Superconvergence of Legendre projection methods for the eigenvalue problem of a compact integral operator

Journal of Computational and Applied Mathematics 235 (2011) 2380–2391

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Superconvergence of Legendre projection methods for the eigenvalue problem of a compact integral operator Bijaya Laxmi Panigrahi, Gnaneshwar Nelakanti ∗ Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-721302, India

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Article history: Received 4 June 2010 Received in revised form 22 October 2010 Keywords: Eigenvalues Eigenvectors Legendre polynomials Compact operator Superconvergence rates Integral equations

In this paper, we consider the Galerkin and collocation methods for the eigenvalue problem of a compact integral operator with a smooth kernel using the Legendre polynomials of degree ≤ n. We prove that the error bounds for eigenvalues are of the order O (n−2r ) and the gap between the spectral subspaces are of the orders O (n−r ) in L2 -norm and O (n1/2−r ) in the infinity norm, where r denotes the smoothness of the kernel. By iterating the eigenvectors we show that the iterated eigenvectors converge with the orders of convergence O (n−2r ) in both L2 -norm and infinity norm. We illustrate our results with numerical examples. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Let X be a Banach space with a norm ‖ · ‖. We are interested in the approximation of eigenfunctions (eigenvalues and eigenvectors) of the following eigenvalue problem: find u ∈ X and λ ∈ C \ {0} such that

K u = λu,

‖u‖ = 1,

(1.1)

where K : X → X is a compact linear integral operator with a smooth kernel. Many practical problems in science and engineering are formulated as eigenvalue problems (1.1) of compact linear integral operators K [1]. For many years, numerical solutions of eigenvalue problems have attracted much attention. The Galerkin, petrov-Galerkin, collocation, Nyström and degenerate kernel methods are the commonly used methods for the approximation of eigenfunctions of the compact integral operator K . The analysis for the convergence of the Galerkin, petrov-Galerkin, collocation, Nyström and degenerate kernel methods are well documented in [2,1,3,4]. Let Xn be a sequence of finite dimensional subspaces of X and Pn be either orthogonal or interpolatory bounded projections from X into Xn . Then either in Galerkin (orthogonal projection) or in collocation methods (interpolatory projection), the eigenvalue problem (1.1) is approximated by the following problem: find un ∈ Xn and λn ∈ C \ {0} such that

Pn K un = λn un ,

‖un ‖ = 1.

(1.2) 1

The iterated eigenvector is defined by u˜ n = λ K un . n Let Xn be a sequence of piecewise polynomial subspaces of X of degree ≤ r − 1 on [−1, 1] and Pn be either orthogonal or interpolatory bounded projections from X into Xn . Then in both the Galerkin and collocation methods, under the suitable conditions on the kernel, it is known that the orders of convergence for the eigenvalues and the iterated eigenvectors are of the orders O (h2r ) and the gap between the spectral subspaces is of the order O (hr ), where h denotes the norm of the

∗

Corresponding author. Tel.: +91 03222 283656. E-mail addresses: [email protected] (B.L. Panigrahi), [email protected], [email protected] (G. Nelakanti).

0377-0427/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2010.10.038

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partition (see [2,1,3,4]). However, using piecewise polynomials as basis functions, the computation of a sufficiently accurate eigenvalues and eigenvectors may require the use of much finer partition of the domain [−1, 1] and hence the size of the matrix eigenvalue problem corresponding to the eigenvalue (1.2) becomes very large as the norm of the partition h becomes smaller. On the other hand, it is well known that the matrices in these matrix eigenvalue problems are dense, and it requires huge computation to generate these matrices. In this paper, we consider the Galerkin and collocation methods for the eigenvalue problem (1.1) using global polynomial basis functions rather than piecewise polynomial basis functions. Obviously, low degree polynomials imply small matrix eigenvalue problems, something which are highly desirable in practical computations. However, if Pn denotes either orthogonal or interpolatory projection from X into a subspace Xn of global polynomials of degree ≤ n, then ‖Pn ‖∞ is unbounded. Therefore, there exists at least one u ∈ C ([−1, 1]) such that ‖Pn u − u‖∞ 9 0 as n → ∞. It is the purpose of this paper to obtain similar superconvergence rates for the eigenvalues and eigenvectors using global polynomial bases as in the case of piecewise polynomial bases. We use the Legendre polynomials as basis functions to find the approximate eigenvalues and eigenvectors. The Legendre polynomial bases are easy to generate recursively and they have nice property of orthogonality. Hence the Legendre basis functions are less expensive computationally in solving the matrix eigenvalue problem corresponding to the eigenvalue problem (1.2) in comparison to piecewise polynomial basis functions. By choosing Xn as a sequence of the Legendre polynomial subspaces of X of degree ≤ n, and Pn either orthogonal or interpolatory projection operator from X into Xn , we obtain similar superconvergence rates for the eigenvalues and eigenvectors as in the case of piecewise polynomial basis functions. In fact we show that, in both the Legendre–Galerkin and Legendre collocation methods the eigenvalues converge with the order of convergence O (n−2r ) and the gap between the spectral subspaces are of the orders O (n−r ) in L2 -norm and O (n1/2−r ) in the infinity norm. Further we show that in both the Legendre–Galerkin and Legendre collocation methods the iterated eigenvector converges with the order of convergence O (n−2r ) in both L2 -norm and infinity norm, where r denotes the smoothness of the kernel. In Section 2 we set up a theoretical framework for the eigenvalue problem using global polynomial bases. In Section 3 we apply the theory presented in Section 2 to the Legendre–Galerkin methods and the Legendre collocation methods to obtain superconvergence rates. In Section 4 we present numerical results. Throughout the paper we assume that c is a generic constant. 2. An abstract framework In this section, we present an abstract framework of projection methods for solving eigenvalue problems for a compact linear integral operator using the polynomial bases. Consider the following integral operator defined on X = L2 ([−1, 1]) or C ([−1, 1]) by

K u(t ) =

1

∫

K (t , s)u(s) ds,

s ∈ [−1, 1],

(2.1)

−1

where kernel K (., .) ∈ C ([−1, 1] × [−1, 1]). Then K : X → X is a compact operator. Let BL(X) denote the space of all bounded linear operators from X into X. The resolvent set of K is given by

ρ(K ) = {z ∈ C : (K − zI )−1 ∈ BL(X)}. The spectrum of K , σ (K ) is defined as σ (K ) = C \ ρ(K ). We are interested in the following eigenvalue problem: find u ∈ X and λ ∈ C \ {0} such that

K u = λ u,

‖u‖ = 1.

(2.2)

Assume λ be the eigenvalue of K with algebraic multiplicity m and ascent ℓ. Let Γ ⊂ ρ(K ) be a simple closed rectifiable curve such that σ (K ) ∩ intΓ = {λ}, 0 ̸∈ intΓ , where intΓ denotes the interior of Γ . Now we describe the projection methods for the eigenvalue problem (2.2) using the Legendre polynomial bases. To do this, ‖u‖L2 =



1

2 −1 |u(t )| dt

1/2

for u ∈ L2 ([−1, 1]). Let Xn denote the sequence of the Legendre polynomial subspaces of

X of degree ≤ n, and Pn be the projection operator from X into Xn satisfying the following conditions: (C1) The set of operators {Pn : n ∈ N} is uniformly bounded in L2 -norm, i.e., there exists a positive constant M independent of n such that ‖Pn ‖L2 ≤ M, for all n ∈ N. (C2) Operators Pn converge pointwise to I on X, i.e., for any u ∈ X, ‖Pn u − u‖L2 → 0 as n → ∞. Further, we assume that for any u ∈ C r [−1, 1], the space of r times continuously differentiable functions on [−1, 1], the following error bounds hold.

‖u − Pn u‖L2 ≤ cn−r ‖u(r ) ‖L2 ,

(2.3)

‖u − Pn u‖∞ ≤ cnβ−r ‖u(r ) ‖L2 ,

(2.4)

where c is a constant independent of n and β < r. From assumption (2.4), we notice that ‖Pn u − u‖∞ 9 0, as n → ∞ for any u ∈ C ([−1, 1]).

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Now the projection method for the eigenvalue problem (2.2) is defined as follows: find un ∈ Xn and λn ∈ C \ {0}, such that

Pn K un = λn un ,

‖un ‖ = 1.

(2.5)

Next we discuss the existence and error bounds for the approximated eigenfunctions. To do this, we set the following notations. For any function u ∈ C r [−1, 1], denote ‖u‖r ,∞ = max{‖u(j) ‖∞ : 0 ≤ j ≤ r }, where u(j) denotes the jth derivative of u. If K (., .) ∈ C r ([−1, 1] × [−1, 1]), then R(K ) ∈ C r [−1, 1] and we denote K (i,j) (t , s) =

∂ i+j K ( t , s) ∂ t i ∂ sj

and

‖K ‖r ,∞ = max{‖K (i,j) (., .)‖∞ : 0 ≤ i ≤ r , 0 ≤ j ≤ r }. Set Kt (s) = K (t , s), for s, t ∈ [−1, 1]. Note that for K (t , .) ∈ C j ([−1, 1]), j = 0, 1, 2, . . . , r and u ∈ C [−1, 1], we have

  ∂j K (t , s)u(s) ds j −1 ∂ t  j ∫ 1 ∂   ≤ max  j K (t , s) |u(s)| ds s,t ∈[−1,1] ∂ t −1 √ ≤ 2‖K ‖j,∞ ‖u‖L2 ≤ 2‖K ‖j,∞ ‖u‖∞ .

∫  |(K u) (t )| =  (j)

1

From this, it follows that for j = 0, 1, 2 . . . , r,

‖(K u)(j) ‖∞ ≤

√

‖(K u)(j) ‖L2 ≤

√

2‖K ‖j,∞ ‖u‖L2 ≤ 2‖K ‖j,∞ ‖u‖∞ ,

(2.6)

and

√

2‖(K u)(j) ‖∞ ≤ 2‖K ‖j,∞ ‖u‖L2 ≤ 2 2‖K ‖j,∞ ‖u‖∞ .

(2.7)

First we prove the existence of approximated eigenfunctions. To this end, we first show that the projection operator Pn K converges to K in both L2 -norm and infinity norm. Theorem 2.1. Pn K is norm-convergent to K both in L2 -norm and infinity norm. Proof. Using the estimates (2.3), (2.4) and (2.7), we see that

‖(I − Pn )K u‖L2 ≤ cn−r ‖(K u)(r ) ‖L2 ≤ cn−r ‖K ‖r ,∞ ‖u‖L2 , and

‖(I − Pn )K u‖∞ ≤ cnβ−r ‖(K u)r ‖L2 ≤ cnβ−r ‖K ‖r ,∞ ‖u‖∞ . From these estimates, it follows that,

‖K − Pn K ‖L2 → 0 and ‖K − Pn K ‖∞ → 0 for r > β as n → ∞. This completes the proof.



Since Pn K is norm-convergent to K in both L2 -norm and infinity norm, the spectrum of Pn K inside Γ consists of m eigenvalues say λn,1 , λn,2 , . . . , λn,m counted accordingly to their algebraic multiplicities inside Γ (see, [1,5]). Let

λˆ n =

λn,1 + λn,2 + · · · + λn,m m

,

ˆ n . Let denote their arithmetic mean and we approximate λ by λ S

P =− PnS = −

1

∫

2π i Γ ∫ 1 2π i

Γ

(K − z I)−1 dz , (Pn K − z I)−1 dz ,

be the spectral projections of K and Pn K , respectively, associated with their corresponding spectra inside Γ . To discuss the closeness of eigenfunctions of the original operator K and those of the approximate operator Pn K , we recall the concept of the gap between the spectral subspaces. For nonzero closed subspaces Y1 and Y2 of X, and for p = 2 or ∞, let

δp (Y1 , Y2 ) = sup{distp (y, Y2 ) : y ∈ Y1 , ‖y‖p = 1},

(2.8)

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then

δˆp (Y1 , Y2 ) = max{δp (Y1 , Y2 ), δp (Y2 , Y1 )}

(2.9)

denotes the gap between the spectral subspaces in L2 norm (p = 2) and infinity norm (p = ∞). We quote the following lemmas, which give the error bounds for the eigenfunctions. Lemma 2.2 ([3]). If X is a Banach space, K , Pn K ∈ BL(X) for n ∈ N and Pn K is norm-convergent to K in L2 -norm, then for sufficiently large n, there exists a constant c independent of n such that for j = 1, 2, . . . , m,

|λ − λˆ n | ≤ c ‖(KPn − K )K ‖L2 , |λ − λn,j |ℓ ≤ c ‖(KPn − K )K ‖L2 . Lemma 2.3 ([3]). Let X be a Banach space, K , Pn K ∈ BL(X) for n ∈ N and let Pn K be norm-converges to K both in L2 and infinity norm. Let R(P S ) and R(PnS ) be the ranges of the spectral projections P S and PnS , respectively. Then for sufficiently large n, there exists a constant c independent of n such that for j = 1, 2, . . . , m,

δˆ2 (R(P S ), R(PnS )) ≤ c ‖(Pn K − K )K ‖L2 , δˆ∞ (R(P S ), R(PnS )) ≤ c ‖(Pn K − K )K ‖∞ . In particular, for any un ∈ R(PnS ), we have

‖un − P S un ‖L2 ≤ c ‖(Pn K − K )K ‖L2 ‖un − P S un ‖∞ ≤ c ‖(Pn K − K )K ‖∞ . In the following lemma we give the error bounds for the iterated eigenvectors u˜ n = K un , un ∈ R(PnS ). Lemma 2.4 ([3,4]). Let X be a Banach space and let K , Pn K ∈ BL(X) with Pn K is norm-convergent to K in both L2 and infinity norm. Let R(P S ) and R(PnS ) be the ranges of the spectral projections P S and PnS , respectively. Then for sufficiently large n, there exists a constant c independent of n such that,

δ2 (K R(PnS ), R(P S )) ≤ c ‖(KPn − K )K ‖L2 , δ∞ (K R(PnS ), R(P S )) ≤ c ‖(KPn − K )K ‖∞ . In particular, for any un ∈ R(PnS ), u˜ n = K un , we have

‖˜un − P S u˜ n ‖L2 = ‖K un − P S K un ‖L2 ≤ c ‖(KPn − K )K ‖L2 ‖˜un − P S u˜ n ‖∞ = ‖K un − P S K un ‖∞ ≤ c ‖(KPn − K )K ‖∞ . 3. Legendre projection methods In this section, we apply the abstract framework developed in the last section to the Legendre–Galerkin and Legendre collocation methods for the eigenvalue problem (2.2) and obtain the superconvergence rates for the eigenfunctions. To do this, we choose the Legendre polynomials {ψ0 , ψ1 , ψ2 , . . . , ψn } as an orthonormal basis for the subspace Xn , where

ψ0 (x) = 1,

ψ1 (x) = x,

x ∈ [−1, 1],

and for i = 1, 2, . . . , n − 1,

(i + 1)ψi+1 (x) = (2i + 1)xψi (x) − iψi−1 (x),

x ∈ [−1, 1].

3.1. Legendre–Galerkin methods Let PnG : X → Xn be the orthogonal projection defined by

PnG u =

n − ⟨u, ψj ⟩ψj ,

u ∈ X,

(3.1)

j =0

1

where ⟨u, ψj ⟩ = −1 u(t )ψj (t )dt. Using the orthogonal projection PnG , the approximation scheme (2.5) leads to the Legendre–Galerkin method, that is, find ∑n G un = j=0 αj ψj ∈ Xn and λn ∈ C \ {0} such that

PnG K un = λGn un ,

‖un ‖ = 1,

(3.2)

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which is equivalent to, n −

n −

αj ⟨K ψj , ψi ⟩ = λGn

j =0

αj ⟨ψj , ψi ⟩,

i = 0, 1, 2, . . . , n.

j =0

By solving the above eigenvalue problem, we obtain the eigenvalue λGn and [α0 , α1 , . . . , αn ]T and using un = j=0 αj ψj , we obtain the Galerkin eigenvector. Next we discuss the existence of the Galerkin eigenfunctions and their superconvergence results. To this end, we quote the following proposition and lemma which follows from the analysis of ([6], pp. 283–287).

∑n

Proposition 3.1 ([6,7]). Let PnG : X → Xn denote the orthogonal projection defined by (3.1). Then the projection PnG satisfies (C 1) and (C 2), that is, there hold (i) {PnG : n ∈ N} is uniformly bounded in L2 -norm. (ii) There exists a constant C > 0 such that for any n ∈ N and u ∈ X,

‖PnG u − u‖L2 ≤ C inf ‖u − φ‖L2 . φ∈Xn

Lemma 3.2 ([6,8]). Let PnG be the orthogonal projection defined by (3.1). Then for any u ∈ C r [−1, 1], there hold

‖u − PnG u‖L2 ≤ cn−r ‖u(r ) ‖L2 , ‖u − PnG u‖∞ ≤ cn

3 −r 4

‖u(r ) ‖L2 ,

‖u − PnG u‖∞ ≤ cn

1 −r 2

V (u(r ) ),

(3.3) (3.4) (3.5)

where c is a constant independent of n and V (u

(r )

(r )

) denotes the total variation of u .

From the estimate (3.4), we see that ‖ − u‖∞ 9 as n → ∞ for any u ∈ C ([−1, 1]). However, in the following theorem, we prove that PnG K converges to K both in L2 -norm and infinity norm.

PnG u

Theorem 3.3. Let K be a compact linear integral operator with a kernel K (., .) ∈ C r +1 ([−1, 1] × [−1, 1]) and PnG be the orthogonal projection defined by (3.1). Then there hold

‖K − PnG K ‖L2 = O (n−r ),  1  ‖K − PnG K ‖∞ = O n 2 −r .

(3.6) (3.7)

Proof. Using the estimates (3.3) and (2.7), we see that

‖(I − PnG )K u‖L2 ≤ cn−r ‖(K u)(r ) ‖L2 ≤ cn−r ‖K ‖r ,∞ ‖u‖L2 ,

(3.8)

which completes the proof of (3.6). Using the estimates (3.5), we obtain 1

‖K u − PnG K u‖∞ ≤ cn 2 −r V ((K u)(r ) ).

(3.9)

Now by the definition of total variation and Mean value theorem, we have V ((K u)(r ) ) = sup P

n −

|(K u)(r ) (xi ) − (K u)(r ) (xi−1 )|

i=1

 n ∫ 1 −   (r ,0) (r ,0)  = sup [K ( xi , s) − K (xi−1 , s)]u(s) ds  P −1 i=1  n ∫ 1 −    [K (r +1,0) (ξi , s)]u(s) ds(xi − xi−1 ) = sup   P i=1

−1

n

≤ sup P

−

|xi − xi−1 |

i=1

≤ 2‖K ‖r +1,∞ ‖u‖∞ ,

sup

ξ ,s∈[−1,1]

|K (r +1,0) (ξ , s)| ‖u‖∞ (3.10)

for all partitions P = {x0 , x1 , . . . , xn } of [−1, 1], where ξi ∈ (xi−1 , xi ), i = 1, 2, . . . , n. Thus combining the estimate (3.10) with (3.9), leads to (3.7). This completes the proof. 

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From Theorem 3.3, we see that ‖K − PnG K ‖L2 → 0 and ‖K − PnG K ‖∞ → 0 as n → ∞ for r > 12 . Hence all the results of the abstract framework developed in the last section are applicable to the Legendre–Galerkin method (3.2). Let λGn,1 , λGn,2 , . . . λGn,m be the eigenvalues of PnG K inside Γ and let λˆ Gn denote their arithmetic mean. Let PnS ,G denote the spectral projection of PnG K corresponding to its spectra inside Γ . Next we discuss the superconvergence results for the approximated eigenfunctions to those of the exact eigenfunctions. To do this, first we prove the following theorem. Theorem 3.4. Let K be a compact linear integral operator with a kernel K (., .) ∈ C r ([−1, 1] × [−1, 1]) and PnG be the orthogonal projection defined by (3.1). Then there hold

‖K (PnG − I)K ‖L2 = O (n−2r ),

(3.11)

‖K (PnG − I)K ‖∞ = O (n−2r ),

(3.12)

Proof. Using the estimates (3.3) and (3.8) and Schwarz’s inequality, we have

|K (

PnG

∫  − I)K u(t )| = 

1

K (t , s)(

PnG

−1

  − I)K u(s)ds

= |⟨Kt (.), (PnG − I)K u⟩| = |⟨(PnG − I)Kt (.), (PnG − I)K u⟩| ≤ ‖(PnG − I)Kt (.)‖L2 ‖(PnG − I)K u‖L2 ≤ cn−r ‖(Kt (.))(r ) ‖L2 n−r ‖K ‖r ,∞ ‖u‖L2 ≤ cn−2r (‖K ‖r ,∞ )2 ‖u‖L2 √ ≤ c 2n−2r (‖K ‖r ,∞ )2 ‖u‖∞ . Thus we obtain

‖K (PnG − I)K u‖∞ = sup |K (PnG − I)K u(t )| t ∈[−1,1]

≤ cn−2r (‖K ‖r ,∞ )2 ‖u‖L2 √ ≤ c 2n−2r (‖K ‖r ,∞ )2 ‖u‖∞ ,

(3.13) (3.14)

which completes the proof of (3.12). Now using the estimate (3.13), we obtain

‖K (PnG − I)K u‖L2 ≤

√

2‖K (PnG − I)K u‖∞ ≤ c

This completes the proof of (3.11).

√

2n−2r (‖K ‖r ,∞ )2 ‖u‖L2 .



In the following theorems, we give the superconvergence rates for the eigenvalues and eigenvectors. Theorem 3.5. Let K be a compact integral operator with a kernel K (., .) ∈ C r ([−1, 1]×[−1, 1]) and PnG K be norm-convergent

ˆ Gn be the arithmetic mean of to K in L2 -norm. Assume λ be the eigenvalue of K with algebraic multiplicity m and ascent ℓ. Let λ the eigenvalues λn,j , for j = 1, 2, . . . , m, then |λ − λˆ Gn | = O (n−2r ), |λ − λGn,j |ℓ = O (n−2r ). Proof. By using Lemma 2.2 and Theorem 3.4, for j = 1, 2, . . . , m, it follows that

|λ − λˆ n | ≤ c ‖(KPnG − K )K ‖L2 = O (n−2r ), |λ − λn,j |ℓ ≤ c ‖(KPnG − K )K ‖L2 = O (n−2r ). This completes the proof.



Theorem 3.6. Let K be a compact integral operator with a kernel K (., .) ∈ C r +1 ([−1, 1] × [−1, 1]) and PnG K norm-converges to K in L2 -norm. Let R(P S ) and R(PnS ,G ) be the ranges of the spectral projections of P S and PnS ,G respectively. Then

δˆ2 (R(PnS ,G ), R(P S )) = O (n−r ), δ2 (K R(PnS ,G ), R(P S )) = O (n−2r ).

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In particular, for any un ∈ R(PnS ,G ), we have

‖un − P S un ‖L2 = O (n−r ), ‖K un − P S K un ‖L2 = O (n−2r ). Proof. It follows from Lemma 2.3 and Theorem 3.3 that,

δˆ2 (R(PnS ,G ), R(P S )) ≤ c ‖(PnG K − K )K ‖L2 = O (n−r ). In particular, for any un ∈ R(PnS ,G ), we have

‖un − P S un ‖L2 ≤ c ‖(PnG K − K )K ‖L2 = O (n−r ). Now using Lemma 2.4 and Theorem 3.4, we obtain

δ2 (K R(PnS ,G ), R(P S )) ≤ c ‖(KPnG − K )K ‖L2 = O (n−2r ). In particular, for any un ∈ R(PnS ,G ), u˜ n = K un , the iterated eigenvector, we have

‖˜un − P S u˜ n ‖L2 = ‖K un − P S K un ‖L2 ≤ c ‖(KPnG − K )K ‖L2 = O (n−2r ). This completes the proof.



In the following theorem we give the superconvergence rates for the eigenvectors and the iterated eigenvectors in the infinity norm. Theorem 3.7. Let K be a compact integral operator with a kernel K (., .) ∈ C r ([−1, 1] × [−1, 1]) and PnG K norm-converges to K in infinity norm. Let R(P S ) and R(PnS ,G ) be the ranges of the spectral projections of P S and PnS ,G respectively. Then

 1  δˆ∞ (R(PnS ,G ), R(P S )) = O n 2 −r , δ∞ (K R(PnS ,G ), R(P S )) = O (n−2r ). In particular, for any un ∈ R(PnS ,G ), we have

 1  ‖ un − P S un ‖ ∞ = O n 2 − r , ‖K un − P S K un ‖∞ = O (n−2r ). Proof. It follows from Lemma 2.3 and Theorem 3.3 that,

 1  δˆ∞ (R(PnS ,G ), R(P S )) ≤ c ‖(PnG K − K )K ‖∞ = O n 2 −r . In particular, for any un ∈ R(PnS ,G ), we have

 1  ‖un − P S un ‖∞ ≤ c ‖(PnG K − K )K ‖∞ = O n 2 −r . Now using Lemma 2.4 and Theorem 3.4, we obtain

δ∞ (K R(PnS ,G ), R(P S )) ≤ c ‖(KPnG − K )K ‖∞ = O (n−2r ). In particular, for any un ∈ R(PnS ,G ), we have

‖˜un − P S u˜ n ‖∞ = ‖K un − P S K un ‖∞ ≤ c ‖(KPnG − K )K ‖∞ = O (n−2r ). This completes the proof.



Remark 1. From Theorem 3.5, we observe that in the Legendre–Galerkin method, the approximated eigenvalues converges to the exact eigenvalues with the order O (n−2r ) and from Theorems 3.6 and 3.7, we notice that  the gap  between the spectral subspaces in L2 -norm is of the order O (n−r ) whereas in the infinity norm it is of the order O n1/2−r , r > 1/2. This shows that, in the Legendre–Galerkin method the eigenvectors converge faster in L2 -norm than in the infinity norm. However, the iterated eigenvectors converge with the order O (n−2r ) both in L2 -norm and infinity norm. Thus in the Legendre–Galerkin method, we obtain the superconvergence rates for the iterated eigenvectors and for the eigenvalues. 3.2. Legendre collocation methods In this subsection, we consider the spectral collocation method and establish the superconvergence rates for the eigenfunctions.

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Let X = C [−1, 1]. Let {τ0 , τ1 , . . . , τn } be the zeros of the Legendre polynomial of degree n + 1 and define the interpolatory projection PnC : X → Xn by

PnC u ∈ Xn ,

PnC u(τi ) = u(τi ), i = 0, 1, . . . , n, u ∈ X.

(3.15)

According to the analysis of ([6], pp. 289, [7,8]), we have the following proposition and lemma. Proposition 3.8. The following hold (i) {PnC : n ∈ N} is uniformly bounded in L2 -norm. (ii) There exists a constant c > 0 such that for any n ∈ N and u ∈ X,

‖PnC u − u‖L2 ≤ c inf ‖u − φ‖L2 . φ∈Xn

Lemma 3.9. Let PnC : X → Xn be the interpolatory projection defined (3.15). Then for any u ∈ C r [−1, 1], there exists a constant c independent of n such that

‖u − PnC u‖L2 ≤ cn−r ‖u(r ) ‖L2 ,

n ≥ r.

(3.16)

From the above proposition and lemma we observe that the interpolatory projection PnC satisfies assumptions C1 and C2. Using the interpolatory projection operator PnC , the approximation scheme (2.5) leads to the Legendre collocation ∑n method. That is the Legendre collocation method for the eigenvalue problem (2.1) is defined as follows: find un = j =0

βj ψj ∈ Xn and λCn ∈ C \ {0} such that PnC K un = λCn un ,

(3.17)

which is equivalent to, n − j=0

βj K ψj (τi ) = λCn

n −

βj ψj (τi ),

i = 0, 1, 2, . . . , n.

j =0

By solving the above eigenvalue problem we obtain the eigenvalue λCn and [β0 , β1 , . . . , βn ]T and using un = obtain the collocation eigenvector. Next we discuss the existence of collocation eigenfunctions and their superconvergence results. Noting that

∑n

j =0

βj ψj , we

  23/2 ‖PnC ‖∞ = 1 + √ n1/2 + B0 + O n−1/2 , π

(3.18)

where B0 is a bounded constant (see, [9–12]), we see that ‖PnC u − u‖∞ 9 as n → ∞ for any u ∈ C ([−1, 1]). Since

‖(I − PnC )u‖∞ ≤ (1 + ‖PnC ‖∞ ) inf ‖u − χ‖∞ χ∈Xn

1 2

1

≤ cn n ‖u ‖∞ ≤ cn 2 −r ‖u(r ) ‖∞ . The estimate (3.19) shows that ‖(I − PnC )u‖∞ → 0 as n → ∞ for u ∈ C r ([−1, 1]), r > 1/2. −r

(r )

(3.19)

Theorem 3.10. Let K be a compact linear integral operator with a kernel K (., .) ∈ C r ([−1, 1] × [−1, 1]) and PnC be the interpolatory projection defined by (3.15). Then there hold

‖K − PnC K ‖L2 = O (n−r ),  1  ‖K − PnC K ‖∞ = O n 2 −r .

(3.20) (3.21)

Proof. Using the estimate (3.16) and (2.7) we obtain

‖(I − PnC )K u‖L2 ≤ cn−r ‖(K u)r ‖∞ ≤ cn−r ‖K ‖r ,∞ ‖u‖L2 .

(3.22)

This completes the proof of (3.20). Now using the estimate (3.19) and (2.6) we get 1

1

‖(I − PnC )K u‖∞ ≤ cn 2 n−r ‖(K u)(r ) ‖∞ ≤ cn 2 −r ‖K ‖r ,∞ ‖u‖∞ . This completes the proof of (3.21).

(3.23)



From Theorem 3.10, we see that ‖K − PnC K ‖L2 → 0 and ‖K − PnC K ‖∞ → 0 as n → ∞ for r > 12 . Hence all the results of the abstract framework developed in the last section are applicable to the Legendre collocation method (3.17). ˆ Cn denote their arithmetic mean. Let PnS ,C denote the Let λCn,1 , λCn,2 , . . . , λCn,m be the eigenvalues of PnC K inside Γ and let λ spectral projection of PnC K corresponding to its spectra inside Γ .

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The following lemma gives the relation between the interpolatory projection and orthogonal projection, which is useful in obtaining the superconvergence results for the eigenfunctions in the Legendre collocation method. Lemma 3.11 ([13]). Let PnC and PnG be the orthogonal and interpolatory projection defined by (3.1) and (3.15), respectively. Then for any u ∈ C r [−1, 1], r ≥ n + 1 and y ∈ X, there hold

⟨(I − PnC )u, y⟩ = ⟨(I − PnG )v, (I − PnG )yδ n+1 u⟩,

(3.24)

where δ (n+1) denote the (n + 1)th divided difference and v(s) =

∏n

i =0

(s − τi ), s ∈ [−1, 1].

Theorem 3.12. Let K be a compact linear integral operator with a kernel K (., .) ∈ C n+r +1 ([−1, 1] × [−1, 1]) and PnC be the interpolatory projection defined by (3.15). Then there hold

‖K (PnC − I)K ‖L2 = O (n−2r ),

(3.25)

‖K (

(3.26)

PnC

− I ) K ‖∞ = O ( n

−2r

).

Proof. Using the estimate (3.3) and (3.8) and Lemma 3.11, we have

∫  |K (PnC − I)K u(t )| = 

1

−1

 

K (t , s)(PnC − I)K u(s) ds

= |⟨Kt (.), (PnC − I)K u⟩| = |⟨(I − PnG )v, (I − PnG )Kt (.)δ n+1 K u⟩| ≤ ‖(I − PnG )v‖L2 ‖(I − PnG )Kt (.)δ n+1 K u‖L2 ≤ cn−2r ‖v r ‖L2 ‖(Kt (.)δ n+1 K u)r ‖L2 ≤ cn−2r ‖v r ‖L2 ‖K ‖r ,∞ ‖K ‖n+r +1,∞ ‖u‖L2 ≤ cn−2r ‖v r ‖L2 ‖K ‖r ,∞ ‖K ‖n+r +1,∞ ‖u‖∞ . Hence we obtain

‖K (PnC − I)K u‖∞ = sup |K (PnC − I)K u(t )| t ∈[−1,1]

≤ cn−2r ‖v r ‖L2 ‖K ‖r ,∞ ‖K ‖n+r +1,∞ ‖u‖L2 ≤ cn

−2r

‖v ‖L2 ‖K ‖r ,∞ ‖K ‖n+r +1,∞ ‖u‖∞ . r

(3.27) (3.28)

This completes the proof of (3.26). Now using the estimate (3.27), we obtain

‖K (PnC − I)K u‖L2 ≤

√

2‖K (PnC − I)K u‖∞ ≤ c

This completes the proof of (3.25).

√

2n−2r ‖v r ‖L2 ‖K ‖r ,∞ ‖K ‖n+r +1,∞ ‖u‖L2 .



In the following theorem we give the superconvergence rates for the eigenvalues in the Legendre collocation method. Theorem 3.13. Let K be a compact integral operator with a kernel K (., .) ∈ C n+r +1 ([−1, 1] × [−1, 1]) and PnC K norm-

ˆ Cn be the arithmetic converges to K in L2 -norm. Assume λ be the eigenvalue of K with algebraic multiplicity m and ascent ℓ. Let λ C mean of the eigenvalues λn,j , for j = 1, 2, . . . , m, |λ − λˆ Cn | = O (n−2r ), |λ − λCn,j |ℓ = O (n−2r ). Proof. By using Lemma 2.2 and Theorem 3.12 that for j = 1, 2, . . . , m,

|λ − λˆ Cn | ≤ c ‖(KPnC − K )K ‖L2 = O (n−2r ), |λ − λCn,j |ℓ ≤ c ‖(KPnC − K )K ‖L2 = O (n−2r ). This completes the proof.



Theorem 3.14. Let K be a compact integral operator with a kernel K (., .) ∈ C n+r +1 ([−1, 1] × [−1, 1]) and PnC K be normconvergent to K in L2 -norm. Let R(P S ) and R(PnS ,C ) be the ranges of the spectral projections of P S and PnS ,C , respectively.

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Then there hold,

δˆ2 (R(PnS ,C ), R(P S )) = O (n−r ), δ2 (K R(PnS ,C ), R(P S )) = O (n−2r ). In particular, for any un ∈ R(PnS ,C ), u˜ n = K un , we have

‖un − P S un ‖L2 = O (n−r ), ‖˜un − P S u˜ n ‖L2 = O (n−2r ). Proof. It follows from Lemma 2.3 and Theorem 3.10 that,

δˆ2 (R(PnS ,C ), R(P S )) ≤ c ‖(PnC K − K )K ‖L2 = O (n−r ). In particular, for any un ∈ R(PnS ,C ), we have

‖un − P S un ‖L2 ≤ c ‖(PnC K − K )K ‖L2 = O (n−r ). Now using Lemma 2.4 and Theorem 3.12 that,

δ2 (K R(PnS ,C ), R(P S )) ≤ c ‖(KPnC − K )K ‖L2 = O (n−2r ). In particular, for any un ∈ R(PnS ,C ), u˜ n = K un , the iterated eigenvector, we have

‖˜un − P S u˜ n ‖L2 = ‖K un − P S K un ‖L2 ≤ c ‖(KPnC − K )K ‖L2 = O (n−2r ). This completes the proof.



In the following theorem we give the superconvergence rates for the eigenvectors in the Legendre collocation method in the infinity norm. Theorem 3.15. Let K be a compact integral operator with a kernel K (., .) ∈ C n+r +1 ([−1, 1] × [−1, 1]) and PnC K normconvergent to K in infinity norm. Let R(P S ) and R(PnS ,C ) be the ranges of the spectral projections of P S and PnS ,C respectively. Then there hold,

 1  δˆ∞ (R(PnS ,C ), R(P S )) = O n 2 −r , δ∞ (K R(PnS ,C ), R(P S )) = O (n−2r ). In particular, for any un ∈ R(PnS ,C ), u˜ n = K un , we have

 1  ‖un − P S un ‖∞ = O n 2 −r , ‖˜un − P S u˜ n ‖∞ = O (n−2r ). Proof. It follows from Lemma 2.3 and Theorem 3.10 that,

 1  δˆ∞ (R(PnS ,C ), R(P S )) ≤ c ‖(PnC K − K )K ‖∞ = O n 2 −r . In particular, for any un ∈ R(PnS ,C ), we have

 1  ‖un − P S un ‖∞ ≤ c ‖(PnC K − K )K ‖∞ = O n 2 −r . Now using Lemma 2.4 and Theorem 3.12 that,

δ∞ (K R(PnS ,C ), R(P S )) ≤ c ‖(KPnC − K )K ‖∞ = O (n−2r ). In particular, for any un ∈ R(PnS ,C ), we have

‖˜un − P S u˜ n ‖∞ = ‖K un − P S K un ‖∞ ≤ c ‖(KPnC − K )K ‖∞ = O (n−2r ). This completes the proof.



Remark 2. From Theorem 3.13, we observe that in the Legendre collocation method, the approximated eigenvalues converge to the exact eigenvalues with the order O (n−2r ) and from Theorems 3.14 and 3.15, we notice that  the gap  between the spectral subspaces in L2 -norm is of the order O (n−r ) whereas in the infinity norm it is of the order O n1/2−r , r > 1/2. This shows that, in the Legendre collocation method the eigenfunctions converge faster in L2 -norm than in the infinity norm. However, the iterated eigenfunctions converge with the order O (n−2r ) both in L2 -norm and infinity norm. Thus in the Legendre collocation method, we obtain the superconvergence rates for the iterated eigenfunctions and for the eigenvalues.

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Remark 3. In both the Legendre–Galerkin and Legendre collocation methods, the approximated eigenvalues converge to the exact eigenvalues with the order O (n−2r ) whereas the gap  between the spectral subspaces in L2 -norm is of the order O (n−r ) and in the infinity norm it is of the order O n1/2−r , r > 1/2 and the iterated eigenfunctions converge with the order O (n−2r ) both in L2 -norm and infinity norm. This shows that we obtained same superconvergence rates for the eigenfunction both in the Legendre–Galerkin and Legendre collocation methods. Remark 4. From Theorems 3.5–3.7, and from Theorems 3.13–3.15, we notice that in the Legendre–Galerkin and Legendre collocation methods, we obtained similar superconvergence rates for the eigenvalues and the iterated eigenfunctions. This makes our method economical computationally while obtaining the similar superconvergence rates for the eigenfunctions. Remark 5. For eigenvalue problem (1.1) of compact integral operator K with weakly singular kernels of the type

  m(t , s) , K (t , s) = |t − s|α  log |t − s|,

if 0 < α < 1, if α = 1,

where m(t , s) is a smooth function in both the variables, we expect that the convergence rates for the eigenvalues and the iterated eigenfunctions are of the order O (n−r −α ) and for the eigenvector is of the order O (n−r ). We investigate these superconvergence results in our future papers. 4. Numerical results In this section we discuss the numerical results. Consider the eigenvalue problem (1.1),

K u = λu,

0 ̸= λ ∈ C , ‖u‖ = 1,

(4.1)

for the following integral operator

(K u)(t ) =

∫

1

K (t , s)u(s) ds,

s ∈ [−1, 1]

−1

for various smooth kernels K (t , s). As the exact eigenelements of K are not known, for the sake of comparison, we replace the integral operator by Nyström ∑mr operator (K N u)(t ) = j=1 wj K (t , sj )u(sj ), obtained by using composite Gaussian quadrature with r points associated with a uniform partition of [−1, 1] with m intervals. Let λ be the eigenvalue of K N and P S be the associated spectral projection. We choose the Legendre polynomials {ψ0 , ψ1 , . . . , ψn } as an orthonormal basis subspace Xn , where

ψ0 (x) = 1,

ψ1 (x) = x,

x ∈ [−1, 1],

and for i = 1, 2, . . . , n − 1,

(i + 1)ψi+1 (x) = (2i + 1)xψi (x) − iψi−1 (x),

x ∈ [−1, 1].

ˆ n , un in the For different kernels, and for different values of n, we compute the approximate eigenfunctions λ Legendre–Galerkin and the Legendre collocation methods and we compute the iterated eigenvectors u˜ n = K un . The computed errors in both L2 -norm and infinity norm of the approximated eigenfunctions to those of the exact eigenfunctions are presented in Tables 1–6. Kernel-1 2 K (t , s) = e−|t −s| , t , s ∈ [a, b] = [−1, 1]. Kernel-2 K ( t , s) = 

1 1 + |t − s|2

,

t , s ∈ [a, b] = [−1, 1].

Kernel-3 K (t , s) = sin(2π t + π s) + cos(3ts) + (ts2 /3),

t , s ∈ [a, b] = [−1, 1].

From Tables 1–6 we observe that our theoretical results agree with the numerical results. Table 1 Legendre-Galerkin method, Kernel-1. n

|λ − λˆ n |

‖un − P S un ‖L2

‖˜un − P S u˜ n ‖L2

‖un − P S un ‖∞

‖˜un − P S u˜ n ‖∞

3 4 5 6 7

1.437756e−04 1.538258e−07 1.538317e−07 4.024114e−12 9.518852e−08

1.039643e−02 3.357451e−04 3.357451e−04 1.143706e−06 1.149902e−06

2.992167e−04 9.379449e−07 9.379449e−07 3.650024e−08 2.580956e−08

2.059417e−02 8.374203e−04 8.374203e−04 4.346552e−06 4.245631e−06

3.266590e−04 9.193688e−07 9.193686e−07 1.933504e−10 1.003981e−08

B.L. Panigrahi, G. Nelakanti / Journal of Computational and Applied Mathematics 235 (2011) 2380–2391

2391

Table 2 Legendre collocation method, Kernel-1. n

|λ − λˆ n |

‖un − P S un ‖L2

‖˜un − P S u˜ n ‖L2

‖un − P S un ‖∞

‖˜un − P S u˜ n ‖∞

3 4 5 6 7

2.150529e−04 4.000998e−06 1.116253e−08 1.488820e−08 9.399530e−10

1.418168e−02 7.277263e−04 5.589205e−04 2.905153e−06 2.381763e−06

4.727752e−04 1.368549e−05 1.495173e−06 4.214684e−08 2.980232e−08

2.722194e−02 2.341816e−03 1.388991e−03 1.417987e−05 1.055197e−05

5.291304e−04 1.531072e−05 1.477999e−06 3.377033e−08 4.308435e−09

Table 3 Legendre–Galerkin method, Kernel-2. n

|λ − λˆ n |

‖un − P S un ‖L2

‖˜un − P S u˜ n ‖L2

‖un − P S un ‖∞

‖˜un − P S u˜ n ‖∞

3 4 5 6 7

8.087394e−06 5.820764e−09 5.820764e−09 1.918081e−09 6.424670e−10

2.208895e−03 5.905367e−05 5.905367e−05 1.059847e−05 1.059863e−05

2.956274e−05 1.139665e−07 1.139665e−07 2.806327e−09 1.421554e−09

4.732227e−03 1.155571e−04 1.155571e−04 2.996543e−05 2.997580e−05

3.249466e−05 1.555100e−07 1.555100e−07 4.144714e−09 2.515099e−09

Table 4 Legendre collocation method, Kernel-2. n

|λ − λˆ n |

‖un − P S un ‖L2

‖˜un − P S u˜ n ‖L2

‖un − P S un ‖∞

‖˜un − P S u˜ n ‖∞

3 4 5 6 7

1.957074e−05 1.038312e−06 1.573168e−07 1.567022e−08 6.535962e−10

7.985850e−02 1.000042e−02 1.038667e−03 8.295434e−05 6.279661e−06

5.064799e−05 7.230142e−07 7.661248e−07 4.758760e−09 4.391999e−09

1.965561e−01 2.701832e−02 3.044223e−03 2.570204e−04 2.045097e−05

4.896022e−05 7.058225e−07 7.404598e−07 4.654926e−09 4.244772e−09

Table 5 Legendre–Galerkin method, Kernel-3. n

|λ − λˆ n |

‖un − P S un ‖L2

‖˜un − P S u˜ n ‖L2

‖un − P S un ‖∞

‖˜un − P S u˜ n ‖∞

3 4 5 6 7

1.494841e−03 7.427825e−04 2.737794e−05 1.865086e−05 2.035589e−07

1.774942e−01 1.610524e−01 5.938286e−02 5.655554e−02 1.075926e−02

2.035907e−02 1.497854e−02 5.824926e−04 3.518707e−04 5.395670e−06

3.027500e−01 2.426784e−01 1.574389e−01 1.317833e−01 3.609940e−02

2.371489e−02 1.540188e−02 6.370518e−04 3.605281e−04 5.821020e−06

Table 6 Legendre collocation method, Kernel-3. n

|λ − λˆ n |

‖un − P S un ‖L2

‖˜un − P S u˜ n ‖L2

‖un − P S un ‖∞

‖˜un − P S u˜ n ‖∞

3 4 5 6 7

6.919437e−02 1.828686e−03 1.468511e−03 1.762851e−04 1.753217e−05

3.117711e−01 2.759208e−01 1.409414e−01 1.090953e−01 2.962398e−02

1.751205e−01 9.421395e−02 2.107186e−02 3.869351e−03 4.261602e−04

5.110174e−01 4.687127e−01 3.722607e−01 2.273863e−01 1.197079e−01

2.172225e−01 9.748949e−02 2.320230e−02 4.147933e−03 4.595235e−04

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