _J-, j = I , .. . , p 1, (4.2)pi l Springer


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(4.3)is from the Neumann boundary condition.

can show that (4.2) is equivalent to the following:(4.4)

(-, .)* is the same quadrature rule defined in Sect. 2. Let up be the Chebyshev expansionu of (4.1), it is straightforward to verify the following identity for

.. . ,p ,( w ~ - u ~ , ~ ) K2(W p -U p ,k)*

= (u' u ~ , D K2(U up , k)* K2[(u, k) (u, d*]= K2(U - up , k)* K2[(u, d (u, k)*]' (4.5)

/ E Pp interpolate u at p + 1 Chebyshev points X j, j 0,1, .. . , p, then

we can show that the differenceup is of higher order comparing with u up under Condition M and the non-degenerate

p. As a consequence, U - up is the dominant part of the errorp and similar error estimates as in Theorem 3.1 can be obtained for this case.

u( -1 ) = 0 = u(l) areIn this case, we are seeking for wp E Pp n HJ 1,1] in the form

of (3.1). Note that 1/1,. (l) = 0 for m ?: 2. The collocation is (4.2) without constraintt boundaryThe similar results have been proved for the Legendre spectral collocation method

The super-geometric error estimate for the one-dimensional wave equation has beenThe extension to the two dimensional setting is not straightforward due to the corner

of polygonal domains. However, when the solutions are analytic as for someof the one-dimensional results in this paper is fea

We interpret the collocation method at the Chebyshev points X j cos(jn!n) as ais almost equivalent to (up to some higher-order terms)

Springer


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246 J Sci Comput (2008) 34: 237-246a Petrov-Galerkin method via the Chebyshev numerical integration (which is exact for polynomials of degree::: 2p - I), rather than the Clenshaw-Curtis quadrature (which is exactfor polynomials of degree::: p) , [6, 17].Acknowledgement The author is grateful to anonymous referees for their constructive comments.

ReferencesI. Aisworth, M.: Discrete dispersion re lation for hp-version finite element approximat ion at high wave

numher. SIAM ]. Numer. Anal. 42(2), 553- 575 (2004)2. Bernardi, C., Maday, Y: Spectral Methods. In: Ciarlet, P.G., Lions, J.-L. (eds.) Handbook of NumericalAnalysis, vol. 5, pp. 209-485. North-Holland, Amsterdam (1997)3. Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd edn. Dover, New York (2001)4. Canuto, c., Hussaini, M.Y., Quarteroni, A., Zang, T.A . Spectral Methods in Fluid Dynamics. Springer,New York (1998)S. Canuto, C., Hussaini, M.Y, Quarteroni, A., Zang, T.A: Spectral Methods: Fundamentals in Single Do

mains. Springer, New York (2006)6. Clenshaw, C.W . Curtis, AR.: A method of numerical integration on an automatic computer. Numer.Math. 2. 197-205 (] 960)7. Davis, PJ., Rabinowitz, P.: Methods of Numericallntegr ation, 2nd edn. Academic Press, Boston (1984)8. Gottlieb, D . Orszag, T.A: Numerical Analysis of Spectral Methods: Theory and Applications. SIAM,

Philadelphia (1977)9. Hesthaven, 1.S., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time Dependent Problems. CambiidgeUniversity Press, London (2007)10. Karniadakis, G.E., Sherwin, S.L Spectrallhp Element Methods for CFD. Oxford University Press, New

York (1999)11. Mason, J.C., Handscomb, D.C.: Chebyshev Polynomials. Chapman and Hall/CRC, Boca Raton (2003)12. Peyret, R.: Spectral Methods for Incompressible Viscous Flow. Springer, Kew York (2002)13. Phillips, G.M.: Interpolation and Approximation by Polynomials. Springer, New York (2003)14. Shen, J.: A new dual-Petrov-Galerkin method for third and higher odd-order differential equations:application to the KDV equation. SIAM 1. Numer. Anal. 41(5), 1595-1619 (2003)15. Shen, 1, Tang, T.: Spectral and High-Order Methods with Applications. Science Press, Beijing (2006)16. Trefethen, L.N.: Spectral Methods in Matlab. SIAM, Philadelphia (2000)17. Trefethen, L.N.: Is Gauss quadrature better than Clenshaw-Curtis? SI AM Review (2007, to appear)18. Zhang, Z.: Superconvergence of spectral collocation and p-version methods in one dimensional problems. Math. Compo 74, 1621-1636 (2005)

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super convergence of a chebyshev spectral collocation method

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