ON THE CONVERGENCE OF THE FOURIER APPROXIMATION FOR EIGENVALUES AND EIGENFUNCTIONS OF DISCONTINUOUS PROBLEMS

Transcription

1 SIAM J. UMER. AAL. Vol. 40, o. 6, pp c 003 Society for Industrial and Applied Mathematics O THE COVERGECE OF THE FOURIER APPROXIMATIO FOR EIGEVALUES AD EIGEFUCTIOS OF DISCOTIUOUS PROBLEMS M. S. MI AD D. GOTTLIEB Abstract. In this paper, we consider a model eigenvalue problem with discontinuous coefficients in order to study the convergence of the Fourier methods applied to this problem. We prove that the rate of convergence of the Fourier Galerkin method is third order for the eigenvalues and order.5 for the eigenfunctions. For the Fourier collocation method we obtained only second order accuracy. We also show that the Fourier collocation method can be improved by a preprocessing of the coefficients. The theory is confirmed by numerical results. Key words. minmax principle discontinuous problems, Fourier Galerkin method, Fourier collocation method, AMS subject classifications. 6515, 655, 6535 PII. S Introduction. The paper is motivated by an issue arising in the use of spectral methods in nonlinear optics. The Fourier methods when applied to problems in nonlinear optics are extremely fast, and if the problem is smooth they provide high order accuracy. However, when different media are considered, the coefficients are only piecewise smooth and the accuracy is lost. In order to understand the phenomenon, and as a first step to improve the accuracy of the Fourier schemes in those circumstances, we consider in this paper a model eigenvalue problem with piecewise constant coefficients and study the convergence of the Fourier Galerkin and Fourier collocation methods to the eigenvalues and the eigenfunctions of this problem. The surprising fact is that the order of convergence of the eigenvalues obtained by the Fourier Galerkin method is cubic. When the Fourier collocation method is applied, the results are only second order. Those results are proven and supported by numerical computations. It turns out that, by preprocessing the discontinuous coefficients, one can improve the accuracy of the collocation method. In fact, if one uses the point values of the finite Fourier series of the coefficients instead of the point values of the coefficients themselves, one recovers third order accuracy for the eigenvalues and order.5 for the eigenfunction. The paper is organized as follows. In section, we present the problem and show some of the eigenvalues and eigenfunctions. In section 3, we rewrite the problem in its variational form and quote some relevant facts. In section 4, we discuss the Fourier Galerkin method and prove the order of accuracy. Section 5 is devoted to the Fourier collocation method and the error estimates of this method. In section 6, we show how to improve the accuracy of the collocation method. Received by the editors February 1, 00; accepted for publication (in revised form) June 4, 00; published electronically January 7, 003. This work was supported by AFOSR grant F , DOE grant DE-FG0-98ER5346, and SF grant DMS Division of Applied Mathematics, Brown University, Providence, RI 091 (msmin@cfm.brown. edu, dig@cfm.brown.edu). 54

2 DISCOTIUOUS EIGEVALUE PROBLEMS 55 We regard this paper as the first step toward recovering exponential accuracy for this problem.. The discontinuous eigenvalue problem. Consider the following eigenvalue problem with a piecewise constant coefficient: (.1) d u = λɛ(x)u for x (, π), dx where ɛ(x) =1in(, 0) and ɛ(x) =β in [0,π), β 1. The Hp[, π] eigenfunction u l (x) (the p stands for periodic) is given by (.) u l (x) = { C cos( λl x)+βd sin( λ l x), x 0, C cos(β λ l x)+d sin(β λ l x), 0 x π, where the constants C, D and the eigenvalue λ l are determined by the demand that the system C(cos λπ cos β λπ)+d( βsin λπ sin β λπ) =0, C(sin λπ + β sin β λπ)+d(βcos λπ β cos β λπ) =0 has a nontrivial solution. Considering the case β =, for y = cos λπ, the eigenvalues λ satisfy the equation (y 1)(9y +9y +)=0, and so there are families of eigenvalues determined by (.3) cos λπ =1, 1 3, or 3. The first five analytic eigenvalues (with six digits of precision) and the corresponding eigenvectors are shown in Figure 1. For comparison, we also carry the same procedure for β = 3, where the analytic eigenvalues are determined by (.4) cos λπ = ±1 or ± 1 4. In this paper, we examine the rate of convergence of the Fourier methods (Galerkin and collocation) as a first step in an effort to improve the rate of convergence and be able to also apply the Fourier methods for this discontinuous problem. 3. The variational formulation. We define two inner products: (3.1) a(u, v) = u (x)v (x)dx, (3.) (u, v) = u(x)v(x)ɛ(x)dx. Following Strang and Fix [7, p. 0], the eigenvalue problem (.1) can be presented in the following variational form: finding a scalar λ and a function u H 1 p[, π] such that (3.3) a(u, v) =λ(u, v)

3 56 M. S. MI AD D. GOTTLIEB λ 1 = λ = λ 3 = λ 4 = λ 5 = u Fig. 1. The first five analytic eigenvalues and eigenfunctions. for all v in the Hilbert space Hp[, 1 π]. ote that a(u, v) is Hermitian. Our proofs will use extensively the minmax principle [3]. Theorem 3.1. Let λ l denote the eigenvalues of (.1) and S l be any l-dimensional subspace of Hp[, 1 π]. Then, for λ 1 λ λ l..., (3.4) λ l = min max S l Hp 1 [,π] v S l. In this paper, we will use sharper characterizations of the eigenvalues. Lemma 3.. Let λ i be arranged in an ascending order and define E i,j = span{u i,...,u j }, where u i is the eigenfunction corresponding to the eigenvalue λ i. Then (3.5) (3.6) λ l = max v E k,l, k l, λ l = min v E l,m, l m. 4. Fourier Galerkin method. It is natural to consider the Fourier method to approximate the periodic problem. Here, we introduce the Fourier Galerkin method applied to the variational formulation for approximating the eigenvalues and eigenfunctions. Let P be the space of the trigonometric polynomials of degree / defined as (4.1) P = span{e ikx / k /}. In this subspace, we look for λ and u such that (4.) in other words, (4.3) a(u,v )=λ (u,v ) for all v P ; (u (x)) (v (x)) dx = λ u (x)v (x)ɛ(x)dx.

4 DISCOTIUOUS EIGEVALUE PROBLEMS umerical scheme and its results. The approximate eigenfunction u in the subspace P is expanded by (4.4) u = k= (û ) k e ikx. Substituting u into the variational formulation (4.) with v = e inx, and denoting the vector of the coefficients by û, we get a generalized eigenvalue problem in a matrix form as (4.5) Kû = λ Mû, where (4.6) K nk = (k n)e i(k n)x dx and M nk = e i(k n)x ɛ(x)dx. Solving the matrix eigenvalue problem (4.5) computationally using a proper eigensolver, we obtain the approximate lth eigenvalues, λ l (l ), and the set of orthogonal vectors û l =[(û l ) /,...,(û l ) / ] T which is used to approximate the lth eigenfunction u l as a finite Fourier series u l. In Tables 1 and, the orders of the relative errors for λ l λ l and the discrete L -errors of u l u l are provided for the first five eigenvalues in ascending order and for the associated eigenfunctions. We note the surprising fact that the Galerkin approximation to the eigenvalue problem (.1) converges with third order accuracy for the eigenvalues and order.5 for the eigenfunctions even though the eigenfunctions are only in Hp. In fact, we will show in Lemma 4.5 that the eigenfunctions are in H 5 ɛ p for any ɛ> Error estimates for eigenvalues and eigenfunctions. In this section, we provide the error estimates for the approximate eigenvalues and eigenfunctions for the Fourier Galerkin method. We first treat the approximate eigenvalues. Let P u be the th order truncated Fourier series of u. (We will denote also P = P.) It is clear that it satisfies (4.7) a(u P u, v ) = 0 for all v P. It is true that the minmax principle is also valid for the Galerkin procedure: (4.8) λ l = min max S l P v S l. Lemma 4.1. Let λ l procedure. Then be the approximation to λ l which is obtained by the Galerkin λ l λ l λ l max v E 1,l (Pv,Pv). Proof. Due to the minmax principle (3.4) and (4.8), we have λ l = min max S l Hp 1 [,π] v S l min S l P max v S l = λ l.

5 58 M. S. MI AD D. GOTTLIEB Table 1 The relative errors of eigenvalues for the case β =and the discrete L -errors of u i u i the Fourier Galerkin method. for λ i λ (λ i λ i) i λ i Order u i u i l Order (-4) (-4) (-5) (-5) (-6) (-5) (-7) (-6) (-8) (-7) (-5) (-4) (-5) (-4) (-6) (-5) (-7) (-6) (-8) (-7) (-4) (-3) (-5) (-4) (-6) (-5) (-7) (-6) (-8) (-6) (-4).4795(-3) (-5) (-4) (-5) (-5) (-6) (-5) (-7) (-6) (-4) 3.493(-3) (-6) (-4) (-7) (-5) (-9) (-6) (-10) (-7) 3.48 Table The relative errors of eigenvalues for the case β =3and the discrete L -errors of u i u i the Fourier Galerkin method. for λ i λ (λ i λ i) i λ i Order u i u i l Order (-4) (-4) (-5) (-4) (-6) (-5) (-7) (-6) (-8) (-6) (-4).137(-3) (-5) (-4) (-6) (-5) (-7) (-5) (-7) (-6) (-5).1107(-3) (-6) (-4) (-8) (-5) (-9) (-5) (-11) (-6) (-3).6579(-) (-4) (-3) (-5) (-4) (-6) (-5) (-7) (-5) (-3) (-) (-4) (-3) (-5) (-4) (-6) (-5) (-7) (-5).4430

6 DISCOTIUOUS EIGEVALUE PROBLEMS 59 Thus only the upper bound of the approximate eigenvalue is left to be investigated. Let PE 1,l be spanned by Pu 1,...,Pu l. Then it is clear that PE 1,l is the l- dimensional subspace of P. Using the minmax principle (4.8), (4.9) λ l max v PE 1,l = max v E 1,l a(pv,pv) (Pv,Pv). ote that in [7] we have (4.10) =a(pv,pv)+a(v Pv,Pv)+a(v Pv,v Pv). From (4.7), we know that a(v Pv,Pv) always vanishes for all Pv in the space P. Then we have a(pv,pv). Thus, λ l max v E 1,l (Pv,Pv) = max v E1, l (Pv,Pv) λ l max v E 1,l (Pv,Pv). The last inequality is a by-product of (3.5). Thus the lemma is proven. The issue is how close (P v, P v)isto for v E 1,l. One would expect the second order accuracy in because of the smoothness of the eigenfunctions u i. However, we will show that it is really third order. We start by examining the Fourier coefficients of the eigenfunctions. Lemma 4.. The Fourier coefficients (û l ) k of the eigenfunction u l decay as O(k 3 ); in fact, (4.11) { (û l ) k Ck 3 u l (0) + 1 k u l(0) + λ } l k u l, where u l is the L -norm of u l. Proof. Letting u l = u for simplicity, and using the fact that u (4.1) û k = 1 π is continuous, ue ikx dx = 1 u e ikx dx = 1 πik πk u e ikx dx. Substituting u = λɛu into (4.1) and, for convenience, using the notation µ = β 1, we have û k = 1 πk = λ πik 3 = λ πik 3 λɛue ikx dx = λ ( 0 πk ( µ{( 1) k u(π) u(0)} + 0 ue ikx dx + β u e ikx dx + β ( µ[( 1) k u(π) u(0)] + µ ik [( 1)k u (π) u (0)] + 1 ik Therefore, the lemma is proven. We are ready now for the next lemma. Lemma 4.3. max v E1,l (Pv,Pv) 1+Cl 3, where the constant C is independent of and l. 0 0 ) ue ikx dx ) u e ikx dx ) λɛue ikx dx.

7 60 M. S. MI AD D. GOTTLIEB Proof. We first note that (4.13) (Pv,Pv) = 1 1 (v,v) (P v,p v) (v,v). Since v is in E 1,l, it can be represented by v = l i=1 α iu i. We also have Thus we get (Pv,Pv)= (v Pv,v) (v Pv,v)+(v Pv,v Pv). (Pv,Pv) (v Pv,v) = l i,j=1 α i ᾱ j (u i Pu i,u j ) ( l i=1 α i ) l max (u i Pu i,u j ). i,j=1,...,l owwehave (u i Pu i,u j ) = ˆ (u i ) k e ikx, (uˆ j ) n e inx k > n= = ˆ (u i ) ˆ k (u j ) n e i(k n)x ɛ(x)dx k > n= (uˆ i ) k (u ˆ (β 1) (( 1) k n 1) j ) n k n k > + k > n= n k (uˆ i ) k (u ˆ j ) n (β +1)π. n= n=k Recalling (4.11), where û k decays like O(k 3 ) at least, we get (u i Pu i,u j ) C 3, where C is a positive constant. Finally, we have (Pv,Pv) = 1 1 (v,v) (P v,p v) (v,v) (Pv,Pv) 1+ 1+Cl 3. Thus the lemma is proven. We can now state the following theory. Theorem 4.4. Let λ l be the Fourier Galerkin approximation to the eigenvalue λ l. Then λ l λ l Clλ l 3,

8 DISCOTIUOUS EIGEVALUE PROBLEMS 61 where the constant C depends only on the values of u i (0), u i (0), and the L -norm of u i for all i l. ow we are ready to treat the eigenvectors. Following Strang and Fix [7, p. 34], we can state u l u l u l P u l, where P u l is the finite Fourier series of u l. following estimate. Lemma 4.5. For the right-hand side we have the u l P u l C.5. Proof. By the Parseval equality, and using Lemma 4., which states the Fourier coefficients of u l decay cubicly, we get u l P u l = (û l ) k 1 k > C k 6 1 k > C.5. Thus the lemma is proven. We can therefore conclude the following theorem. Theorem 4.6. Let u l be the lth eigenfunction, and let u l Fourier Galerkin approximation (4.); then be the solution of the (4.14) u l u l C.5. The numerical results presented in Tables 1 and conform to the theory. 5. Fourier collocation method. Let I be the space of the trigonometric polynomial of degree /, defined as (5.1) I = span{(cos(kx) 0 k /) (sin(kx) 1 k / 1)}. For an even integer >0, we consider the set of points (5.) x j = + πj, j =0,...,. The discrete approximations of the inner products (3.1) and (3.) are defined by (5.3) (5.4) a(u, v) h = π (u, v) h = π 1 1 u (x j )v (x j ), u(x j )v(x j )ɛ(x j ).

9 6 M. S. MI AD D. GOTTLIEB Alternatively, defining c j =1,0 j, and c = c 0 =, we can redefine (5.5) (5.6) a(u, v) h = π (u, v) h = π u (x j )v (x j ) 1, c j u(x j )v(x j )ɛ(x j ) 1. c j Remark 5.1. ote that the bilinear form a(u, v) h coincides with the inner product a(u, v) for trigonometrical polynomials of the right order: (5.7) a(u, v) h = a(u, v) for all u, v I. This is a result of the exactness of the quadrature formula if u v is up to degree 1 [1]. One can observe that the highest degree for u v is obtained when choosing cos( x) for both u and v. However, {cos( x)} = sin( x) and sin( x) vanishes at the grid points x j so that the quadrature formula still remains valid also for the case of highest degree. Thus (5.7) is true for any u, v in I. Remark 5.. Equation (5.6) can be rewritten as h = π ( v(x 0) ɛ(x 0 )+ v(x + π 1 v(x j ) ɛ(x j )+ +1 ) ɛ(x )) + π 1 j=1 v(x j ) ɛ(x j ) π ( v(x ) ɛ(x )+ v(x ) ɛ(x )). The first two terms can be identified as the trapezoidal rule [6] for 0 v(ξ) ɛ(ξ)dξ, whereas the other two terms are the same rule for 0 v(ξ) ɛ(ξ)dξ. We can therefore state { } (5.8) h C max max x<0 ( v ɛ), max 0 x π ( v ɛ) umerical scheme and its results. The collocation methods can be defined as finding λ c and u c I such that (5.9) a(u c,v c ) h = λ c (u c,v c ) h for all v c I. There are several ways to realize the abstract definition of the collocation methods, and we will quote one of them: u c can be presented using the Lagrange trigonometrical polynomials as interpolation polynomials [4] as follows: (5.10) where (5.11) u c = 1 u c (x j )l j (x), l j (x) = 1 [ sin (x x ] j) cot [ x xj ].

10 Taking v c (x) =l n (x), we have DISCOTIUOUS EIGEVALUE PROBLEMS 63 a(u, v) h = π = π = π 1 i= u c (x j )l j(x i )l n(x i ) 1 u c (x j ) i=0 u c (x j )D nj, l j(x i )l n(x i ) where D = D D and D is the first order differentiation matrix for even grid points [], [4], [5]. Also, (u, v) h = π = π = π 1 i= u c (x j )l j (x i )l n (x i )ɛ(x i ) 1 u c (x j ) i=0 u c (x j )A jn, l j (x i )l n (x i )ɛ(x i ) where A = diag{ɛ(x 0 ),...,ɛ(x 1 )}. Then we solve the matrix equation (5.1) D u c = λ c Au c to get the approximate eigenvalues λ c and eigenfunctions u c =[u c (x 0 ),...,u c (x 1 )] T. Remark 5.3. In order to make (5.1) compatible with definition (5.6), we should replace ɛ(x 0 ) by the average ɛ(x0)+ɛ(x ). The variational formulation with odd grids can be obtained in a similar way. In Tables 3 and 4, we present the relative error for λ c l λ l. It is clear that we see second order accuracy with even grids as well as with odd grids as increases. The discrete L -error of u l u c l converges with second order accuracy with even and odd grids as increases. 5.. Error estimates for eigenvalues and eigenfunctions. Here we provide error estimates for the approximate eigenvalues and eigenfunctions of the Fourier collocation method. We first consider the approximate eigenvalues. Let S l be any l-dimensional subspace of I. Lemma 5.1. Let λ c l be the approximation to λ l obtained by the collocation procedure, and let S l be any l-dimensional subspace of I. Then (5.13) λ c h l max. v S l h Proof. The space S l is spanned by the eigenfunctions u c k 1,...,u c k l for k 1 < < k l. It follows that k l l. ow (5.14) a(u c k l,u c k l ) h (u c k l,u c k l ) h = λ c k l λ c l.

11 64 M. S. MI AD D. GOTTLIEB Table 3 Fourier collocation with even grids: The relative errors of eigenvalues for the case β =and the discrete L -errors of u i u c i. λ i λ c (λ c i λ i) i λ i Order u i u c i l Order (-3) (-3) (-3) (-3) (-4) (-4) (-4) (-5) (-5) (-5) (-) 7.396(-3) (-3) (-3) (-4) (-4) (-4) (-4) (-5) (-5) (-) (-) (-3) (-3) (-3) (-4) (-4) (-4) (-5) (-5) (-) (-) (-3) (-3) (-3) (-3) (-4) (-4) (-5) (-5) (-3) (-) (-5) (-) (-7) (-3) (-8) (-4) (-10) (-4).001 Table 4 Fourier collocation with odd grids: The relative errors of eigenvalues for the case β =and the discrete L -errors of u i u c i. λ i λ c (λ c i λ i) i λ i Order u i u c i l Order (-3) (-) (-4) (-3) (-4) (-4) (-5) (-4) (-6) (-5) (-3) (-3) (-4) (-3) (-4) (-4) (-5) (-4) (-5) (-5) (-3) 1.675(-) (-3) (-3) (-4) (-3) (-5) (-4) (-5) (-5) (-3) (-) (-3) (-3) (-4) (-3) (-5) (-4) (-5) (-4)

12 DISCOTIUOUS EIGEVALUE PROBLEMS 65 This concludes the proof. We are now ready to estimate λ c l Lemma 5.. from above. (5.15) λ c l λ l (1 + C ), where C is independent of (but may depend linearly on l). Proof. Let J (= J) be the orthogonal projection (in the usual L sense) of H to I. Let S l = JE 1,l = span{ju 1,...,Ju l } in Lemma 5.1 to get Then λ c l h a(jv,jv) h max = max. v JE 1,l h v E 1,l (Jv,Jv) h (5.16) λ c l max v E 1,l a(jv,jv) h (Jv,Jv) (Jv,Jv) (Jv,Jv) h. From Lemma 3., we have max v E 1,l = λ l. Also from the exactness of the trapezoidal rule and the fact that J is an orthogonal projection, it is true that a(jv,jv) h = a(jv,jv). Due to Lemma 4.3 (with the same proof for J replacing P ), we have Also, Remark 5. gives max v E 1,l (Jv,Jv) 1+Cl 3. (Jv,Jv) (Jv,Jv) h 1+C, and so the lemma is proven. We will now try to get a lower bound for λ c l. Define Ec 1,l = span{uc 1,...,u c l }. From the minmax theorem, we have It is also clear that λ l max v E c 1,l. λ c h l = max. v E c 1,l h We can now state the following lemma. Lemma 5.3. (5.17) λ l λ c l (1 + C ),

13 66 M. S. MI AD D. GOTTLIEB where C is independent of (but may depend linearly on l). Proof. We start from λ l max v E c 1,l h = max h v E c 1,l h h. Since v I, we have = h. estimate Also, because of the trapezoidal rule h C, and therefore the lemma is proven. We can now conclude the following theorem. Theorem 5.4. Let λ c l be the approximation to λ l obtained by the collocation procedure. Then (5.18) λ c l λ l Cλ l, where C is independent of. We now turn to the eigenfunctions. The set u c 1,u c,...,u c forms an orthogonal basis for I. Then we can express the orthogonal projection Ju l of u l into the subspace I as the following: (5.19) Ju l = (Ju l,u c j) h u c j. j=1 By subtracting the following variational formulations, we have λ l (u l,u c j)=a(u l,u c j), λ c j(ju l,u c j) h = a(ju l,u c j) h = a(ju l,u c j), (λ c j λ l )(Ju l,u c j) h = a(ju l,u c j) a(u l,u c j) λ l [(Ju l,u c j) h (u l,u c j)] = a(u l Ju l,u c j)+λ l [(u l,u c j) (Ju l,u c j) h ]. Since a(u l Ju l,u c j )=0,wehave (Ju l,u c j) h λ l λ c j λ l (u l,u c j) (Ju l,u c j) h λ l λ c j λ l { (u l,u c j) (u l,u c j) h + (u l,u c j) h (Ju l,u c j) h }. From the Schwarz inequality and (u c j,uc j ) h =1,wehave (u l Ju l,u c j) h u l Ju l h u c j h C,

14 DISCOTIUOUS EIGEVALUE PROBLEMS 67 where u h = (u, u) h. Using the trapezoidal rule as in (5.8), we have (u l,u c j) (u l,u c j) h C. Then, following [7], Ju l βu c l h = (Ju l,u c j ) h ρc, j=1 j l where β = (Ju l,u c l ) h and ρ is a separation constant for the eigenvalues as in [7, pp ]. From (5.8) and u l = u c l h =1, Then, following [7], we have u l h u l + C = u c l h + C. u l u c l h u l Ju l h + Ju l βu c l h + βu c l u c l h C. We can therefore conclude the following theorem. Theorem 5.5. Let u l be the lth eigenfunction, and let u c l Fourier collocation approximation (5.9); then be the solution of the u l u c l h C. Due to the equivalence of the norms in finite space, the discrete L -error of u l u c l also converges with O( ). 6. Accuracy enhancement for the collocation method. A simple trick can be used in order to enhance the accuracy of the Fourier collocation method. We expand the discontinuous coefficient function ɛ(x) in the finite Fourier series represented by ɛ (x) = (ˆɛ ) k e ikx, k= where the Fourier coefficients are defined as (ˆɛ ) k = 1 ɛ(x)e ikx dx. π ow, instead of (5.4), defining (6.1) (u, v) h = π 1 u(x j )v(x j )ɛ (x j ) in the variational formulation (5.9), we have the scheme as follows: (6.) D u c = λ c Au c, where A = diag{ɛ (x 0 ),...,ɛ (x 1 )} and D is the same as defined in (5.1). The numerical results are presented in the Tables 5 and 6 for even and odd grids, respectively. The accuracy is now the same accuracy as for the Galerkin method! An analysis for this will appear in a future paper.

15 68 M. S. MI AD D. GOTTLIEB Table 5 The accuracy enhancement for collocation with even grids: The relative errors of eigenvalues for the case β =and the discrete L -errors of u i u c i using ɛ (x). λ i λ c (λ c i λ i) i λ i Order u i u c i l Order (-4) (-4) (-5) (-4) (-6) (-5) (-7) (-6) (-8) (-7) (-6) (-4) (-6) (-4) (-6) (-5) (-7) (-6) (-8) (-7) (-4) (-3) (-5) (-4) (-6) (-5) (-7) (-5) (-8) (-6) (-3) (-3) (-4) (-4) (-5) (-4) (-6) (-5) (-7) (-6) (-) 3.983(-) (-4) (-3) (-5) (-4) (-7) (-5) (-8) (-6) Table 6 The accuracy enhancement for collocation with odd grids: The relative errors of eigenvalues for the case β =and the discrete L -errors of u i u c i using ɛ (x). λ i λ c (λ c i λ i) i λ i Order u i u c i l Order (-4) (-4) (-5) (-4) (-6) (-5) (-7) (-6) (-8) (-7) (-5) (-4) (-6) (-4) (-6) (-5) (-7) (-6) (-8) (-7) (-5) (-3) (-5) (-4) (-6) (-5) (-7) (-5) (-8) (-6) (-3) (-3) (-4) (-4) (-5) (-4) (-6) (-5) (-7) (-6).4930

16 DISCOTIUOUS EIGEVALUE PROBLEMS 69 Acknowledgments. The authors would like to thank Bertil Gustafsson, Seymour Parter, and Wai-Sun Don for several useful suggestions and discussions regarding this paper. REFERECES [1] C. Canuto, M. Y. Hussaini, A. Quarteroni, andt. A. Zang, Spectral Methods in Fluid Dynamics, Springer Ser. Comput. Phys., Springer-Verlag, ew York, [] D. Funaro, Polynomial Approximation of Differential Equations, Springer-Verlag, ew York, [3] G. H. Golub andc. Van Loan, Matrix Computations, The Johns Hopkins University Press, Baltimore, MD, [4] D. Gottlieb, M. Y. Hussaini, ands. A. Orszag, Theory and Applications of Spectral Methods, in Spectral Methods for Partial Differential Equations, R. Voigt, D. Gottlieb, and M.Y. Hussaini, eds., SIAM, Philadelphia, 1984, pp [5] D. Gottlieb ands. A. Orszag, umerical Analysis of Spectral Methods: Theory and Application, CMBS-SF Regional Conf. Ser. Appl. Math. 6, SIAM, Philadelphia, [6] E. Isaacson andh. B. Keller, Analysis of umerical Methods, John Wiley & Sons, ew York, [7] G. Strang andg. Fix, An Analysis of The Finite Element Method, Prentice Hall, Englewood Cliffs, J, 1973.