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

- Received : February 14, 2012
- Accepted : April 16, 2014
- Published : September 30, 2014

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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.
K
defined on
by
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
.
We are interested in the following eigenvalue problem: find
and
λ
∈ ℂ − {0} such that
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.
K
defined on
by
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
ℓ
. 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
be the uniform partitions of finite intervals [
a, b
] and [
c, d
], respectively, defined by
and
for
m
= 0, 1, 2, . . . ,
M
− 1 and
n
= 0, 1, 2, . . . ,
N
− 1. These partitions define a grid for
D
,
: 0 ≤
m
≤
M
− 1, 0 ≤
n
≤
N
− 1}. Set
and
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,
is the finite element space of dimension
MNpq
, which is the tensor product space of univariate spline spaces
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
s_{m,i}
=
x_{m}
+
τ_{i}h
and
t_{n,j}
=
y_{n}
+
θ_{j}k
, be the collocation points on [
x_{m}
,
x
_{m+1}
],
m
= 0, 1, . . . ,
M
−1, and [
y_{n}
,
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
and
be the interpolatory projection with respect to the nodes {
s_{m,i}
} and {
t_{n,j}
}, respectively, that is, for
u
∈
L
^{∞}
([
a, b
]),
then there holds(
[3]
), ∥
P_{h}
∥
_{∞}
≤
c
_{1}
< ∞, ∥
P_{k}
∥
_{∞}
≤
c
_{2}
< ∞ and for any
u
∈
C^{p}
[
a, b
] and
u
∈
C^{q}
[
c, d
], there holds,
Note that
As a consequence, from (6), it follows that
Let
ϕ_{im}
,
ψ_{jn}
denote the Lagrange polynomials of degree
p
− 1 and
q
− 1 on the subintervals [
x_{m}
,
x
_{m+1}
],
m
= 0, 1, . . . ,
M
− 1 and [
y_{n}
,
y
_{n+1}
],
n
= 0, 1, . . . ,
N
− 1 respectively, where,
j
= 0, 1, . . .
q
− 1,
i
= 0, 1, . . . ,
p
− 1. Then it follows that
and
Then we have
Now the collocation method for solving the eigenvalue problem (2) is defined as follows: find
∥
u_{hk}
∥ = 1 and
λ_{hk}
∈ ℂ − {0} such that
for
i
′ = 0, 1, . . .
p
−1,
m
′ = 0, 1, . . .
M
−1,
j
′ = 0, 1, . . .
q
−1,
n
′ = 0, 1, . . .
N
−1. Using
the equation (9) can be converted to the matrix eigenvalue problem. The iterated eigenvector is defined by
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],
be the numerical quadrature with weights
and quadrature points
c_{i}, i
= 0, 1, . . . ,
k
′ − 1,
d_{j}, 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
where
x_{m,i}
=
x_{m}
+
c_{i}h
,
i
= 0, 1, . . . ,
k
′−1, and
y_{n,j}
=
y_{n}
+
d_{j}k
,
j
= 0, 1 . . . ,
l
′−1, be the quadrature points on the subintervals [
x_{m}
,
x
_{m+1}
],
m
= 0, 1, . . . ,
M
− 1 of [
a, b
] and [
y_{n}
,
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
Let
be the Nyström operator defined for
Now replacing the integral operator
K
by the Nyström operator (14), the matrix eigenvalue problem leads to the discrete collocation method,
By solving this discrete matrix eigenvalue problem (15), we find the eigenvalue
and
β
= [
β_{ij}, i
= 0, 1 . . . ,
p
−1,
j
= 0, 1, . . . ,
q
−1]. Then the discrete collocation eigenvector is defined by,
The discrete matrix eigenvalue problem (15) can be written in operator form as
Next we define the iterated discrete collocation eigenvector by
Clearly we see that
and
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
) =
K_{s,t}
(
x, y
). For
K
(., ., ., .) ∈
C
^{(i,j)}
(
D
) ×
C
^{(i′,j′)}
(
D
), denote
For any
α, β, γ, δ
∈ ℕ, we set
Then we have
and
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 K_{hk} be the Nyström operator defined by (14), then for any u
∈
C
^{(2k′,2l′)}
(
D
),
the following holds
where c is independent of h and k.
Definition 2.2
(
[1]
). Let
be a Banach space and,
T
and
T_{n}
are bounded linear operators from
into
. Then {
T_{n}
} is said to be ν-convergent to
T
, if
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
be a Banach space and S
⊂
is a relatively compact set. Assume that T and T_{n} are bounded linear operators from
into
satisfying
∥
T_{n}
∥ ≤
c for all n
∈ ℕ,
and for each x
∈
S
, ∥
T_{n}x−Tx
∥ → 0
as n
→ ∞,
where c is a constant independent of n. Then
∥
T_{n}x
−
Tx
∥ → 0
uniformly for all
x ∈
S
.
Theorem 2.4.
P_{h}P_{k}K_{hk}
and
K_{hk}P_{h}P_{k}
are
ν-convergent to K
.
Proof
. Since
K_{hk}
,
P_{h}
and
P_{k}
are uniformly bounded, it follows that ∥
P_{h}P_{k}K_{hk}
∥
_{∞}
≤ ∥
P_{h}
∥
_{∞}
∥
P_{k}
∥
_{∞}
∥
K_{hk}
∥
_{∞}
< ∞ and ∥
K_{hk}P_{h}P_{k}
∥
_{∞}
≤ ∥
K_{hk}
∥
_{∞}
∥
P_{h}
∥
_{∞}
∥
P_{k}
∥
_{∞}
< ∞. Now using (8) and Theorem 2.1, we see that
This shows that
P_{h}P_{k}K_{hk}
point wise converges to
K
.
Let
be a closed unit ball in
. 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
Since
P_{h}P_{k}
is bounded and
K_{hk}
compact,
S
′ = {
P_{h}P_{k}K_{hk}x
:
x
∈
B
} is a relatively compact set. Thus
as
h, k
→ 0. Combining all these results leads to the first result that
P_{h}P_{k}K_{hk}x
is
ν
-convergent to
K
. The proof of
K_{hk}P_{h}P_{k}
is
ν
-convergent to
K
follows by similar steps as in above.
Since
P_{h}P_{k}K_{hk}
and
K_{hk}P_{h}P_{k}
are
ν
-convergent to
K
, the spectrum of both
P_{h}P_{k}K_{hk}
and
K_{hk}P_{h}P_{k}
inside Γ consists of
m
eigenvalues say
counted accordingly to their algebraic multiplicities inside Γ with ascent
ℓ
(cf.,
[3
,
14]
). Let
denote their arithmetic mean and we approximate
λ
by
Let
be the spectral projections of
K
associated with their corresponding spectra inside Γ. Similarly,
be the spectral projections of
P_{h}P_{k}K_{hk}
and
K_{hk}P_{h}P_{k}
, respectively. Let
be the ranges of the spectral projections
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
let
then
is known as the gap between
We quote the following three Lemmas, which give the error bounds for the eigenelements.
Theorem 2.5
(
[1]
,
[13]
).
Let P_{h}P_{k}K_{hk} be ν-convergent to K. Then for sufficiently large M,N, there exists a constant c independent of M,N, we have
Theorem 2.6
(
[13]
,
[16]
).
Let K_{hk}P_{h}P_{k} be ν-convergent to K. Then for sufficiently large M,N, there exists a constant c independent of M,N, we have
Theorem 2.7
(
[1]
,
[13]
).
If K_{hk}P_{h}P_{k} 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
,
K
. To do this, first we prove the following Lemma.
Lemma 3.1.
Let K_{hk} 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
Proof
. Using the estimates (6), we have
and
Combining these estimates with the identity (7), proof of (22) follows.
To prove the estimate (23), let us denote
Since
H
(
t
) and
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
,
Since
P_{h}
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 [
x_{m}
,
x
_{m+1}
],
m
= 0, 1, . . . ,
M
− 1, we have for
s
∈ [
x_{m}
,
x
_{m+1}
],
t
∈ [
c, d
],
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
P_{k}
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 [
y_{n}
,
y
_{n+1}
],
n
= 0, 1, . . . ,
N
− 1, we have for
t
∈ [
y_{n}
,
y
_{n+1}
],
s
∈ [
a, b
],
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
For the first term in the above, for any (
s, t
) ∈
D
, using (26), we obtain
where
g_{i}
(
x_{m,i}
,
y_{n,j}
) =
K_{s,t}
(
x_{m,i}
,
y_{n,j}
)
δ
^{(p,0)}
u
(
x_{m,i}
,
y_{n,j}
). The Taylor’s expansion of
g_{i}
(
x_{m,i}
,
y_{n,j}
) =
g_{i}
(
x_{m}
+
c_{i}h
,
y_{n,j}
) at the point
x_{m}
is given by
where
ξ_{m}
∈ [
x_{m}
,
x
_{m+1}
]. Using (30) in the estimate (29), we obtain
Using the estimate (24) in (31), it follows that
On the similar mechanism, for the second term in (28), we can prove that
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
Using this in the third term of (28), we have
where
g_{i,j}
(
x_{m,i}
,
y_{n,j}
) =
K_{s,t}
(
x_{m,i}
,
y_{n,j}
)
δ
^{(p,q)}
u
(
x_{m,i}
,
y_{n,j}
). The Taylor’s series expansion for
g_{i,j}
(
x_{m,i}
,
y_{n,j}
) =
g_{i,j}
(
x_{m}
+
c_{i}h
,
y_{n}
+
d_{j}k
) at the point
x_{m}
and
y_{n}
is given by
where
r
_{1}
=
p
+
q
. Using (34) in the above equation and by adjustment with the the estimates (24) and (25), we obtain
where
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
K_{hk}
be the Nyström operator defined by (14). Then the following hold
Proof
. Replacing
u
by
Ku
in (20), and using the estimate (19), we obtain
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
this proves the estimate (37). Again replacing
u
by
Ku
in (23), then we obtain
this proves the estimate (38).
Theorem 3.3.
Assume that all the conditions of theorem 3.2 hold. Then the following hold
Proof
. Since
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
{
K_{hk}P_{h}P_{k}
}
and
{
P_{h}P_{k}K_{hk}
}
be a sequence of bounded operators on
,
which converges to K in ν
−
convergence. Then
In particular, for any
we have
For j
= 1, 2, . . . ,
m
,
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.
K_{hk}P_{h}P_{k}
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
where B_{i}
(
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
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 P_{h} and P_{k} be the interpolatory projections defined by (4) and (5), respectively. Then for any
(
x, y
) ∈
I_{mn}, the following holds
where
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
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
K_{hk}
defined by (14) and the Lemma 4.3, we have
Now we consider
I
_{1}
,
Using Euler-Maclaurin summation formula in
I
_{1}
, we obtain
where
and
S
(
B_{b}
) is the numerical quadrature of
B_{a}
defined as in (11). Since
τ
_{0}
,
τ
_{1}
, . . . ,
τ
_{p−1}
are symmetric points in the interval [0, 1], i.e.,
τ_{j}
= 1 −
τ
_{p−1−j}
for 0 ≤
j
≤
p
− 1, we have
and
Using these estimates we obtain Φ
_{μ}
(1−
c_{i}
) = (−1)
^{μ}
Φ
_{μ}
(
c_{i}
),
i
= 0, 1, 2, . . . ,
k
′−1. Also note that
B_{a}
(
c_{i}
) = (−1)
^{a}B_{a}
(1 −
c_{i}
),
c_{i}
= 1 −
c
_{k′−1−i}
and
w
_{k′−i−1}
=
w_{i}
, for 0 ≤
i
≤
k
′ − 1. Hence we have
From this, it follows that
A_{μ,a}
= 0, when
μ
+
a
= odd. Since the quadrature rule (10) is exact for polynomial of degree less than 2
k
′ and
is orthogonal to all polynomial of degree less than
p
and
is a polynomial of degree
μ
−
p
, then for
μ
+
a
< 2
p, k
′ ≥
p
, we have
and
Thus we obtain
Since
d_{j}
,
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 −
d_{j}
and
for
j
= 0, 1, . . . ,
l
′ − 1. Noting that the Bernoulli polynomials have the property that
B_{b}
(1 −
d
) = (−1)
^{b}B_{b}
(
d
), we have
From this, it follows that
S
(
B_{b}
) 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,
Thus we have
Combining the estimates (46) and (45) with (44) we obtain
where,
Similarly we can prove that
where,
Similarly for
I
_{3}
, we can prove that
where,
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
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
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
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 K_{hk} 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
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
be the arithmetic mean of the eigenvalues
Then the following holds
where
Q
_{2p}
=
C
_{2p}
P^{S}
−
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
into two halves which makes up a new partitions denoted by
Here
and
D
= [
a, b
] × [
a, b
]. We then have following asymptotic expansion for eigenvalue approximation with respect to this new partitions,
From the asymptotic expansions (47) and (48), the Richardson extrapolation for the eigenvalue approximation is defined by
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
is defined by (49). Then the following error estimate holds
K
(3) for various smooth kernels
K
(
s, t, x, y
).
Let
be the space of piecewise constant functions (p=q=1) on [0, 1]×[0, 1] with respect to the initial uniform partitions
with
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
and
respectively.
For different kernels and for different values of
M
, we compute the discrete collocation eigen vector
iterated eigen vector
and eigenvalue
and approximated eigenvalue in Richardson extrapolation
Denote
where
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
Eigenvalue error bounds
Example.
K
(
s, t, x, y
) =
s
sin(
t
) +
xe^{y}
, [
a, b
] = [0, 1], [
c, d
] = [0, 1].
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

Eigenvalue problem
;
compact integral operator
;
Discrete collocation methods
;
Richardson extrapolation

1. Introduction

Consider the following integral operator
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2. Discrete and Iterated discrete collocation methods

Consider the following compact integral operator
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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
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4. Richardson Extrapolation

In this section, we derive an asymptotic error expansions (cf.,
[10]
,
[11]
) for the iterated discrete collocation operator
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5. Numerical Example

Consider the eigenvalue problem (2), for the integral operator
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Eigenvector error bounds

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Eigenvalue error bounds

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BIO

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Citing 'RICHARDSON EXTRAPOLATION OF ITERATED DISCRETE COLLOCATION METHOD FOR EIGENVALUE PROBLEM OF A TWO DIMENSIONAL COMPACT INTEGRAL OPERATOR
'

@article{ E1MCA9_2014_v32n5_6_567}
,title={RICHARDSON EXTRAPOLATION OF ITERATED DISCRETE COLLOCATION METHOD FOR EIGENVALUE PROBLEM OF A TWO DIMENSIONAL COMPACT INTEGRAL OPERATOR}
,volume={5_6}
, url={http://dx.doi.org/10.14317/jami.2014.567}, DOI={10.14317/jami.2014.567}
, number= {5_6}
, journal={Journal of Applied Mathematics & Informatics}
, publisher={Korean Society of Computational and Applied Mathematics}
, author={PANIGRAHI, BIJAYA LAXMI
and
NELAKANTI, GNANESHWAR}
, year={2014}
, month={Sep}