ding shu tang (ijnme) 2005

8/14/2019 Ding Shu Tang (IJNME) 2005

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2005; 63:15131529

Published online 22 March 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1318

Error estimates of local multiquadric-based differential quadrature(LMQDQ) method through numerical experiments

H. Ding1, C. Shu1, , and D. B. Tang2

1Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent,

Singapore 117576, Singapore2Department of Aerodynamics, Nanjing University of Aeronautics and Astronautics, 29 Yudao Jie,

Nanjing 210016, Peoples Republic of China

SUMMARY

In this article, we present an error estimate of the derivative approximation by the local multiquadric-based differential quadrature (LMQDQ) method. Radial basis function is different from the polynomialapproximation, in which Taylor series expansion is not applicable. So, the present analysis is performedthrough the numerical solution of Poisson equation. It is known that the approximation error ofLMQDQ method depends on three factors, i.e. local density of knots h, free shape parameter c andnumber of supporting knots ns. By numerical experiments, their contribution to the approximationerror and correlation were studied and analysed in this paper. An error estimate O((h/c)n) isthereafter proposed, in which n is a positive constant and determined by the number of supportingknots ns. Copyright 2005 John Wiley & Sons, Ltd.

KEY WORDS: radial basis function; RBF; local multiquadric-based differential quadrature

1. INTRODUCTION

In the past decades, the interpolation theory of radial basis function (RBF) has undergone

intensive research, and nowadays RBF plays an increasingly important role in the field of

reconstructing functions from multivariate scattered data. In general, the interpolation theory

of RBF can be described as follows: if an unknown function f (x) is only known at a finite set

of centres xi , i= 1, . . . , N , the approximation of a function f (x) can be written as a linear

combination of N radial basis functions

f (x

)

=

Nj=1 j(

x xj

2)

+(

x) (1)

Correspondence to: C. Shu, Department of Mechanical Engineering, National University of Singapore, 10 KentRidge Crescent, Singapore 117576, Singapore.

E-mail: [email protected]

Received 23 January 2004

Revised 16 August 2004

Copyright 2005 John Wiley & Sons, Ltd. Accepted 11 November 2004


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1514 H. DING, C. SHU AND D. B. TANG

where s are coefficients to be determined, the radial basis function and the additional

polynomial.

The success of RBF interpolant is due to its excellent performance. In 1982, based on

numerical experiments, Franke [1] gave a comprehensive review on the interpolation methods

for scattered data. From the numerical tests, he found that RBFs performed better than othertested methods regarding accuracy, stability, efficiency, memory requirement, and simplicity

of implementation. Among the tested RBFs, multiquadrics (MQ) yields the most accurate

results. Madych et al. [2] have shown that the MQ interpolation scheme converges faster

as the partial dimension increases, and converges exponentially as the density of the nodes

increases.

It is known that a good interpolation scheme has great potential for solving partial differential

equations (PDEs). Kansa [3, 4] made the first attempt to apply RBFs to solve PDEs. Since then,

motivated by attractive properties of RBFs such as high convergence order and naturally mesh-

free, more and more researchers cast their sights on the implementation of RBFs in PDE solvers

and enjoyed considerable successes. Fasshauer [5] proposed an alternate method based on the

Hermite RBFs, which can generate symmetric coefficient matrix and guarantee the solvability of

the related linear equations. Cheng et al. [6] presented a so-called H-c multiquadric collocation

method, which showed exponential convergence by numerical experiments. It is known that

RBF approximation tends to degrade in the boundary neighbouring regions. In this regard,

Chen [7] proposed a new RBF collocation approach based on Kansas method to improve

the solution accuracy near the boundary. Wu [8] gave the convergence proofs for the use of

RBF HermiteBirkhoff in solving PDEs. Schaback et al. [9] provided an error bound for the

RBF-based collocation method. Other great contributions in using RBFs to solve PDEs include

the work of Fornberg [10], Hon and Wu [11], Chen and Hon [12], Chen and Tanaka [13],

Chen et al. [14].

In the previous studies, radial basis functions are usually used in the collocation technique

to solve PDEs in a global manner. In other words, the support of every node covers the

whole domain. Consequently, the system of linear equations arising from the collocation methodusually has very large condition number, and becomes increasingly ill-conditioned as the number

of nodes increases. Therefore, when complex problems are confronted and a large number of

collocation nodes are required to catch the physical details, the problem of ill-conditioning is

almost unavoidable. In this sense, the pure global collocation method is difficult to be applied for

practical problems. To remove the drawbacks mentioned above, Kansa and Hon [15] proposed

a domain decomposition method, which can reduce many orders of magnitude of the condition

number of resultant matrix. For the same reason, Shu et al. [16] proposed a local RRF-

based differential quadrature (LRBFDQ) method. The method can be employed with a large

number of nodes and requires no extra effort on the division of the computational domain.

It has been proved to be a robust and effective method by solving buoyancy-driven natural

convection problem [16]. However, due to the lack of theoretical analysis, there is no error

estimate available for this method to date. The objective of this paper is to present a priorerror estimate for the derivative approximation by LRBFDQ method. The performed analysis of

error estimate is based on the numerical experiments. This is because the conventional Taylor

series expansion for error estimate used in the finite difference scheme is not applicable in

the RBF approximation. At this time, the theoretical error analysis for derivative approximation

by RBF is not available.

Copyright 2005 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2005; 63:15131529


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incorporated the shape parameter as O(eacc/ h), where a is a positive constant. For the RBF-

based numerical scheme, Wu [8] provided the convergence proof that the convergence order is

of O(hd+1), where h is the density of the collocation points and d is the spatial dimension.

In this study, we attempt to establish an error estimate for the LMQDQ method by numerical

experiments.

2.2. Derivative approximation by LMQDQ

As its name implies, the LMQDQ method is based on the multiquadric RBFs and differential

quadrature (DQ) technique. The concept of DQ was first proposed by Bellman et al. [18] to

approximate the derivative of a smooth function. From the viewpoint of derivative approxima-

tion, the essence of the DQ method is that the partial derivative of any dependent variable can

be approximated by a weighted linear sum of functional values at all discrete points along one

co-ordinate. In other words, the DQ approximation of the mth order derivative of a function

f(x) at xi can be expressed as

m

f

xm

x=xi

=N

j=1

w(m)i, jf (xj), i =1, 2, . . . , N (3)

where xj are the co-ordinates of discrete points in the domain. f (xj) and w(m)i, j are the function

values at these points and the related weighting coefficients.

In the LMQDQ method, we localize the DQ approximation, and extend it to the multi-

dimensional case. Thus, Equation (3) in LMQDQ method is changed to

m

f

xmk x=xi=

ns

j=1 w(mk)i, j fj, i =1, 2, . . . , N (4)

where k is the dimension, ns is the number of supporting points in the local support of node

i, which may vary at different nodes for the real world computation. In this study, i t is a

fixed number so that we can investigate the dependence of solution accuracy on it. It should

be noted that the subscript i represents the global node index while j the local index in the

support of node i.

To determine the weighting coefficients w(m)i, j

in Equation (4), a set of base functions are

required. In the present LMQDQ method, multiquadric radial basis function is chosen as the

base function.

Substituting the set of radial basis functions into Equation (4), the determination of cor-

responding coefficients for the first-order derivative at the reference point xi or node i is

equivalent to solving the following linear equations:

k(xi )

x=

nsj=1

w(1x)i, j k(xj) k =1, 2, . . . , ns (5)

For simplicity, the notation k(x) is adopted to replace (x, xk) in Equation (1).



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ERROR ESTIMATES OF LMQDQ METHOD 1517

The above equations can be rewritten in the matrix form

1(xi )

x

2(xi )

x

...

ns (xi )

x

(xi )x

=

1

(x1

) 1

(x2

) 1

(xns

)

2(x1) 2(x2) 2(xns )

......

. . . ...

ns (x1) ns (x2) ns (xns )

[A]

w(1x)i, 1

w(1x)i, 2

...

w(1x)i, ns

{w}i

(6)

Clearly, there exists a unique solution only if the collocation matrix [A] is non-singular.

The non-singularity of the collocation matrix [A] depends on the properties of used RBFs.Micchelli [19] proved that matrix [A] is conditionally positive definite for MQRBFs. This fact

cannot guarantee the non-singularity of matrix [A]. Hon and Schaback [20] showed that cases

of singularity are quite rare, and not serious objection to a valuable numerical technique.

Therefore, the coefficient vector {w}i can be obtained by

{w}i = [A]1

(xi )

x

(7)

Then, the coefficient vector {w}i can be used to approximate the first-order derivative in the x

direction for any unknown smooth function at node i. The calculation of weighting coefficients

for other derivatives can follow the same procedure.

From the procedure of DQ approximation of derivatives, it can be observed that the weightingcoefficients are only dependent on the selected RBFs and the distribution of the supporting

points in the local support. During the period of numerical simulation, they are only computed

once, and stored for all numerical discretization. Once the coefficients are computed, they will

be stored and used to discretize the partial differential equation in a similar manner as in the

traditional finite difference method. It should be noted that the computed coefficients can be

consistently well applied to linear and non-linear problems. Therefore, it is very convenient to

use LMQDQ method to solve complex non-linear problems such as NavierStokes equations in

fluid mechanics. From the above, we can also see that the implementation of LMQDQ method

is very simple and straightforward.

2.3. Determination of local support for interior nodes

As discussed in the previous section, the derivative approximation at each interior node is

performed within its local support. Therefore, the local support of interior nodes must be

determined and prepared before doing numerical discretization. In this section, three concrete

approaches used to determine the local support for each interior node are briefly described.

The most common approach is that the practitioner provides the shape and size of the local

support explicitly. For example, the local support can be a circle in two-dimension and a sphere



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Reference knot

Supporting knots

Non-supporting knots

Figure 1. Supporting knots around a reference knot.

in three-dimension, i.e. for a central node i, its local support is defined as

Si = {j :0 < xjxi 2 < ri } (8)

where ri is the radius of the circle or sphere, and represents the size of the local support.

A two-dimensional local support of this type is shown in Figure 1. The second approach restricts

the number of supporting points within the local support to a given value. The supporting points

are chosen by its distance to the central node, and the nearer one has higher priority. These

two approaches work well when the nodes in the domain are uniformly distributed or the local

density of nodes varies smoothly. However, when the nodes are unevenly distributed locally,

these two methods may not be appropriate anymore. For example, when the nodes in the

domain have a track-like distribution, these two approaches will capture too many supporting

points in one direction and not enough in the others. The third approach can optimize the

choice of local support by implementing local Delaunay triangulation among the points which

surround the interior node xi . However, this method requires much more computational effort as

compared with the first and second methods. In the present study, since the nodes are uniformlydistributed and the number of supporting points is considered as one important factor for the

error estimate in LMQDQ method, the second approach is adopted to determine the region of

local support.

2.4. Solution procedure of LMQDQ method

The general solution procedure of LMQDQ method for the PDEs is shown below:

(1) set up the node distribution in the domain;

(2) determine the local support for the interior nodes;

(3) calculate the weighting coefficients for the related derivatives in the partial differential

equations at the interior nodes;

(4) discretize the partial differential equations with the computed weighting coefficients;

(5) solve the resultant algebraic equations.

For the treatment of boundary conditions involved with derivatives, there are two approaches

to discretize the derivatives. The first approach is that one can calculate the corresponding

weighting coefficients on the boundary nodes similar to the interior node. This approach has

the advantage of consistent discretization as the interior node, but suffers from the decrease of



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accuracy due to the biased support. The second approach is to generate so-called locally orthog-

onal grids around the boundaries, then discretize the derivatives by one-side finite difference

schemes. For more details of this approach, one can refer to Reference [16].

3. NUMERICAL RESULTS AND DISCUSSION

3.1. Numerical experiments

In this section, numerical experiments are carried out by solving a sample problem to study

the convergence properties of LMQDQ method. In the present study, we restrict ourselves to

two-dimensional Poisson equation in a unit square domain (0 x 1, 0 y 1).

The governing equation is

2

u

x2 +

2

u

y2 =f( x,y) (9)

The Dirichlet condition presents on all the four boundaries, i.e. uboundary =uexact. The source

function f( x,y) on the right side of Equation (9) is determined from the given analytical

solution, which is also used to measure the numerical error. Relative error is taken to measure

the accuracy of numerical results, which is defined as

=

Ni=1

(unum uexact)2 Ni=1

(uexact )2

(10)

To investigate the solution dependence of LMQDQ method, four functions are selected as the

analytical solutions of above Poisson equation. They are specified as follows:

u1= 0.75 exp

(9x 2)2 +(9y 2)2

4

+0.75 exp

(9x+ 1)2

49

9y+ 1

10

+ 0.5exp

(9x 7)2 +(9y 3)2

4

0.2 exp((9x 4)2 (9x 7)2)

u2=

1x

2

6 1

y

2

6+1000(1x)3x3(1y)3y3 +y6

1

x

2

6+x6

1

y

2

6u3= sin(x) sin(y)

u4= x2 +y2

(11)

The four functions are displayed in Figure 2. Among them, function u1 is taken from Franke [1]

and function u2 is taken from Lyche et al. [21]. Functions u3 and u4 are provided by

the authors.

In principle, LMQDQ method can be implemented on scattered nodes and requires no mesh.

However, in the present study, we aim to study the effect of mesh size, shape parameter and

number of supporting points on the numerical error. The use of non-uniform mesh may not be



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0

0.2

0.4

0.6

0.8

1

u

0

0.2

0.4

0.6

0.8

1

u

00.25

0.50.751x

0

0.2

0.4

0.6

0.8

1

y

0

0.2

0.4

0.6

0.8

1

y

X

Y

Z

X

Y

Z

00.2

0.40.6

0.8

1

x

0

0.2

0.4

0.6

0.8

1

u

00.250.5

0.751 x

0

0.2

0.4

0.6

0.8

1

y

X

Y

Z

0

0.2

0.4

0.6

0.8

1

u

00.250.50.751 x

0

0.2

0.4

0.6

0.8

1

y

X

Y

Z

(a) (b)

(d)(c)

Figure 2. Perspective view of solution functions: (a) function 1 (u1); (b) function 2 (u2);(c) function 3 (u3); and (d) function 4 (u4).

appropriate because in this case, the mesh spacing is different at different locations. Therefore,the two-dimensional Poisson equation is solved on the uniform mesh in the present work.

Employing the weighting coefficients for the second-order derivatives to discretize the Poisson

equation (9) at an interior node i, gives

Nij=1

(w2xi, j+w2yi, j)uj =fi (12)



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h

error

0.01 0.015 0.02 0.025

10-4

10-3

function 1

function 2

function 3

function 4

Slope=3.7

Figure 3. Convergence rate of relative error versus mesh size h.

where w2xi, j and w2yi, j denote the weighting coefficients at node i associated with the second-

order derivatives in the x and y directions, respectively. The subscript i represents the global

node index while j is the local node index in the support of node i. To solve this linear and

sparse system of algebraic equations, successive over-relaxation (SOR) iteration method is used

in this study. The convergence criterion is set to 108

, which is considered small enough toobtain the converged solution.

3.2. Numerical error versus mesh size h

To study the convergence rate of relative error versus the mesh size h, five uniform meshes

are employed, i.e. 4141, 6161, 8181 and 101101. The value of shape parameter c

is chosen as 0.15, and the number of supporting points for each interior node is fixed to 18.

The solutions are illustrated in Figure 3 in the form of relative error versus mesh size h

in the loglog scale.

It can be observed in Figure 3 that the numerical errors for all the four cases are straight lines

and parallel to each other. This implies that LMQDQ method accomplishes so-called super-

convergence, i.e. an error estimate of O(hn). Moreover, the same convergence rate shown by

all the four cases indicates that the convergence rate with respect to mesh size h is independent

on the solution function.

3.3. Numerical error versus shape parameter c

The numerical experiments of testing the convergence property with respect to shape parameter c

are carried out on a uniform mesh of 41 41. The number of supporting points within the



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Shape parameter c

Error

0.1 0.2 0.3 0.4

10-5

10-4

10-3

10-2

10-1

function 1

function 2

function 3

function 4

slope=3.75

Figure 4. Convergence rate of relative error versus shape parameter c.

local support of each interior node is also restricted to 18. The value of shape parameter c

ranges from 0.05 to 0.45. The numerical solutions are illustrated in Figure 4 in the form of

versus shape parameter c in the loglog scale.

It is interesting to see in Figure 4 that in general, the numerical solutions show a super

convergence as the value of shape parameter c increases. However, for the four different

solution functions, the plotted lines are not parallel to each other, especially when the shapeparameter c has a large value. It implies that the convergence rate with respect to shape

parameter is sensitive to the solution functions within a certain range of shape parameter.

However, it can also be observed from Figure 4 that among the solution functions, functions 3

and 4 keep parallel pattern within the whole range of tested shape parameters. The functions 3

and 4 are the trigonometric and polynomial functions, respectively. As observed from Figure 2,

they are of less complexity as compared with functions 1 and 2 in terms of functional variation.

It indicates that the convergence rate of solution functions of less complexity may have less

variation in amplitude. Approximately, a convergence rate of 3.75 is estimated with respect to

shape parameter c, as shown by the dash-dot line in Figure 4. Since the optimization of shape

parameter c needs no additional computation cost and can improve the accuracy of solution, it

is one of the reasons to explain why the MQ-based methods are so attractive in solving PDEs.

On the other hand, it may be a risky factor in the practical applications due to the fact that it

is an arbitrary number.

3.4. Numerical error versus number of supporting points

The study of accuracy variation with different number of supporting points is carried out on a

uniform mesh of 41 41. Similar to the test of mesh size h, the value of shape parameter c



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Number of supporting points

Error

10 15 20 25 30

10-4

10-3

10

-2

function 1

function 2

function 3

function 4

Figure 5. Relative error versus number of supporting points.

is fixed to 0.2. The number of supporting points within the local support of each interior

node varies from 9 to 30. The numerical solutions are shown in Figure 5 in the form of

relative error norm versus number of supporting points. From Figure 5, it can be seen that,

when the number of supporting points is increased, in general, the accuracy of solutions isgradually improved. However, the relative error is not going down smoothly as the number

of supporting point increases. Two jumps of relative error are observed in the plot, and they

divide the convergence lines into three regimes, which can be described by the number of

supporting points: ns < 11, 12ns 25 and ns 28. In each regime, the relative error shows

slight variation and the numerical solution can be considered to have the similar accuracy.

From Figure 5, all the four solution functions also show the similar tendency of convergence

as the number of supporting points increases. It implies that the contribution of the number of

supporting points to the accuracy is independent of solution function. This is a very interesting

phenomenon, which is in line with polynomial approximation. We will give a detailed discussion

about this in the following section.

3.5. Relationships between numerical error and three factors

In order to estimate the approximation error, it is necessary to understand the relationships

between the numerical error and the three factors. To fulfill this goal, numerical experiments

are designed in such a manner that one factor is fixed and the other two are variable. Then,

from the corresponding variation of relative error, we can investigate the correlation of two

variable factors.



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h

Error

0.015 0.02 0.025

h0.015 0.02 0.025

10-5

10-4

10-3

10-2

c=0.05c=0.10c=0.15c=0.18c=0.20c=0.22

c=0.05c=0.10c=0.15c=0.18c=0.20c=0.22

c=0.05c=0.10c=0.15c=0.18c=0.20c=0.22

h

0.015 0.02 0.025

h0.015 0.02 0.025

10-5

10-4

10-3

10-2

c=0.05c=0.10c=0.15c=0.18c=0.20c=0.22

Error

10-4

10-3

10-2

10-1

Error

Error

10-6

10-5

10-4

10-3

10-2

(a) (b)

(d)(c)

Figure 6. Convergence rate of relative error versus mesh size for various shape parameter c:(a) function 1 (u1); (b) function 2 (u2); (c) function 3 (u3); and (d) function 4 (u4).

3.5.1. Dependence of numerical error on shape parameter and mesh size. To study the depen-

dence of numerical error on the shape parameter c and mesh size h, the number of supporting

points is fixed at 16. The numerical solutions are shown in Figures 6(a) (d) for the four

solution functions, respectively. In each figure, the numerical results are plotted in the form

of relative error versus mesh size in the loglog scale and a group of convergence lines are

drawn according to the same value of shape parameter. The mean values of convergence rate ofrelative error versus the mesh size h are listed in Table I. From Figures 6(a) (d), it is obviously

observed that the symbols representing the accuracy of solution with the same shape parameter

c are in perfect alignment, especially for the cases using large value of c. It is also clear to

see that the lines standing for different shape parameters are parallel to each other. In other

words, they have the same convergence rate. This can also be confirmed by the mean value

of convergence rate listed in Table I. Therefore, from the viewpoint of convergence rate, we



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Table I. Mean value of convergence rate with number ofsupporting points ns = 18.

Solution functions

Shape parameter u1 u2 u3 u4

c = 0.05 2.92 2.75 2.79 2.92c = 0.10 3.48 3.33 3.40 3.49c = 0.15 3.68 3.61 3.68 3.72c = 0.18 3.74 3.70 3.76 3.78c = 0.20 3.76 3.74 3.79 3.81c = 0.22 3.77 3.76 3.79 3.83

Table II. Mean value of convergence rate with shapeparameter c = 0.12.

Solution functionsNumber ofsupporting points u

1 u

2 u

3 u

4ns =6 1.87 1.88 1.86 1.96ns =8 1.84 1.88 1.85 1.87ns =12 3.65 3.67 3.66 3.69ns =20 3.56 3.57 3.56 3.53ns =24 3.54 3.56 3.56 3.52ns =30 4.92 4.82 4.97 4.94ns =34 4.99 4.77 4.99 4.97

can say that the contributions of shape parameter c and mesh size h are utterly independent.

From Table I, it can be seen that the convergence rates for the different solution function with

various shape parameter c almost have the same value (3.7), which confirms our previous

finding for the convergence property with respect to mesh size h, i.e. the convergence rate with

respect to mesh size h is independent of the solution function.

3.5.2. Relationship between numerical error, mesh size and number of supporting points. For

the investigation of relationship between the numerical error, the number of supporting points nsand the mesh size h, the value of shape parameter c is fixed at 0.12, which is suitable for all

the cases considered. The numerical solutions are illustrated in Figures 8(a) (d) in the form

of relative error versus h in the loglog scale. The corresponding mean values of convergence

rate are listed in Table II.

Comparing LMQDQ with the traditional finite element method, we can see that the number

of supporting points plays a similar role as the collocation points in the finite element method.

In the finite element method, the use of more collocation points means implementation of higher

order polynomials for function approximation. In the LMQDQ method, the number of supportingpoints equals to the number of MQ RBFs used for function approximation. It is known that a

polynomial interpolant of degree k requires (k +1)(k+2)/2 collocation points in two-dimensional

function approximation, and achieves an accuracy of O(hk(m+n)+1) for a partial derivative

m+n

u/xmyn. Taking the second-order derivative as an example, we can see that the second

order of accuracy (k = 4, m = 2, n = 0 or k = 4, m = 0, n = 2) requires 15 collocation

points, while the third order of accuracy (k = 5, m = 2, n = 0 or k = 5, m = 0, n = 2)



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requires 21 collocation points. This implies that when the number of collocation points is

increased from 15 to 20, the order of accuracy for the numerical results cannot be improved,

which keeps the second order. Only when the number of collocation points is increased to

21, the accuracy of numerical results can be improved to the third order. For the LMQDQ

method, we cannot apply the Taylor series expansion to do theoretical analysis. However, itis interesting to see whether the above feature can be held by the LMQDQ method from the

numerical experiment. Our numerical results do show a similar feature for the LMQDQ method.

It can be observed from Figures 7(a) (d) that the convergence lines can be classified into

three groups by the value of slope, with the number of supporting points ranging from 6 to 30.

Specifically, the convergence rate is approximately 1.9 for the scheme with 6 and 8 supporting

points, 3.6 for the scheme with 12, 20, and 24 supporting points, and 4.9 for the scheme with

30 and 34 supporting points. Therefore, an error estimate with respect to the mesh size h and

the number of supporting points can be written as

O (hn

) and n 1.9 for 6ns 9

3.6 for 9< ns 27

4.9 for 27< ns 34

h

Error

0.015 0.02 0.025

h0.015 0.02 0.025

10-4

10-3

10-2

h0.015 0.02 0.025

h0.015 0.02 0.025

10-4

10-3

10-2

Error

10-4

10-3

10-2

Error

Error

10-4

10-3

10-2

(a) (b)

(d)(c)

Figure 7. Convergence rate of relative error versus mesh size for various number of supporting points:(a) function 1 (u1); (b) function 2 (u2); (c) function 3 (u3); and (d) function 4 (u4).



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The above results are in line with the analysis of polynomial approximation. The accuracy

of numerical results can be greatly improved at some critical number of supporting points.

The above results also reveal that LMQDQ method has a similar convergence rate as the

finite element method associated with polynomial approximation if using the same number of

supporting points. From Table II, it can be seen that the convergence rates are independent ofsolution functions.

3.5.3. Relationship between numerical error, shape parameter and number of supporting points.

The relationship between the numerical error, the number of supporting points ns and the shape

parameter c is studied on the mesh of 41 41, i.e. mesh size h equals to 0.025. The numerical

solutions are illustrated in Figures 8(a)(d) in the form of relative error versus shape parameter

in the loglog scale. The corresponding mean values of convergence rate are listed in Table III.

From Figures 8(a)(d), it is clear that the convergence lines for various number of supporting

points are very similar to those in Figures 7(a)(d), i.e. the convergence lines can be classified

into three groups with respect to the slope. Moreover, the critical numbers of supporting points,

which indicate the changing of convergence rate, also coincide with those in the number of

supporting points versus mesh size. From the comparison of Tables II and III, it can be ob-

served that regardless of difference between shape parameter and mesh size, the convergence

rates for the same number of supporting points are approximately the same. It implies that the

number of supporting points has similar effects either on the mesh size h or the shape pa-

rameter c. However, it can also be seen from Figures 8(a)(d) that different solution functions

may experience different convergence rate when a large value of shape parameter c is used.

In summary, from the performed numerical experiments, we can see that the accuracy of nu-

merical solutions can be improved either by increasing the value of shape parameter or refining

the mesh size, and their contributions are utterly independent. The use of large number of sup-

porting points may not directly lead to the accuracy improvement in some cases. Instead, it can

improve the convergence rate. Based on the experimental observation, an error estimate can then

be established for the discretization error of Poisson equation using LMQDQ method as follows:

O((h/c)n) and n

1.9 for 6ns 9

3.6 for 9< ns 27

4.9 for 27< ns 34

4. CONCLUSIONS

In this paper, an error estimate is provided for the numerical solution of Poisson equation using

LMQDQ method. The error estimate is based on the numerical experiments on the uniform

meshes. Three factors, i.e. mesh size, shape parameter and number of supporting points whichmay determine the accuracy of numerical solutions, are numerically investigated. It has been

observed that the accuracy of numerical solutions can be improved either by increasing the

value of shape parameter or refining the mesh size, and their contributions to the accuracy

improvement are utterly independent. We also found that, although the use of large number

of supporting points may not directly lead to the accuracy improvement in some cases, it can

accelerate the convergence instead.



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Shape parameter c

Error

Error

0.1 0.2 0.3

Shape parameter c

0.1 0.2 0.3

Shape parameter c

0.1 0.2 0.3

Shape parameter c

0.1 0.2 0.3

10-5

10-4

10-3

10-2

Error

10-5

10-4

10-3

10-2

10-4

10-3

10

-2

10-1

Error

10-5

10-4

10-3

10-2

(a) (b)

(d)(c)

Figure 8. Convergence rate of relative error versus shape parameter c for various number of supportingpoints: (a) function 1 (u1); (b) function 2 (u2); (c) function 3 (u3); and (d) function 4 (u4).

Table III. Mean value of convergence rate with meshsize h= 0.025.

Solution functionsNumber ofsupporting points u1 u2 u3 u4

ns =6 2.14 1.68 1.82 1.91ns =8 2.15 1.85 1.75 1.76ns =12 3.77 3.38 3.60 3.61ns =17 3.73 3.12 3.41 3.63ns =20 3.90 3.04 3.41 3.50ns =24 3.66 3.08 3.35 3.41ns =28 5.18 4.41 4.41 4.30

ns =34 5.05 4.60 4.49 4.56

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