Advanced
RICHARDSON EXTRAPOLATION OF ITERATED DISCRETE COLLOCATION METHOD FOR EIGENVALUE PROBLEM OF A TWO DIMENSIONAL COMPACT INTEGRAL OPERATOR
RICHARDSON EXTRAPOLATION OF ITERATED DISCRETE COLLOCATION METHOD FOR EIGENVALUE PROBLEM OF A TWO DIMENSIONAL COMPACT INTEGRAL OPERATOR
Journal of Applied Mathematics & Informatics. 2014. Sep, 32(5_6): 567-584
Copyright © 2014, Korean Society of Computational and Applied Mathematics
  • Received : February 14, 2012
  • Accepted : April 16, 2014
  • Published : September 30, 2014
Download
PDF
e-PUB
PubReader
PPT
Export by style
Share
Article
Author
Metrics
Cited by
TagCloud
About the Authors
BIJAYA LAXMI PANIGRAHI
GNANESHWAR NELAKANTI

Abstract
In this paper, we consider approximation of eigenelements of a two dimensional compact integral operator with a smooth kernel by dis-crete collocation and iterated discrete collocation methods. By choosing numerical quadrature appropriately, we obtain convergence rates for gap between the spectral subspaces, and also we obtain superconvergence rates for eigenvalues and iterated eigenvectors. We then apply Richardson ex-trapolation to obtain further improved error bounds for the eigenvalues. Numerical examples are presented to illustrate theoretical estimates. AMS Mathematics Subject Classification : 45C05, 45B05.
Keywords
1. Introduction
Consider the following integral operator K defined on
PPT Slide
Lager Image
by
PPT Slide
Lager Image
where kernel K (., ., ., .) ∈ C ( D ) × C ( D ), D = [ a, b ] × [ c, d ] ⊂ ℝ 2 is a given rectangular domain. Then K is a compact linear operator on
PPT Slide
Lager Image
.
We are interested in the following eigenvalue problem: find
PPT Slide
Lager Image
and λ ∈ ℂ − {0} such that
PPT Slide
Lager Image
Many practical problems in science and engineering are formulated as eigenvalue problems (2) of compact linear integral operators K (cf., [3] ). For many years, numerical solution of eigenvalue problems have attracted much attention. During the last some years, significant work has been done in the numerical analysis of the one-dimensional eigenvalue problem for the compact integral operator K . The Galerkin, petrove-Galerkin, collocation, Nyström and degenerate kernel methods are the commonly used methods for the approximation of eigenelements of the compact integral operator K . The analysis for the convergence of Galerkin, petrove-Galerkin, collocation, Nystr¨om and degenerate kernel methods for the one dimensional eigenvalue problems are well documented in ( [1] , [3] , [12] , [13] , [14] ). By replacing the various integrals appearing in these methods by numerical quadrature leads to discrete methods. In ( [9] ) discrete and iterated discrete Galerkin methods and in ( [4] ) discrete and iterated discrete collocation methods were discussed for the one dimensional eigenvalue problem (2) with a smooth kernel.
In [15] , we were interested to solve the eigenvalue problem of a two dimensional compact integral operator with smooth kernel taking the help of discrete Galerkin and iterated discrete Galerkin methods and obtained the error bounds for approximated eigenelements. Further, to improve the convergence rates for the eigenvalues, we derived an asymptotic expansion for the iterated discrete Galerkin operator and then using Richardson extrapolation we improved the convergence rates for the eigenvalues. Meanwhile, to do so, we replace the various integrals arise in L 2 inner product, when the projection is an orthogonal projection and the two-dimensional integral operator K by using Gauss quadrature rule, which improves the computational cost to generate the matrix eigenvalue problem. To avoid such, discrete collocation method receive favorable attention due to lower computational cost in generating matrix eigenvalue problem. In fact, in comparison to discrete Galerkin and discrete petrove Galerkin methods, the discrete collocation method requires much less computational effort in evaluation of its entries defined by integrals. This motivates us to do this work.
In section-2, we develop discrete collocation, iterated discrete collocation methods and theoretical frame work for the eigenvalue problem using interpolatory projections. In section-3, we discuss the convergence rates for the approximated eigenfunctions to the exact eigenfunctions. In section-4, we discuss Richardson extrapolation for eigenvalue problem to improve convergence rates. In section-5, we present numerical results, which agree with the theoretical results. Throughout the paper, we assume c as the generic constant.
2. Discrete and Iterated discrete collocation methods
Consider the following compact integral operator K defined on
PPT Slide
Lager Image
by
PPT Slide
Lager Image
where the kernel K (., ., ., .) ∈ C ( D ) × C ( D ), D = [ a, b ] × [ c, d ] ⊂ ℝ 2 .
We are interested in the eigenvalue problem (2). Assume λ be the eigenvalue of K with algebraic multiplicity m and ascent . Let Γ ⊂ ρ ( K ) be a simple closed rectifiable curve such that σ ( K ) ∩ int Γ = { λ }, 0 ≠ int Γ, where int Γ denotes the interior of Γ. Now we describe the collocation method for the eigenvalue problem (2).
Let
PPT Slide
Lager Image
be the uniform partitions of finite intervals [ a, b ] and [ c, d ], respectively, defined by
PPT Slide
Lager Image
and
PPT Slide
Lager Image
for m = 0, 1, 2, . . . , M − 1 and n = 0, 1, 2, . . . , N − 1. These partitions define a grid for D ,
PPT Slide
Lager Image
: 0 ≤ m M − 1, 0 ≤ n N − 1}. Set
PPT Slide
Lager Image
and
PPT Slide
Lager Image
m = 0, 1, . . . , M − 1 and n = 0, 1, . . . , N − 1. For any given positive integer p and q , let P p−1,q−1 denotes the space of polynomials of degree p − 1 in x and q − 1 in y , then for 0 ≤ m M − 1, 0 ≤ n N − 1,
PPT Slide
Lager Image
is the finite element space of dimension MNpq , which is the tensor product space of univariate spline spaces
PPT Slide
Lager Image
on [ c, d ]. The use of superscript (-1) in the notation for the above finite element space is to emphasize that it is not a subspace of C ( D ).
Let sm,i = xm + τih and tn,j = yn + θjk , be the collocation points on [ xm , x m+1 ], m = 0, 1, . . . , M −1, and [ yn , y n+1 ], n = 0, 1, . . . , N −1, respectively, where τ 0 , τ 1 , . . . , τ p−1 and θ 0 , θ 1 , . . . , θ q−1 are zeros of Legendre polynomials of degree p and q , respectively on [0, 1].
Let
PPT Slide
Lager Image
and
PPT Slide
Lager Image
be the interpolatory projection with respect to the nodes { sm,i } and { tn,j }, respectively, that is, for u L ([ a, b ]),
PPT Slide
Lager Image
PPT Slide
Lager Image
then there holds( [3] ), ∥ Ph c 1 < ∞, ∥ Pk c 2 < ∞ and for any u Cp [ a, b ] and u Cq [ c, d ], there holds,
PPT Slide
Lager Image
Note that
PPT Slide
Lager Image
As a consequence, from (6), it follows that
PPT Slide
Lager Image
Let ϕim , ψjn denote the Lagrange polynomials of degree p − 1 and q − 1 on the subintervals [ xm , x m+1 ], m = 0, 1, . . . , M − 1 and [ yn , y n+1 ], n = 0, 1, . . . , N − 1 respectively, where, j = 0, 1, . . . q − 1, i = 0, 1, . . . , p − 1. Then it follows that
PPT Slide
Lager Image
and
PPT Slide
Lager Image
Then we have
PPT Slide
Lager Image
Now the collocation method for solving the eigenvalue problem (2) is defined as follows: find
PPT Slide
Lager Image
uhk ∥ = 1 and λhk ∈ ℂ − {0} such that
PPT Slide
Lager Image
for i ′ = 0, 1, . . . p −1, m ′ = 0, 1, . . . M −1, j ′ = 0, 1, . . . q −1, n ′ = 0, 1, . . . N −1. Using
PPT Slide
Lager Image
the equation (9) can be converted to the matrix eigenvalue problem. The iterated eigenvector is defined by
PPT Slide
Lager Image
To solve the matrix eigenvalue problem and the iterated eigenvector, we need to evaluate various integrals arising from the integral operator K . In practice, numerical quadrature has to be used to compute these integrals. This leads to discrete methods. To do this, let for ƒ, g C [0, 1],
PPT Slide
Lager Image
PPT Slide
Lager Image
be the numerical quadrature with weights
PPT Slide
Lager Image
and quadrature points ci, i = 0, 1, . . . , k ′ − 1, dj, j = 0, 1, . . . , l ′ − 1, chosen as Gauss points in [0, 1] which satisfy 0 < c 0 < c 1 < · · · < c k′−1 < 1 and 0 < d 0 < d 1 , . . . ,< d l′−1 < 1 having degree of precision 2 k ′ − 1, 2 l ′ − 1, respectively. Then the composite Gauss quadrature rule for any ƒ C ([ a, b ]), g C ([ c, d ]) is given by
PPT Slide
Lager Image
PPT Slide
Lager Image
where xm,i = xm + cih , i = 0, 1, . . . , k ′−1, and yn,j = yn + djk , j = 0, 1 . . . , l ′−1, be the quadrature points on the subintervals [ xm , x m+1 ], m = 0, 1, . . . , M − 1 of [ a, b ] and [ yn , y n+1 ], n = 0, 1, . . . , N −1 of [ c, d ], respectively. Now using (12) and (13), we define the composite quadrature rule for g C ( D ) by
PPT Slide
Lager Image
Let
PPT Slide
Lager Image
be the Nyström operator defined for
PPT Slide
Lager Image
PPT Slide
Lager Image
Now replacing the integral operator K by the Nyström operator (14), the matrix eigenvalue problem leads to the discrete collocation method,
PPT Slide
Lager Image
By solving this discrete matrix eigenvalue problem (15), we find the eigenvalue
PPT Slide
Lager Image
and β = [ βij, i = 0, 1 . . . , p −1, j = 0, 1, . . . , q −1]. Then the discrete collocation eigenvector is defined by,
PPT Slide
Lager Image
The discrete matrix eigenvalue problem (15) can be written in operator form as
PPT Slide
Lager Image
Next we define the iterated discrete collocation eigenvector by
PPT Slide
Lager Image
Clearly we see that
PPT Slide
Lager Image
and
PPT Slide
Lager Image
This is the iterated discrete collocation method.
Next we discuss the convergence of approximated eigenvalues and eigenvectors to the exact eigenvalues and eigenvectors of the integral operator K . To do this, first we set the following notations: Set K ( s, t, x, y ) = Ks,t ( x, y ). For K (., ., ., .) ∈ C (i,j) ( D ) × C (i′,j′) ( D ), denote
PPT Slide
Lager Image
For any α, β, γ, δ ∈ ℕ, we set
PPT Slide
Lager Image
PPT Slide
Lager Image
Then we have
PPT Slide
Lager Image
and
PPT Slide
Lager Image
In the following theorem we give the error bounds for the Nystr¨om operator.
Theorem 2.1 ( [15] ). Let K be an integral operator with a kernel K (., ., ., .) ∈ C (2k′,2l′) ( D ) × C (2k′,2l′) ( D ) and Khk be the Nyström operator defined by (14), then for any u C (2k′,2l′) ( D ), the following holds
PPT Slide
Lager Image
where c is independent of h and k.
Definition 2.2 ( [1] ). Let
PPT Slide
Lager Image
be a Banach space and, T and Tn are bounded linear operators from
PPT Slide
Lager Image
into
PPT Slide
Lager Image
. Then { Tn } is said to be ν-convergent to T , if
PPT Slide
Lager Image
We quote the following lemma which is useful in proving the existence of eigenvalue and eigenvectors in discrete and iterated discrete collocation methods.
Lemma 2.3 ( [2] ). Let
PPT Slide
Lager Image
be a Banach space and S
PPT Slide
Lager Image
is a relatively compact set. Assume that T and Tn are bounded linear operators from
PPT Slide
Lager Image
into
PPT Slide
Lager Image
satisfying Tn ∥ ≤ c for all n ∈ ℕ, and for each x S , ∥ Tnx−Tx ∥ → 0 as n → ∞, where c is a constant independent of n. Then Tnx Tx ∥ → 0 uniformly for all x ∈ S .
Theorem 2.4. PhPkKhk and KhkPhPk are ν-convergent to K .
Proof . Since Khk , Ph and Pk are uniformly bounded, it follows that ∥ PhPkKhk ≤ ∥ Ph Pk Khk < ∞ and ∥ KhkPhPk ≤ ∥ Khk Ph Pk < ∞. Now using (8) and Theorem 2.1, we see that
PPT Slide
Lager Image
This shows that PhPkKhk point wise converges to K .
Let
PPT Slide
Lager Image
be a closed unit ball in
PPT Slide
Lager Image
. Since K is a compact operator, the set S = { Kx : x B } is a relatively compact set in X . By Lemma 2.3, we have
PPT Slide
Lager Image
Since PhPk is bounded and Khk compact, S ′ = { PhPkKhkx : x B } is a relatively compact set. Thus
PPT Slide
Lager Image
as h, k → 0. Combining all these results leads to the first result that PhPkKhkx is ν -convergent to K . The proof of KhkPhPk is ν -convergent to K follows by similar steps as in above.
Since PhPkKhk and KhkPhPk are ν -convergent to K , the spectrum of both PhPkKhk and KhkPhPk inside Γ consists of m eigenvalues say
PPT Slide
Lager Image
counted accordingly to their algebraic multiplicities inside Γ with ascent (cf., [3 , 14] ). Let
PPT Slide
Lager Image
denote their arithmetic mean and we approximate λ by
PPT Slide
Lager Image
Let
PPT Slide
Lager Image
be the spectral projections of K associated with their corresponding spectra inside Γ. Similarly,
PPT Slide
Lager Image
be the spectral projections of PhPkKhk and KhkPhPk , respectively. Let
PPT Slide
Lager Image
be the ranges of the spectral projections
PPT Slide
Lager Image
respectively.
To discuss the closeness of eigenfunctions of the integral operator K and those of the approximate operators, we recall (cf., [3] ) the concept of gap between the spectral subspaces. For nonzero closed subspaces let
PPT Slide
Lager Image
let
PPT Slide
Lager Image
then
PPT Slide
Lager Image
is known as the gap between
PPT Slide
Lager Image
We quote the following three Lemmas, which give the error bounds for the eigenelements.
Theorem 2.5 ( [1] , [13] ). Let PhPkKhk be ν-convergent to K. Then for sufficiently large M,N, there exists a constant c independent of M,N, we have
PPT Slide
Lager Image
Theorem 2.6 ( [13] , [16] ). Let KhkPhPk be ν-convergent to K. Then for sufficiently large M,N, there exists a constant c independent of M,N, we have
PPT Slide
Lager Image
Theorem 2.7 ( [1] , [13] ). If KhkPhPk is ν-convergent to K then for sufficiently large M,N, there exists a constant c independent of M,N such that for j = 1, 2...., m ,
PPT Slide
Lager Image
3. Convergence Rates
In this section we discuss the convergence rates for the approximated eigenvalues and eigenvectors to the exact eigenvalues and exact eigenvectors of the integral operator K . To do this, first we prove the following Lemma.
Lemma 3.1. Let Khk be the Nyström operator defined by (14) with a kernel K (., ., ., .) ∈ C (2k′,2l′) ( D C (2k′,2l′) ( D ), k ′ ≥ p , l ′ ≥ q . Then for u C (2p,2q) ( D ), the following hold
PPT Slide
Lager Image
PPT Slide
Lager Image
Proof . Using the estimates (6), we have
PPT Slide
Lager Image
and
PPT Slide
Lager Image
Combining these estimates with the identity (7), proof of (22) follows.
To prove the estimate (23), let us denote
PPT Slide
Lager Image
Since H ( t ) and
PPT Slide
Lager Image
are orthogonal polynomials of degree p and q , respectively, and the numerical quadratures defined by (10) and (11) have degree of precision 2 k ′ and 2 l ′, respectively, it follows that, for k ′ ≥ p and l ′ ≥ q ,
PPT Slide
Lager Image
PPT Slide
Lager Image
Since Ph is the interpolatory projection interpolating at u ( s, t ) in the first variable s at the points s m,0 , s m,1 , . . . , s m,p−1 in the subintervals [ xm , x m+1 ], m = 0, 1, . . . , M − 1, we have for s ∈ [ xm , x m+1 ], t ∈ [ c, d ],
PPT Slide
Lager Image
where δ (p,0) u ( s, t ) = [ s m,0 , s m,1 , . . . , s m,p−1 , s ; t ] u be the Newton divided difference of u in first variable. Similarly, since Pk is the interpolatory projection interpolating at u ( s, t ) in the second variable t at the points t n,0 , t n,1 , . . . , t n,q−1 in the subintervals [ yn , y n+1 ], n = 0, 1, . . . , N − 1, we have for t ∈ [ yn , y n+1 ], s ∈ [ a, b ],
PPT Slide
Lager Image
where δ (0,q) u ( s, t ) = [ s ; t n,0 , t n,1 , . . . , t n, q−1 , t ] u be the Newton divided difference of u in second variable. Now using the identity (7), we have
PPT Slide
Lager Image
For the first term in the above, for any ( s, t ) ∈ D , using (26), we obtain
PPT Slide
Lager Image
where gi ( xm,i , yn,j ) = Ks,t ( xm,i , yn,j ) δ (p,0) u ( xm,i , yn,j ). The Taylor’s expansion of gi ( xm,i , yn,j ) = gi ( xm + cih , yn,j ) at the point xm is given by
PPT Slide
Lager Image
where ξm ∈ [ xm , x m+1 ]. Using (30) in the estimate (29), we obtain
PPT Slide
Lager Image
Using the estimate (24) in (31), it follows that
PPT Slide
Lager Image
On the similar mechanism, for the second term in (28), we can prove that
PPT Slide
Lager Image
Let δ (p,q) u ( s, t ) = [ s m,0 , s m,1 , . . . , s m,p−1 , s ; t n,0 , t n,1 , . . . , t n,q−1 , t ] u be p and q th Newton divided difference of u in first and second variables, respectively. Then we have
PPT Slide
Lager Image
Using this in the third term of (28), we have
PPT Slide
Lager Image
where gi,j ( xm,i , yn,j ) = Ks,t ( xm,i , yn,j ) δ (p,q) u ( xm,i , yn,j ). The Taylor’s series expansion for gi,j ( xm,i , yn,j ) = gi,j ( xm + cih , yn + djk ) at the point xm and yn is given by
PPT Slide
Lager Image
where r 1 = p + q . Using (34) in the above equation and by adjustment with the the estimates (24) and (25), we obtain
PPT Slide
Lager Image
where
PPT Slide
Lager Image
is a constant independent of h and k . Combining the estimates (28), (32) and (35), the result (23) follows. This completes the proof.
Theorem 3.2. Let K be a compact integral operator with a kernel K (., ., ., .) ∈ C (2k′,2l′) ( D ) × C (2k′,2l′) ( D ), k ′ ≥ p, l ′ ≥ q and Khk be the Nyström operator defined by (14). Then the following hold
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
Proof . Replacing u by Ku in (20), and using the estimate (19), we obtain
PPT Slide
Lager Image
where c is a constant independent of h and k . This completes the proof of (36). Now replacing u by Ku in (22) and using the estimate (19), we see that
PPT Slide
Lager Image
this proves the estimate (37). Again replacing u by Ku in (23), then we obtain
PPT Slide
Lager Image
this proves the estimate (38).
Theorem 3.3. Assume that all the conditions of theorem 3.2 hold. Then the following hold
PPT Slide
Lager Image
PPT Slide
Lager Image
Proof . Since
PPT Slide
Lager Image
the proof follows from the above Theorem 3.2.
In the following Theorem we give the superconvergence results for the eigenvalues and eigenvectors
Theorem 3.4. Suppose K is a compact integral operator with a kernel function K (., ., ., .) ∈ C (2k′,2l′) ( D ) × C (2k′,2l′) ( D ), k ′ ≥ p, l ′ ≥ q , and suppose that λ be the eigenvalue of K with algebraic multiplicity m and ascent ℓ. Let { KhkPhPk } and { PhPkKhk } be a sequence of bounded operators on
PPT Slide
Lager Image
, which converges to K in ν convergence. Then
PPT Slide
Lager Image
In particular, for any
PPT Slide
Lager Image
we have
PPT Slide
Lager Image
For j = 1, 2, . . . , m ,
PPT Slide
Lager Image
Proof . The proof follows directly using the Theorems 2.5, 2.6, 2.7 and 3.3.
Remark : From Theorem 3.4, we observe that discrete collocation eigenvectors converges with the order of convergence O (max{ h min{p,2k′} , k min{q,2l′} }) where as iterated discrete collocation eigenvectors and eigenvalues converges with the order of convergence O (max{ h {min{2p,2k′} , k {min{2q,2l′} }). This shows that iterated discrete eigenvectors gives superconvergence results over the discrete collocation eigenvectors. By choosing the degree of precisions of the numerical quadrature rules sufficiently large, i.e., 2 k ′ ≥ 2 p and 2 l ′ ≥ 2 q on [ a, b ] and [ c, d ], respectively, we obtain the superconvergence results for the eigenvalues and eigenvectors in the discrete collocation and iterated discrete collocation methods.
4. Richardson Extrapolation
In this section, we derive an asymptotic error expansions (cf., [10] , [11] ) for the iterated discrete collocation operator KhkPhPk and an asymptotic error expansion of arithmetic mean of approximate eigenvalues. We then apply Richardson extrapolation to obtain improved error bounds for the eigenvalues.
Lemma 4.1. (Euler-MacLaurin summation formulae) ( [7] ).
Let f ( x, y ) ∈ C r+1 ( D ), 0 ≤ τ ≤ 1, 0 ≤ θ ≤ 1. Then
PPT Slide
Lager Image
where Bi ( t ) are Bernoulli polynomials of degree i.
Theorem 4.2 ( [15] ). Let K be a compact integral operator with a kernel K (., ., ., .) ∈ C ( D ) × C r+1 ( D ) and Khk be the Nyström operator defined by (14), then there holds
PPT Slide
Lager Image
where D 2i , ε 2j , and F 2i , 2j are bounded linear operators independent of h and k .
Lemma 4.3 ( [7] ). Assume that u ( x, y ) ∈ C r+1 ( D ). Let Ph and Pk be the interpolatory projections defined by (4) and (5), respectively. Then for any ( x, y ) ∈ Imn, the following holds
PPT Slide
Lager Image
where
PPT Slide
Lager Image
PPT Slide
Lager Image
Proposition 4.4. Assume that the kernel K (., ., ., .) ∈ C r+1 ( D ) × C r+1 ( D ) and u C r+1 ( D ), then the following holds
PPT Slide
Lager Image
where R 2i , 2j , S 2i , 2j and T 2i , 2j are bounded linear operators on C r+1 ( D ).
Proof . Using the definition of Khk defined by (14) and the Lemma 4.3, we have
PPT Slide
Lager Image
Now we consider I 1 ,
PPT Slide
Lager Image
Using Euler-Maclaurin summation formula in I 1 , we obtain
PPT Slide
Lager Image
where
PPT Slide
Lager Image
and S ( Bb ) is the numerical quadrature of Ba defined as in (11). Since τ 0 , τ 1 , . . . , τ p−1 are symmetric points in the interval [0, 1], i.e., τj = 1 − τ p−1−j for 0 ≤ j p − 1, we have
PPT Slide
Lager Image
and
PPT Slide
Lager Image
Using these estimates we obtain Φ μ (1− ci ) = (−1) μ Φ μ ( ci ), i = 0, 1, 2, . . . , k ′−1. Also note that Ba ( ci ) = (−1) aBa (1 − ci ), ci = 1 − c k′−1−i and w k′−i−1 = wi , for 0 ≤ i k ′ − 1. Hence we have
PPT Slide
Lager Image
From this, it follows that Aμ,a = 0, when μ + a = odd. Since the quadrature rule (10) is exact for polynomial of degree less than 2 k ′ and
PPT Slide
Lager Image
is orthogonal to all polynomial of degree less than p and
PPT Slide
Lager Image
is a polynomial of degree μ p , then for μ + a < 2 p, k ′ ≥ p , we have
PPT Slide
Lager Image
and
PPT Slide
Lager Image
Thus we obtain
PPT Slide
Lager Image
Since dj , j = 0, 1, . . . , l ′ − 1 are the Gauss points in the interval (0, 1) and the quadrature rule (3) is an symmetric quadrature rule, we have d l′−j−1 = 1 − dj and
PPT Slide
Lager Image
for j = 0, 1, . . . , l ′ − 1. Noting that the Bernoulli polynomials have the property that Bb (1 − d ) = (−1) bBb ( d ), we have
PPT Slide
Lager Image
From this, it follows that S ( Bb ) is zero when b is odd. Also we have S ( B 0 ) = 1. Now since the degree of precision of this quadrature rule (10) is 2 l ′−1, it follows that for 1 ≤ b ≤ 2 l ′ − 1,
PPT Slide
Lager Image
Thus we have
PPT Slide
Lager Image
Combining the estimates (46) and (45) with (44) we obtain
PPT Slide
Lager Image
where,
PPT Slide
Lager Image
Similarly we can prove that
PPT Slide
Lager Image
where,
PPT Slide
Lager Image
Similarly for I 3 , we can prove that
PPT Slide
Lager Image
where,
PPT Slide
Lager Image
Now combining the estimates for I 1 , I 2 and I 3 with (43), we obtain the following asymptotic expansion This completes the proof.
Theorem 4.5. Let K be a compact integral operator with the kernel K (., ., ., .) ∈ C r+1 ( D ) × C r+1 ( D ) with p = k and q = l ′. Then the following holds
PPT Slide
Lager Image
where U 2i , V 2j and W 2i,2j are bounded linear operators on C r+1 ( D ).
Proof . Combining Theorems 4.4 and 4.2, we have
PPT Slide
Lager Image
where U 2i = R 2i,0 D 2i , V 2j = S 0,2j ε 2j , and W 2i,2j = R 2i,2j + S 2i,2j T 2i,2j F 2i,2j are bounded linear operators on C r+1 ( D ). This completes the proof.
In rest of the paper, we choose the domain D = [ a, b ]×[ a, b ] and the partition
PPT Slide
Lager Image
Also we choose the numerical quadratures (10) and (11) to be same, i.e., k ′ = l ′. Then we have the following corollary.
Corollary 4.6. Let Khk be the Nystr¨om operator defined by (14) with p = q = k ′ = l ′. Assume that the kernel K (., ., ., .) ∈ C 2p+2 ( D ) × C 2p+2 ( D ). Then the following holds
PPT Slide
Lager Image
where C 2p = U 2p + V 2p is a bounded linear operators on C 2p+2 ( D ).
Richardson Extrapolation For Eigenvalue problem: By the similar way as followed in [15] , we obtain the following theorem which gives an asymptotic error expansion of the arithmetic mean of eigenvalues by iterated discrete collocation method.
Theorem 4.7 ( [15] ). Let λ be the eigenvalue of K with algebraic multiplicity m and
PPT Slide
Lager Image
be the arithmetic mean of the eigenvalues
PPT Slide
Lager Image
Then the following holds
PPT Slide
Lager Image
where Q 2p = C 2p PS K U 2p is a bounded linear operator independent of h.
According to the asymptotic expansion (47), the Richardson extrapolation for eigenvalue approximation should be the following. We first divide each subinterval with respect to the partitions of
PPT Slide
Lager Image
into two halves which makes up a new partitions denoted by
PPT Slide
Lager Image
PPT Slide
Lager Image
Here
PPT Slide
Lager Image
and D = [ a, b ] × [ a, b ]. We then have following asymptotic expansion for eigenvalue approximation with respect to this new partitions,
PPT Slide
Lager Image
From the asymptotic expansions (47) and (48), the Richardson extrapolation for the eigenvalue approximation is defined by
PPT Slide
Lager Image
In the following Theorem we give the superconvergence rates for the eigenvalue approximation using Richardson extrapolation.
Theorem 4.8. Assume that conditions of Theorem 4.7 hold and the Richardson extrapolation
PPT Slide
Lager Image
is defined by (49). Then the following error estimate holds
PPT Slide
Lager Image
5. Numerical Example
Consider the eigenvalue problem (2), for the integral operator K (3) for various smooth kernels K ( s, t, x, y ).
Let
PPT Slide
Lager Image
be the space of piecewise constant functions (p=q=1) on [0, 1]×[0, 1] with respect to the initial uniform partitions
PPT Slide
Lager Image
with
PPT Slide
Lager Image
We choose numerical quadrature as the one-point composite Gaussian quadrature formula which is exact for all polynomials of degree less than 2, that is k = l = 1.
The quadrature points and weights are given by
PPT Slide
Lager Image
and
PPT Slide
Lager Image
respectively.
For different kernels and for different values of M , we compute the discrete collocation eigen vector
PPT Slide
Lager Image
iterated eigen vector
PPT Slide
Lager Image
and eigenvalue
PPT Slide
Lager Image
and approximated eigenvalue in Richardson extrapolation
PPT Slide
Lager Image
Denote
PPT Slide
Lager Image
where
PPT Slide
Lager Image
is the step length. For M = 2, 4, 8, 16, we compute α, β, γ and δ which are listed in the following Table.
Since k ′ = l ′ = 1, we get the theoretical convergence of the order of eigenvector is 1, the orders of the iterated eigenvector and eigenvalues are 2, and the order of eigenvalue in Richardson extrapolation is 4. In the following Table 1 and Table 2 , the numerical results agrees with the theoretical results.
Eigenvector error bounds
PPT Slide
Lager Image
Eigenvector error bounds
Eigenvalue error bounds
PPT Slide
Lager Image
Eigenvalue error bounds
Example. K ( s, t, x, y ) = s sin( t ) + xey , [ a, b ] = [0, 1], [ c, d ] = [0, 1].
BIO
Bijaya Laxmi Panigrahi received M.Sc. from Sambalpur University, India, and Ph.D from Indian Institute of Technology Kharagpur, India. She is currently working as a Lecturer in the Department of Mathematics, Sambalpur University. Her research interests include spectral approximations of integral operators.
Department of Mathematics, Sambalpur University, Jyoti Vihar, Odisha-768019, India.
e-mail: bijayalaxmi rs@maths.iitkgp.ernet.in
Gnaneshwar Nelakanti received Ph.D. from Indian Institute of Technology Bombay, India. He is currently working as Asst. Professor in Indian Institute of Technology Kharag-pur, India. His research interests include Inverse and ill-posed problems and Approximate solutions of operator equations.
Department of Mathematics, Indian Institute of Technology Kharagpur, West Bengal - 721302, India.
e-mail: gnanesh@maths.iitkgp.ernet.in
References
Ahues M. , Largillier A. , Limaye B.V. 2001 Spectral computations for bounded operators Chapman and Hall/CRC New York
Atkinson K.E. 1997 The numerical solution of integral equations of the second kind Cambridge University Press Cambridge, UK
Chatelin F. 1983 Spectral approximation of linear operators Academic Press New York
Chen Z. , Long G. , Nelakanti G. (2009) Richardson extrapolation of iterated discrete projection methods for eigenvalue appoximation J. Comp. and Appl. Math. 223 48 - 61    DOI : 10.1016/j.cam.2007.12.019
Conte S.D. , de Boor C. 2005 Elementary numerical analysis Tata McGraw-Hill
Han G. (1995) Extrapolation of a discrete collocation-type method of Hammerstein equations J. Comp. Appl. Math. 61 73 - 86    DOI : 10.1016/0377-0427(94)00049-7
Han G. , Zhang L. (1994) Asymptotic Error Expansion of Two-Dimensional Volterra Integral Equation by Iterated Collocation Appl. Math. Comp. 61 269 - 285    DOI : 10.1016/0096-3003(94)90050-7
Isaacson E. , Keller H.B. 1966 Analysis of Numerical Methods John Wiley and Sons
Kulkarni R.P. , Nelakanti G. (2002) Spectral approximation using iterated discrete Galerkin method Numer. Funct. Opt. 23 91 - 104    DOI : 10.1081/NFA-120003672
Lin Q. , Sloan I.H. , Xie R. (1990) Extrapolation of the iterated-collocation method for integral equations of the second kind SIAM J. Numer. Anal. 27 1535 - 1541    DOI : 10.1137/0727090
McLean W. (1989) Asymptotic error expansions for numerical solutions of integral equations IMA J. Numer. Anal. 9 373 - 384    DOI : 10.1093/imanum/9.3.373
Nelakanti G. (2007) A degenerate kernel method for eigenvalue problems of compact integral operators Adv. Comput. Math. 27 339 - 354    DOI : 10.1007/s10444-005-9005-9
Nelakanti G. , Ph.D Thesis 2003 Spectral approximation for integral operators Indian Insitute of Technology Bombay, India Ph.D Thesis
Osborn J.E. (1975) Spectral approximation for compact operators Math. Comp. 29 712 - 725    DOI : 10.1090/S0025-5718-1975-0383117-3
Panigrahi B.L. , Nelakanti G. (2012) Richardson extrapolation of iterated discrete Galerkin method for eigenvalue problem of a two dimensional compact integral operator J. Sci. Comp. 51 421 - 448    DOI : 10.1007/s10915-011-9516-0
Sloan I.H. (1976) Iterated Galerkin method for eigenvalue problems SIAM J. Numer. Anal. 31 753 - 760    DOI : 10.1137/0713061