Research Article An Adaptive Approach to Solutions...

Research ArticleAn Adaptive Approach to Solutions of Fredholm IntegralEquations of the Second Kind

Nebiye Korkmaz and Zekeriya Guumlney

Department of Secondary Science and Mathematics Education Mugla Sıtkı Kocman University 48000 Mugla Turkey

Correspondence should be addressed to Nebiye Korkmaz nkorkmazmuedutr

Received 5 May 2014 Revised 13 September 2014 Accepted 22 September 2014 Published 17 November 2014

Academic Editor Chun-Gang Zhu

Copyright copy 2014 N Korkmaz and Z Guney This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

As an approach to approximate solutions of Fredholm integral equations of the second kind adaptive hp-refinement is used firstlytogether with Galerkin method and with Sloan iteration method which is applied to Galerkin method solution The linear hatfunctions and modified integrated Legendre polynomials are used as basis functions for the approximationsThe most appropriaterefinement is determined by an optimization problem given by Demkowicz 2007 During the calculations 1198712-projections ofapproximate solutions on four different meshes which could occur between coarse mesh and fine mesh are calculated Dependingon the error values these procedures could be repeated consecutively or different meshes could be used in order to decrease theerror values

1 Introduction and Preliminaries

The aim of this study is to find approximate solutions for theFredholm integral equations of the second kind by applyingadaptive refinement together with Galerkin method andSloan iteration method Our reason to apply adaptive refine-ment is to search for meshes on which wemight obtain betterapproximations to the solution of the problems with themethodsmentioned In Section 2 we explained how to obtaina finer mesh from a given mesh which is called coarse meshand howwe construct the basis functions used for the approx-imation In Section 3we solved the problemgivenwith (10) byGalerkin method and then in order to determine an optimalmesh we solved the optimization problem given with (18) foradaptive refinement In Section 4 as the subsequent step weiterate Galerkin method solution by Sloan iteration methodand we solved the same optimization problem for this casein order to make adaptive refinement Finally in Section 5 wepresented someproblem examples In this sectionwewill givesome basic knowledge about two essential subjects that thisstudy stands on integral equations and adaptive refinement

11 On Integral Equations As the theory of integral equationshas significant importance in mathematics it is also closely

related to various fields of science Many problems suchas ordinary and partial differential equations problems ofmathematical physics can be laid out as integral equationsHochstadt [1]mentioned thatmany existence and uniquenessresults can be derived from the corresponding results fromintegral equations and there is almost no area of appliedmathematics and mathematical physics where integral equa-tions do not play a role Many studies can be found that statesome of these areas of usage of integral equations Rahbarand Hashemizadeh [2] indicated to high applicability ofintegral equations in different areas of applied mathematicsphysics and engineering and they particularly mentionedsome areas in which these equations are widely used suchas mechanics geophysics electricity and magnetism kinetictheory of gases hereditary phenomena in biology quantummechanics mathematical economics and queuing theoryWe can sort some more examples of these areas of usage asfollows automatic control theory network theory and thedynamics of nuclear reactors [3] acoustics optics and lasertheory potential theory radiative transfer theory cardiol-ogy fluid mechanics and statics [4] continuum mechan-ics hereditary phenomena in physics and biology renewaltheory radiation optimization optimal control systemscommunication theory population genetics medicine and

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 629179 13 pageshttpdxdoiorg1011552014629179

2 Abstract and Applied Analysis

mathematical problems of radiative equilibrium the particletransport problems of astrophysics and reactor theory steadystate heat conduction and fracture mechanics [5]

As Pachpatte [3] expressed the beginning of the integralequations can be traced back to N H Abel who found anintegral equation in 1812 starting from a problem in mechan-ics and in 1895 V Volterra emphasized the significance of thetheory of integral equations

Lonseth [6] stated that Fredholm published his distin-guished paper in Acta Mathematica [7] in 1903 in whichhe gave the first detailed account of the existence andmultiplicity of solutions of the following two equations wherethe kernel119870(119904 119905) and119910(119904) are known functions and the valuesof 120582 (proper values) are values to be determined such that acontinuous solution 119909(119904) equiv 0 exists

119909 (119904) minus int

1

0

119870 (119904 119905) 119909 (119905) 119889119905 = 119910 (119904) 0 le 119904 le 1

119909 (119904) minus 120582int

1

0

119870 (119904 119905) 119909 (119905) 119889119905 = 119910 (119904) 0 le 119904 le 1

(1)

In his book Pachpatte [3] stated a few number ofmonographsthat he accepted as an excellent account of integral equationsmay be found in the following Burton [8] Corduneanu [9ndash11] Gripenberg et al [12] Krasnoselskii [13] Miller [14] andTricomi [15]

12 On Adaptive Refinement In 1988 Babuska [16] publishedhis work on the advances in the 119901 and ℎ-119901 versions of thefinite elementmethod and in this study he distinguished threeversions of the finite elementmethod (FEM) as follows the ℎ-version the 119901-version and the ℎ-119901 version The main idea inthe ℎ-version is to refine the size of the meshes while degreesof the polynomials used for approximation are kept fixed(usually 119901 = 1 2) in the 119901-version it is the opposite size ofthemeshes kept fixed but degrees of the polynomials used forapproximation are increased In the ℎ-119901 version both changesare done simultaneously size of the meshes are refined andthe degrees of the polynomials used for approximation areincreased In [16] while the ℎ-version of FEM is introducedto be the standard one the other versions are said to bedeveloped later and the first theoretical papers about the 119901-version and the ℎ-119901 version which appeared in 1981 are givenwith [17] and [18] respectively

In Demkowiczrsquos book [19] which is about computationwith ℎ119901-adaptive finite elements he studied one- and two-dimensional elliptic and Maxwell problems and he men-tioned two major components of the one-dimensional ver-sion of their ℎ119901-algorithm as fine grid solution and optimalmesh selection For the first component a given (coarse)mesh is refined in both ℎ and119901 to obtain a corresponding finemesh and then the problem is solved on this finemesh to findthe fine mesh solution In the latter component he used thisfine mesh solution to determine optimal mesh refinement ofthe coarse mesh by minimizing the projection based inter-polation error solving the following discrete optimizationproblem where 119906 prod

ℎ119901119906 prodℎ119901opt119906 119873ℎ119901opt

and 119873ℎ119901

denote

respectively the solution on the fine mesh the interpolant ofthe fine grid solution on the original mesh interpolant of thefine grid solution on the new optimal mesh to be determinedthe corresponding number of degrees of freedom on thenew optimal mesh to be determined and the correspondingnumber of degrees of freedom on the original mesh

10038171003817100381710038171003817119906 minus prod

ℎ11990111990610038171003817100381710038171003817

2

1198671(0119897)minus100381710038171003817100381710038171003817119906 minus prod

ℎ119901opt119906100381710038171003817100381710038171003817

2

1198671(0119897)

119873ℎ119901opt

minus 119873ℎ119901

997888rarr min (2)

Here the aim of the optimization problem is said tomaximizethe rate of decrease of the interpolation error Asadzadehand Eriksson [20] gave a few number of references [21ndash25]on solving integral equations with FEM in their paper inwhich they have chosen to work on the single layer potentialproblem for Laplacersquos equation with Neumann boundaryconditions in order to be concrete In their paper the studieson solving integral equations with adaptive FEM in thatperiod are givenwith [25ndash28] Adaptive FEMare usually usedto solve partial differential equations but in literature thesemethods are also seen to be used for solving different typeof problems in various branches of science such as hydro-dynamics [29] optimal design [30] elliptic stochastic equa-tions [31] parabolic problems [32] parabolic systems [33]elliptic problems [34] elliptic partial differential equations[35] elliptic boundary value problems [36 37] electrostatics[38] electromagnetic problems [39] biological flows [40]and Laplace eigenvalue problem [41]

2 Refining a Finer Mesh from a Coarse Meshand Construction of Basis Functions

21 Refining a Finer Mesh from a Coarse Mesh Let 119886 = 1199051lt

1199052lt sdot sdot sdot lt 119905

119899= 119887 be the node points of a given (finite element)

mesh which is accepted as the coarsemesh and denote the listof these node points as follows

119871119888= [1199051 1199052 sdot sdot sdot 119905119899] (3)

For each 119894 isin 1 2 119899 minus 1 the interval of the form 119868119894=

[119905119894 119905119894+1] is called an element (or finite element) having two

parameters element length ℎ119894= 119905119894+1minus 119905119894and element local

polynomial of order of approximation 119901119894(119901119894ge 2) What is

meant by element local polynomial of order of approximationcan be explained as follows ldquoIf an element has element localpolynomial of order of approximation of order119901 it means thenonlinear base functions on that element are the polynomialsof degree starting from 2 to 119901rdquo Let 119863119888 be the list of 119899 minus 1numbers of element local polynomial order of approximationof the coarse mesh elements

119863119888= [1199011 1199012 sdot sdot sdot 119901119899minus1] (4)

Firstly dividing each element of thismesh from themiddle (ℎ-refinement) and then increasing the element local polynomialorder of approximation by 1 for each new element (119901-refine-ment) we obtain a finer mesh having new lists of node pointsand the element local polynomial orders of approximationgiven belowwhich are of length 2119899minus1 and 2119899minus2 respectively

Abstract and Applied Analysis 3

119871119891= [1199051

(1199051+ 1199052)

21199052sdot sdot sdot 119905119899minus1

(119905119899minus1+ 119905119899)

2119905119899] (5)

119863119891= [1199011 + 1 1199011 + 1 1199012 + 1 1199012 + 1 sdot sdot sdot 119901119899minus1 + 1 119901119899minus1 + 1] (6)

22 Construction of Basis Functions In this study for eachelement of a mesh two kinds of basis functions are used hatfunctions and bubble functions The reasons why these arecalled so can be explained as follows the linear base functionsare called hat functions because their shapes look like a hatand the nonlinear ones are called bubble functions becausethey vanish at node points as bubbles Considering the coarsemesh given with lists (3) and (4) we explain how to constructthe basis functions on the coarse mesh as sample Basisfunctions of anymesh can be built up similarly Formulationsof number of 119899 hat functions belonging to coarse mesh are asfollows

1205931(119905) =

(1199052minus 119905)

(1199052minus 1199051) if 119905 isin [119905

1 1199052]

0 otherwise

120593119894(119905) =

(119905 minus 119905119894)

(119905119894+1minus 119905119894) if 119905 isin [119905

119894minus1 119905119894]

(119905119894+1minus 119905)

(119905119894+1minus 119905119894) if 119905 isin [119905

119894 119905119894+1]

0 otherwise

119894 = 2 119899 minus 1

120593119899(119905) =

(119905 minus 119905119899minus1)

(119905119899minus 119905119899minus1) if 119905 isin [119905

119899minus1 119905119899]

0 otherwise

(7)

For the construction of bubble functions in the coarse meshthe integrated Legendre polynomials are combined with afunction120595mapping any closed interval (which is an element)[119905119894 119905119894+1] to the domain of integrated Legendre polynomials

that is given as follows

120595 [119905119894 119905119894+1] 997888rarr [minus1 1] 120595 (119905) =

2119905 minus (119905119894+1+ 119905119894)

119905119894+1minus 119905119894

(8)

Let 119871119901denote the integrated Legendre polynomial of degree

119901 Bubble function of degree 119901 in the 119894th (119894 = 1 2 119899 minus 1)element that is used for approximation is defined as follows

120593119894119901(119905) =

119871119901∘ 120595 (119905) if 119905 isin [119905

119894minus1 119905119894]

0 otherwise119894 = 2 119899 119901 ge 2

(9)

3 Galerkin Method Solution andAdaptive Refinement by UsingDemkowiczrsquos Optimization

Firstly we calculate the Galerkin method approximate solu-tion on the fine mesh 119871119891 which was constructed in Section 2

Then in order to decide the optimal mesh we solve an opti-mization problem given byDemkowicz [19] on each elementof the fine mesh For this we need 1198712-projections of the finemesh solution on each element of 119871119891 onto the correspondingelement of the coarse mesh 119871119888 and on the four possibleoptimal mesh refinements of the coarse mesh element Thepossible four optimal mesh refinements are defined clearly inthe latter parts

31 Galerkin Method Solution on the Fine Mesh 119871119891 Considerthe Fredholm integral equation of the second kind

120582119909 (119905) minus int

119887

119886

119870 (119905 119904) 119909 (119904) 119889119904 = 119891 (119905) 119886 le 119905 le 119887 120582 = 0

(10)

and the fine mesh having lists (5) and (6) On this mesh thetotal number of basis functions is 2119899minus1+2(119901

1+1199012+sdot sdot sdot+119901

119899minus1)

which we denote by 119889119891 Let

119906 (119905) =

119889119891

sum

119895=1

119888119895120601119895(119905) 119905 isin [119886 119887] (11)

be the approximate solutionwe are looking forwhere120601119895(119905) are

the basis functions (hat functions and bubble functions) and119888119895(119905) are the coefficients to be calculated for 119895 = 1 2 119889

119891

Substituting (10) in the residual function of the Galerkinmethod [42] we obtain the following equality

119903 (119905) =

119889119891

sum

119895=1

119888119895120582120601119895(119905) minus int

119863

119870 (119905 119904) 120601119895(119905) 119889119904 minus 119910 (119905) 119905 isin 119863

(12)

The residual function is required to satisfy the followingequalities

⟨119903 120601119894⟩ = 0 119894 = 1 119889

119891 (13)

Rearranging (13) for all 119894 isin 1 119889119891

119889119891

sum

119895=1

119888119895[120582int

119887

119886

120601119895(119905) 120601119894(119905) 119889119905 minus int

119887

119886

int

119887

119886

119870 (119905 119904) 120601119895(119904) 120601119894(119905) 119889119904 119889119905]

= int

119887

119886

119891 (119905) 120601119894(119905) 119889119905

(14)

is obtained which is a system of equations and this systemcan be represented in the matrix form by using the matricesdefined as follows


119864 =

[[[[[

[

11986411

sdot sdot sdot 1198641198891198911

11986421

sdot sdot sdot 1198641198891198912

1198641198891198911sdot sdot sdot 119864119889119891119889119891

]]]]]

]

119864119894119895= 120582int

119887

119886

120601119895(119905) 120601119894(119905) 119889119905

119870 =

[[[[[

[

11987011

sdot sdot sdot 1198701198891198911

11987021

sdot sdot sdot 1198701198891198912

1198701198891198911sdot sdot sdot 119870

119889119891119889119891

]]]]]

]

119870119894119895= int

119887

119886

int

119887

119886

119870 (119905 119904) 120601119895(119904) 120601119894(119905) 119889119904 119889119905

119865 = [int

119887

119886

119891(119905)1206011(119905)119889119905 int

119887

119886

119891(119905)1206012(119905)119889119905 sdot sdot sdot int

119887

119886

119891(119905)120601119889119891

(119905)119889119905]

119879

119862 = [1198881 1198882 sdot sdot sdot 119888119889119891]119879

(15)

The system (14) can be expressed as

(119864 minus 119870) sdot 119862 = 119865 (16)

Hence the coefficient matrix 119862 is found as follows

119862 = (119864 minus 119870)minus1sdot 119865 (17)

By substituting these coefficients to the lefthand side of equal-ity (11) the desired approximate solution of the Galerkinmethod is obtained

32 Adaptive Refinement by Using Demkowiczrsquos Optimiza-tion for Galerkin Method Solution on the Fine Mesh 119871119891As explained in the abstract for adaptive refinement wesolve an optimization problem which was originally usedby Demkowicz [19] with Sobolev spaces for solving one-and two-dimensional elliptic and Maxwell problems In thisstudy instead of Sobolev norm 1198712-norm is used and naturallyinner products are 1198712-inner product Under this choice ouroptimization problem turns into

10038171003817100381710038171003817119906 minus Π

ℎ11990111990610038171003817100381710038171003817

2

minus100381710038171003817100381710038171003817119906 minus Π

ℎ119901opt119906100381710038171003817100381710038171003817

2

119873ℎ119901opt

minus 119873ℎ119901

997888rarr min (18)

For determining optimal mesh (18) is solved on each elementof the fine mesh For simplicity we solve the problem onelement of the form 119868 = [119886 119887] as a representative elementfor all elements of the coarse mesh which were in the form119868119894= [119905119894 119905119894+1] (119894 = 1 2 119899 minus 1) In other words we

will start with a mesh consisting of just one element whichis the interval itself Let 119863119888 = [119901] be the list of elementlocal polynomial order of approximation of the element 119868119888Refining this element in both ℎ and 119901we get a fine mesh withthe following new list of node points

119871119891= [119886

119886 + 119887

2119887] 119863

119891= [119901 + 1 119901 + 1] (19)

There are also other possible refinements (just in ℎ or 119901 orin both but with different choice of refinements in 119901) whichproduce the following four possible optimal mesh choices

119871opt= [119886 119887] 119863

opt= [119901 + 1] (20)

or 119871opt= [119886

119886 + 119887

2119887] 119863

opt= [119901 119901] (21)

or 119871opt= [119886

119886 + 119887

2119887] 119863

opt= [119901 119901 + 1] (22)

or 119871opt= [119886

119886 + 119887

2119887] 119863

opt= [119901 + 1 119901] (23)

Consider the fine mesh solution 119906 given with (11) Firstly 1198712-projections of 119906 on the coarse meshΠ

ℎ119901119906 and on the optimal

meshes Πℎ119901opt119906 are calculated During these calculations we

introduced some matrices and notations that we need Let119889119888and 119889opt be the total number of basis functions and let 120585119888

119895

(119895 = 1 2 119889119888) and 120585opt

119895 (119895 = 1 2 119889opt) be the basis

functions of the coarse and optimal mesh cases respectively

Πℎ119901119906 =

119889119888

sum

119895=1

120585119888

119895120601119888

119895 (24)

Πℎ119901opt119906 =

119889opt

sum

119895=1

120585opt119895120601opt119895 (25)

Let 120585119888 = [120585119888

1120585119888

2sdot sdot sdot 120585119888

119889119888

]119879 120585opt = [120585

opt1

120585opt2

sdot sdot sdot 120585opt119889opt]119879

be the coefficient matrices corresponding to (24) and (25)respectively We calculated 1198712-projections of the Galerkinmethod solution (11) as Larson and Bengzon explained


in [43] For calculatingΠℎ119901119906 given with (24) above we define

two matrices 119885119888 and119872119888 that we need as follows

119885119888=

[[[[[

[

119885119888


119888

1119889119891

119885119888


119888

2119889119891

119885119888

1198891198881sdot sdot sdot 119885

119888

119889119888119889119891

]]]]]

]

119885119888

119894119895= int

119887

119886

120601119888

119894(119904) 120601119891

119895(119904) 119889119904

119872119888=

[[[[

[

119872119888


119888

1119889119888

119872119888


119888

2119889119888

119872119888

1198891198881sdot sdot sdot 119872

119888

119889119888119889119888

]]]]

]

119872119888

119894119895= int

119887

119886

120601119888

119894(119904) 120601119888

119895(119904) 119889119904

(26)

We obtain the coefficient matrix 120585119888 as follows

120585119888= (119872119888)minus1

sdot 119885119888sdot 119862 (27)

Substituting these coefficients in (24) we obtain the 1198712-projection function Π

ℎ119901119906 For calculating Π

ℎ119901opt119906 given with

(25) we need two new matrices 119885opt and 119872opt defined asfollows

119885opt=

[[[[[[

[

119885opt11

sdot sdot sdot 119885opt1119889119891

119885opt21


119885opt119889opt1

sdot sdot sdot 119885opt119889opt119889119891

]]]]]]

]

119885opt119894119895= int

119887

119886

120601opt119894(119904) 120601119891

119895(119904) 119889119904

119872opt=

[[[[[[

[

119872opt11

sdot sdot sdot 119872opt1119889opt

119872opt21


119872opt119889opt1

sdot sdot sdot 119872opt119889opt119889opt

]]]]]]

]

119872opt119894119895=int

119887

119886

120601opt119894(119904) 120601119888

119895(119904) 119889119904

(28)

We obtain the coefficient matrix 120585opt as follows

120585opt= (119872

opt)minus1

sdot 119885optsdot 119862 (29)

Substituting these coefficients in (25) we obtain the 1198712-pro-jection function Π

ℎ119901opt119906

We reformulate the right hand side of the optimizationproblem (18) in terms of the matrices we introduced beforeby using (27) and (29) as follows

(119862119879sdot [(119885

opt)119879

sdot ((119872opt)minus1

)

119879

sdot 119885optminus (119885119888)119879

sdot ((119872119888)minus1

)119879

sdot 119885119888] sdot 119862)

times (119889optminus 119889119888)minus1

(30)

The value (30) is calculated for each of the four possibleoptimal mesh refinements of the coarse mesh element

The case giving the minimum value is replaced with thatelement of the coarse mesh Starting from the first elementof the coarse mesh we repeat this procedure for each coarsemesh element respectively Joining all these replaced ele-ments at node points of the coarsemesh we obtain adaptivelyrefined new mesh which is the optimal mesh we are trying toachieve During this process to guarantee the continuity ofthe approximate solution at node points of the coarse meshthe boundary conditions at node points of the coarse meshhave to be fixed In other words the values of the fine meshsolution 119906 and its 119871

2-projections is forced to take the same

value at the node points of the coarse meshWe start with the 1198712-projection function Π

ℎ119901119906 If we

equalize the coefficients of the 1198712-projection function Πℎ119901119906

with finemesh solution 119906 at node points 119886 and 119887 of the sampleelement 119868119888 = [119886 119887] we get the following results

Πℎ119901119906 (119886) = 119906 (119886)

997904rArr 120585119888

1sdot 120601119888

1(119886) + 0 = 119888

1sdot 120601119891

1(119886) + 0

997904rArr 120585119888

1= 1198881

Πℎ119901119906 (119887) = 119906 (119887)

997904rArr 120585119888

2sdot 120601119888

2(119887) + 0 = 119888

3sdot 120601119891

3(119887) + 0

997904rArr 120585119888

2= 1198883

(31)

In this case calculating the coefficients in the coefficientmatrix 120585119888 except 120585119888

1and 120585119888

2will be sufficient Removing the

first two columns of the matrices given with (26) and thefirst two elements of the matrix 120585119888 and solving the remainingsystem brings the desired coefficients Since the structure ofthe optimal element given with (20) is similar to the coarsemesh the calculation for this case is similar with the one donefor obtaining the coefficient matrix 120585119888 For this optimal case120585opt1

= 1198881and 120585opt

2= 1198883 Deleting the first two columns of

the matrices given with (28) and the first two elements ofthe matrix 120585opt and solving the remaining system brings theremaining coefficients of the projection function Π

ℎ119901opt

For the remaining three optimal cases given with (20)(21) and (22) we follow the same way

Πℎ119901opt119906 (119886) = 119906 (119886)

997904rArr 120585opt1sdot 120601

opt1(119886) + 0 = 119888

1sdot 120601119891

1(119886) + 0

997904rArr 120585opt1= 1198881

Πℎ119901opt119906 (119887) = 119906 (119887)

997904rArr 120585opt3sdot 120601

opt3(119887) + 0 = 119888

3sdot 120601119891

3(119887) + 0

997904rArr 120585opt3= 1198883

(32)


For these cases of optimal meshes the first and the thirdcolumns of matrices (28) and the first and the third elements(120585

opt1

and 120585opt3) of the matrix 120585opt are deleted and the remain-

ing system is solved in order to get the remaining coefficientsof the projection Π

ℎ119901opt

4 Sloan Iteration Solution andAdaptive Refinement by UsingDemkowiczrsquos Optimization

TheFredholm integral equation of the second kind givenwithformula (10) can be reformulated as

119909 =1

120582(119891 + 119911) (33)

where 119911 = K119909 = int119887

119886119870(119905 119904)119909(119904)119889119904 119905 isin [119886 119887] Atkinson [44]

defined iterated projection solution 119909119899for a given projected

method solution 119909119899as

119909119899=1

120582(119891 +K119909

119899) (34)

and beside this he mentioned that although such iterationsare found in the literature in many places Sloan [45] firstrecognized the importance of doing one such iteration andin his honor 119909

119899is often called the Sloan iterate We express

(34) in a more clear and general way as follows

119909 (119905) =1

120582[int

119887

119886

119870 (119905 119904) 119909 (119904) 119889119904 + 119891 (119905)] 119905 isin [119886 119887] (35)

Substituting the Galerkin method solution on the fine meshgiven with formula (11) to the right hand side of formula (35)the iterated solution on the fine mesh is obtained as follows

119909 (119905) =1

120582

[

[

119891 (119905) +

119889

sum

119895=1

119888119895int

119887

119886

119870 (119905 119904) 120601119895(119904) 119889119904]

]

119905 isin [119886 119887]

(36)

We solve the optimization problemgivenwith (18) for iteratedsolution as it was solved for Galerkinmethod solution In thiscase119906 in (18) is taken as the iterated solution119909 givenwith (36)

We define two matrices 119885119888 and 119872119888 needed in the

calculation of Πℎ119901119906 given with (24) as follows

119885119888=

[[[[[

[

119885119888


119888

1198891198911

119885119888


119888

1198891198912

119885119888

1119889119888

sdot sdot sdot 119885119888

119889119891119889119888

]]]]]

]

119885119888= int

119887

119886

int

119887

119886

119870 (119905 119904) 120601119891

119895(119904) 120601119888

119894(119905) 119889119904 119889119905

119865119888= [int

119887

119886

119891 (119905) 120601119888

1(119905) 119889119905 int

119887

119886

119891 (119905) 120601119888

2(119905) 119889119905 sdot sdot sdot int

119887

119886

119891 (119905) 120601119888

119889119888

(119905) 119889119905]

119879

(37)

We obtain the coefficient matrix 120585119888 as

120585119888=1

120582(119872119888)minus1

sdot (119865119888+ 119885119888sdot 119862) (38)


ℎ119901119906 for the iterated solution We need two

new matrices 119885opt and 119872opt in the calculation of Πℎ119901opt119906

which are given as follows

119885opt=

[[[[[[[[

[

119885opt11


119885opt12


119885opt1119889opt


]]]]]]]]

]

119885opt119894119895= int

119887

119886

int

119887

119886

119870 (119905 119904) 120601119891

119895(119904) 120601

opt119894(119905) 119889119904 119889119905

119865opt= [int

119887

119886

119891 (119905) 120601opt1(119905) 119889119905 int

119887

119886

119891 (119905) 120601opt2(119905) 119889119905 sdot sdot sdot int

119887

119886

119891 (119905) 120601opt119889opt(119905) 119889119905]

119879

(39)


120585opt=1

120582(119872

opt)minus1

sdot (119865opt+ 119885

optsdot 119862) (40)


ℎ119901opt119906 for the iterated solution

As the right hand side of the optimization problem(18) was reformulated in matrix form for Galerkin method


Table 1 Error values of (42) in Example 1

119899 1198661198712 119866max 119878

1198712 119878max

I 3807778247544568119890 + 00 3962758499440351119890 + 00 6142245995845852119890 minus 02 2691129516577462119890 minus 02

II 3681501530476621119890 minus 01 4734800657784639119890 minus 01 1436655818926428119890 minus 05 5651797188477303119890 minus 06

III 3980571071622163119890 minus 02 7073630464941472119890 minus 02 1437092335784317119890 minus 05 5601928975806914119890 minus 06

IV 1231396356019893119890 minus 03 2530443837131857119890 minus 03 5107562613500647119890 minus 06 2459469289561866119890 minus 06

V 1705913430341418119890 minus 03 3875831522837103119890 minus 03 2009899309359689119890 minus 07 6851382039485543119890 minus 08


119873119901 1198661198712 119866max 119878

1198712 119878max

20 3 1919678771148673119890 minus 05 3143456151022406119890 minus 05 2861019689160329119890 minus 11 1624300693947589119890 minus 11




solution the same will be done for the iterated solution Wereformulate the right hand side of the optimization problem(18) in terms of the matrices we introduced before by using(38) and (40) as follows

1

1205822(((119865

opt)119879

+ 119862119879sdot (119885

opt)119879

) sdot ((119872opt)minus1

)

119879

sdot (119865opt+ 119885

optsdot 119862) minus ((119865

119888)119879

+ 119862119879sdot (119885119888)119879

)

sdot ((119872119888)minus1

)119879

sdot (119865119888+ 119885119888sdot 119862))

(41)

As explained for Galerkin method solution at the end ofSection 32 on each element of the coarse mesh expression(41) should be calculated for each of four possible optimalmesh refinement cases and the case giving the minimumvalue is replaced with that element of the coarse mesh Forthe continuity of the approximate solution at node points ofthe coarse mesh the boundary conditions at node points ofthe coarse mesh are fixed during the calculations in the sameway as it was done for the Galerkin method solution

5 Some Applications

Both methods are applied to some problems in [46] on Fred-holm integral equations of the second kind with smoothkernel and discontinuous kernel and results are stated Foreach example we have presented error values in two differentkinds of tables In the first kind we give the error values of theconsecutive solutions where 119899 denotes the repeating orderHere resulting refinedmesh is used as the coarse mesh for thelater turn In the second kind we give the error values whenwe use119873 number of equidistant node points on coarse meshand 119901 as element local polynomial order of approximationat each element of the coarse mesh In both tables while1198661198712 and 119878

1198712 denote the 1198712-errors 119866max and 119878max denote the

maximum absolute error at node points of the fine mesh(obtained from coarsemesh) of theGalerkinmethod solutionand the iterated solution respectively

In this study our main and final goal is to reach betterapproximations by applying adaptive refinement togetherwith Sloan iteration to Galerkin method solutions and toexamine them For this reason in all examples presentedwe give two kinds of graphs with relative errors on log-logscale one with relative error in 1198712-norm and onewith relativeerror in maximum norm on 119910-axis and both with number ofdegrees of freedoms on the 119909-axis via Sloan iteration resultsGraphs clearly show the decrease in relative error whilenumber of degrees of freedoms are increasing For simplicitylog of number of degrees of freedoms is represented byldquolog(dofs)rdquo on the 119909-axis

In order to illustrate the refinement process better weprovide more details for the first example than for the latterones besides the error plots for the Sloan iteration we alsoadd the corresponding graphs for the Galerkin method andshow the mesh refinement for the five consecutive runs

Example 1 The exact solution of the problem

119909 (119904) +1

120587int

120587

minus120587

03

1 minus 064cos2 ((119904 + 119905) 2)119909 (119905) 119889119905

= 25 minus 16 sin (1199042) minus120587 le 119904 le 120587

(42)

is given with 119909(119905) = 172 + (1287) cos(2119905) Let the coarsemesh be given with the lists 119871 = [minus120587 120587] 119863 = [2] Theproblem is solved five times consecutively

As we see in each row of Table 1 Sloan iteration cause adecrease in both119866

1198712 and119866max error values for each run Also

we see a general decrease in all error types especially this ismuchmore clear for errors via Sloan iteration given in the lasttwo columns of the table

When we look to the relative error graphs given withFigures 1 and 2 we see that relative errors via Sloan iterationget smaller values rather than the ones via Galerkin method

In Table 2 we give the error for the computations startingfrom four different initial coarse meshes with 20 30 40 and50 equidistant node points on the interval [minus120587 120587] and withinitial element local polynomial order of 119901 = 2 From this


100

100

10minus1

10minus2

10minus3

10minus4

100

10minus1

10minus2

10minus3

10minus410minus5

101 102

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof)100 101 102

log (dof)

Figure 1 Convergence graphs of relative error in 1198712-norm and maximum norm via Galerkin method for (42) in Example 1

100

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

101 102 100 101 102

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof) log (dof)

Figure 2 Convergence graphs of relative error in 1198712-norm and maximum norm via Sloan iteration for (42) in Example 1

table we see that at just one run of the procedure we canreach much more smaller error values with lower elementlocal polynomial order of approximation by just increasingthe number of nodes In [46] the error at nodes obtained byusing Nystrom method was given with the value 11119890 minus 8Just for this example as a sample we give a graph includinga diagram which shows us the mesh refinement step by stepfor the five consecutive runs via Galerkin method results andthe lists of the obtained optimal mesh at the end of the fiveconsecutive runs for both methods

In Figure 3 we see the optimal mesh selections chosen bythe optimization problem (18) In the first run it chose just todo 119901 refinement and increase the element local polynomialorder of approximation from 2 to 3 which corresponds tochoice (20) In the second run it chose to make both ℎ and 119901refinements together by choosing case (22) In the third runit chose to refine the first element as in (23) and the secondone as in (20) On the fourth run while it chose to refine thefirst and third elements of level III by (22) it chose to refinethe middle element by (20) Finally in the fifth run refining



119899 1198661198712 119866max 119878

1198712 119878max

I 1154995991009992119890 minus 05 4800924769501891119890 minus 05 9079011385069973119890 minus 06 2077330804106659119890 minus 05




[minus120587 120587] p = 3

[minus120587 0] p = 3 [0 120587] p = 4

[minus120587 minus1205874] p = 4 [minus1205874 0] p = 3 [0 120587] p = 5

[minus120587 minus1205874] p = 4 [minus1205874 minus1205872] p = 5 [minus1205872 0] p = 4 [0 1205872] p = 5 [1205872 120587] p = 6

[minus120587 minus71205878] p = 5 [minus71205878 minus31205874] p = 4 [minus31205874 minus1205872] p = 6 [minus1205872 minus1205874] p = 4

[minus1205874 0] p = 5 [0 1205872] p = 6 [1205872 120587] p = 7

I Refinement

II Refinement

III Refinement

IV Refinement

V Refinement

Figure 3 Representation of mesh refinement via Galerkin method for (42) in Example 1

the first element of level IV by (23) the second fourth andfifth elements of level IV by (20) and the third element of levelIV by (22) the optimal mesh at the end of five consecutiveruns given below for the Galerkin method is obtained

For Galerkin method solution

119871 = [minus31416119890 + 00 minus27489119890 + 00 minus23562119890 + 00

minus15708119890 + 00 minus78540119890 minus 01 0

15708119890 + 00 31416119890 + 00 ]

119863 = [5 4 6 4 5 6 7]

(43)

For iterated solution we obtain the final optimal mesh as

119871 = [minus31416119890 + 00 minus15708119890 + 00 minus78540119890 minus 01

0 15708119890 + 00 23562119890 + 00 31416119890 + 00 ]

119863 = [5 5 4 5 5 4]

(44)


minus119909 (119904)

2minus int

1

0

01

001 + (119904 minus 119905)2119909 (119905) 119889119905 = 119891 (119904) 0 le 119904 le 1

(45)

is 119909(119905) = 006 minus 08119905 + 1199052 Let the coarse mesh be given withlists 119871 = [0 1] and119863 = [2] The problem is solved four timesconsecutively

We observe from the rows of Table 3 that Sloan iterationdecreases the 119866

1198712 and 119866max errors as we saw in Example 1

Besides while the error values obtained by the consecutiveruns via Galerkin method do not show much differencethe ones obtained by Sloan iteration are decreasing fasterThe graphs given by Figure 4 for Example 2 clearly show thedecrease in the relative errors in 1198712 and maximum norms asexpected

Likewise we did in our first example we used three dif-ferent initial coarse meshes having 30 40 and 50 equidistantnode points on the interval [0 1] and 119901 = 2 as initial elementlocal polynomial order of approximation at each element ofthese coarse mesh elements for all three situations As inExample 1 we again see in Table 4 that we reach much moresmaller error values with lower element local polynomialorder of approximation by just increasing the number ofnodes In [46] the error at nodes obtained by using Nystrommethod was given with the value 66119890 minus 6

Example 3 The solution of the equation

119909 (119904) minus int

1

0

119896 (119905 119904) 119909 (119905) 119889119905 = (1 minus1

1205872) sin (120587119904) 0 le 119904 le 1

(46)

with

119896 (119905 119904) = 119904 (1 minus 119905) 119904 le 119905

119905 (1 minus 119904) 119905 le 119904(47)

is 119909(119905) = sin(120587119905) Let the coarse mesh be given with the lists119871 = [0 1] and 119863 = [2] The problem is solved seven timesconsecutively


log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof)

10minus4

10minus5

10minus6

10minus7

100 101 102

10minus4

10minus5

10minus6

10minus7

log (dof)100 101 102



119873119901 1198661198712 119866max 119878

1198712 119878max





119899 1198661198712 119866max 119878

1198712 119878max






VI 5397336277740180119890 minus 04 3256397951289958119890 minus 03 8303708583183078119890 minus 05 5990667999111743119890 minus 06

VII 6252849507196142119890 minus 04 4555052326156391119890 minus 03 4721973545142185119890 minus 05 4470476808404733119890 minus 06


119873119901 1198661198712 119866max 119878

1198712 119878max




Table 5 shows that Sloan iteration causes a decrease inthe error values obtained via Galerkin method and they aregetting smaller in each seven consecutive runs Graphs givenby Figure 5 clearly show the decrease in both relative errorsin 1198712 and maximum norms as in the previous examples

By using the same three different coarse meshes used inExample 2 finally we see from Table 6 that we can obtainsmaller errors with lower element local polynomial order of

approximation by just increasing the number of nodes In[46] the error at nodes obtained by using Nystrom methodwas given with the value 17119890 minus 7

6 Conclusions

The two methods presented aimed to find and improveapproximate solutions for Fredholm integral equations of


10minus2 10minus3

10minus4

10minus5

10minus3

10minus4

10minus5 10minus6

100 102 104 100 102 104

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof) log (dof)


the second kind and to observe the effect of adaptive refine-ment on these solutions for both methods Using polynomialtype functions in the approximation of solutions for manykinds of problems in mathematics is one of the most usualmethods Generally in the whole solution interval polyno-mials having the same degree are usedThemain idea behindwhy we preferred to use adaptive refinement in our studyis observing the changes in the results when we change thisgeneral approach When we use adaptively refined meshesthis gives us a chance to use polynomials of different degreein different subintervals of the solution interval which mightcause obtaining better approximations When comparing themaximum absolute error values at nodes of Examples 1 and 2with the results in [46] we saw that our methods are able toreach much smaller error values by increasing the number ofnode points even by using polynomials having lowdegrees forthese examples which have smooth kernels For Example 3when the absolute error values at nodes are compared withthe ones in [46] we see that they are getting closer toeach other as we used more node points again by usingpolynomials of low degrees In our study we also see thatwith Sloan iteration we obtain better approximations ratherthanGalerkinmethod as seen from the examplesThe relativeerror graphs show us that Sloan iteration brings the expecteddecrease in relative error values of 1198712-error and maximumabsolute error at node points of the fine mesh The resultsshowed that generally error values are better when we usepolynomials of degree between 2 and 6 which is an advantagefor decreasing the time we spend to solve the problemsAnother advantage of our methods is that approximatesolutions are found easily by using computer code written inMatlab It is also observed that using polynomials with higherdegrees can cause oscillations in the errors The methods can

be improved not only to get better results but also to solveother problem models by some modifications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors appreciate helpful and valuable comments fromall the referees to improve the paper

References

[1] H Hochstadt Integral Equations Wiley-Interscience NewYork NY USA 1973

[2] S Rahbar and E Hashemizadeh ldquoA computational approachto the fredholm integral equation of the second kindrdquo inProceedings of theWorldCongress onEngineering vol 2 pp 933ndash937 London UK July 2008

[3] B G Pachpatte Inequalities for Differential and Integral Equa-tions Academic Press London UK 1998

[4] C T H Baker The Numerical Treatment of Integral EquationsClarendon Press Oxford UK 1977

[5] W Wang ldquoA new mechanical algorithm for solving the secondkind of Fredholm integral equationrdquo Applied Mathematics andComputation vol 172 no 2 pp 946ndash962 2006

[6] A T Lonseth ldquoApproximate solutions of Fredholm-type inte-gral equationsrdquo Bulletin of the American Mathematical Societyvol 60 pp 415ndash430 1954

[7] I Fredholm ldquoSur une classe drsquoequations fonctionnellesrdquo ActaMathematica vol 27 no 1 pp 365ndash390 1903


[8] T A Burton Volterra Integral and Differential Equations Aca-demic Press New York NY USA 1983

[9] C Corduneanu Integral Equations and Stability of FeedbackSystems Academic Press New York NY USA 1973

[10] C Corduneanu Principles of Differential and Integral EquationsChelsea Bronx County NY USA 2nd edition 1977

[11] C Corduneanu Integral Equations and Applications Cam-bridge University Press Cambridge UK 1991

[12] G Gripenberg S O Londen and O Staffans Nonlinear Vol-terra and Integral Equations Cambridge University Press Cam-bridge UK 1990

[13] A M Krasnoselskii Positive Solutions of Operator EquationNoordhoff Uitgevers Groningen The Netherlands 1964

[14] R K Miller Nonlinear Volterra Integral Equations W ABenjamin Menlo Park Calif USA 1971

[15] F G Tricomi Integral Equations Interscience New York NYUSA 1957

[16] I Babuska ldquoAdvances in the p and h-p versions of the finite ele-ment method A surveyrdquo in Numerical Mathematics Singapore1988 vol 86 of International Series of Numerical Mathematicspp 31ndash46 Birkhauser Basel Switzerland 1988

[17] I Babuska B A Szabo and I N Katz ldquoThe 119901-version of thefinite element methodrdquo SIAM Journal on Numerical Analysisvol 18 no 3 pp 515ndash545 1981

[18] I Babuska andM R Dorr ldquoError estimates for the combined hand p-versions of the finite elementmethodrdquoNumerischeMath-ematik vol 37 no 2 pp 257ndash277 1981

[19] L Demkowicz Computing with hp-Adaptive Finite ElementsI One- and Two-Dimensional Elliptic and Maxwell ProblemsChapman amp HallCRC Boca Raton Fla USA 2007

[20] M Asadzadeh and K Eriksson ldquoOn adaptive finite elementmethods for Fredholm integral equations of the second kindrdquoSIAM Journal on Numerical Analysis vol 31 no 3 pp 831ndash8551994

[21] K Atkinson A Survey of Numerical Methods for the Solutionof Fredholm Integral Equations of the Second Kind Society forIndustrial and Applied Mathematics Philadelphia Pa USA1976

[22] Y Ikebe ldquoThe Galerkin method for the numerical solution ofFredholm integral equations of the second kindrdquo SIAM Reviewvol 14 no 3 pp 465ndash491 1972

[23] J C Nedelec Approximation des quations intgrales en mcaniqueet physique Centre de Mathematiques Ecole PolytechniquePalaiseau France 1977

[24] I H Sloan ldquoA review of numerical methods for fredholm equa-tions of the second kindrdquo in The Application and NumericalSolution of Integral Equations R S Anderssen F Hook andMde Lukas Eds Sijthoff and Noordhoff Australian Nat UnivCanberra Alphen aan den Rijn The Netherlands 1978

[25] W L Wendland ldquoOn some mathematical aspects of boundaryelement methods for elliptic problemsrdquo in The Mathematics ofFinite Elements and Applications J R Whiteman Ed pp 193ndash227 Academic Press London UK 1985

[26] I G Graham R E Shaw and A Spence ldquoAdaptive numericalsolution of integral equationswith application to a problemwitha boundary layerrdquo Congressus Numerantium vol 68 pp 75ndash901989

[27] E Rank I Babuska O C Zienkiewicz J Gago and E ROliveira ldquoAdaptivity and accuracy estimation for finite elementand boundary integral elementmethodsrdquo inAccuracy Estimates

and Adaptive Refinements in Finite Element Computations pp79ndash94 John Wiley amp Sons New York NY USA 1986

[28] D H Yu ldquoA posteriori error estimates and adaptive approachesfor some boundary element methodsrdquo in Boundary ElementMethods C A Brebbia W L Wendland and G Kuhn Edspp 241ndash256 Springer Berlin Germany 1987

[29] M J Berger and P Colella ldquoLocal adaptive mesh refinement forshock hydrodynamicsrdquo Journal of Computational Physics vol82 no 1 pp 64ndash84 1989

[30] J E Schiermeier and B A Szabo ldquoInteractive design based onthe 119901-version of the finite element methodrdquo Finite Elements inAnalysis and Design vol 3 no 2 pp 93ndash107 1987

[31] S Larsen Numerical analysis of elliptic partial differential equa-tions with stochastic input data [PhD thesis] University ofMaryland College Park Md USA 1986

[32] K Eriksson and C Johnson ldquoAdaptive finite element methodsfor parabolic problems I a linearmodel problemrdquo SIAM Journalon Numerical Analysis vol 28 no 1 pp 43ndash77 1991

[33] P K Moore ldquoComparison of adaptive methods for one-dimensional parabolic systemsrdquo Applied Numerical Mathemat-ics vol 16 no 4 pp 471ndash488 1995

[34] L DemkowiczW Rachowicz and P Devloo ldquoA fully automatichp-adaptivityrdquo Journal of Scientific Computing vol 17 no 1ndash4pp 117ndash142 2002

[35] P Houston and E Suli ldquoA note on the design of ℎ119901-adaptivefinite elementmethods for elliptic partial differential equationsrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 2ndash5 pp 229ndash243 2005

[36] L He and A Zhou ldquoConvergence and complexity of adaptivefinite elementmethods for elliptic partial differential equationsrdquoInternational Journal of Numerical Analysis andModeling vol 8no 4 pp 615ndash640 2011

[37] R H W Hoppe and M Kieweg ldquoAdaptive finite elementmethods for mixed control-state constrained optimal controlproblems for elliptic boundary value problemsrdquo ComputationalOptimization and Applications vol 46 no 3 pp 511ndash533 2010

[38] D Pardo L Demkowicz C Torres-Verdın and L TabarovskyldquoA goal-oriented ℎ119901-adaptive finite element method with elec-tromagnetic applications I Electrostaticsrdquo International Jour-nal for Numerical Methods in Engineering vol 65 no 8 pp1269ndash1309 2006

[39] E P Stephan M Maischak and F Leydecker ldquoAn hp-adaptivefinite elementboundary element coupling method for electro-magnetic problemsrdquo Computational Mechanics vol 39 no 5pp 673ndash680 2007

[40] L Botti M Piccinelli B Ene-Iordache A Remuzzi and LAntiga ldquoAn adaptive mesh refinement solver for large-scalesimulation of biological flowsrdquo International Journal for Numer-icalMethods in Biomedical Engineering vol 26 no 1 pp 86ndash1002010

[41] R H Hoppe H Wu and Z Zhang ldquoAdaptive finite elementmethods for the Laplace eigenvalue problemrdquo Journal of Numer-ical Mathematics vol 18 no 4 pp 281ndash302 2010

[42] K E AtkinsonThe Numerical Solution of Integral Equations ofthe Second Kind vol 4 of Cambridge Monographs on Appliedand Computational Mathematics Cambridge University PressCambridge UK 1997

[43] M B Larson and F Bengzon The Finite Element Method The-ory Implementation and Practice Springer New York NYUSA 2010


[44] K E Atkinson ldquoA personal perspective on the history of thenumerical analysis of Fredholm integral equations of the secondkindrdquo inThe Birth of Numerical Analysis pp 53ndash72 World Sci-entific Leuven Belgium 2008

[45] I H Sloan ldquoImprovement by iteration for compact operatorequationsrdquo Mathematics of Computation vol 30 no 136 pp758ndash764 1976

[46] K E Atkinson and L F Shampine ldquoAlgorithm 876 solvingFredholm integral equations of the second kind in MatlabrdquoACM Transactions on Mathematical Software vol 34 no 4 pp1ndash20 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


mathematical problems of radiative equilibrium the particletransport problems of astrophysics and reactor theory steadystate heat conduction and fracture mechanics [5]

As Pachpatte [3] expressed the beginning of the integralequations can be traced back to N H Abel who found anintegral equation in 1812 starting from a problem in mechan-ics and in 1895 V Volterra emphasized the significance of thetheory of integral equations

Lonseth [6] stated that Fredholm published his distin-guished paper in Acta Mathematica [7] in 1903 in whichhe gave the first detailed account of the existence andmultiplicity of solutions of the following two equations wherethe kernel119870(119904 119905) and119910(119904) are known functions and the valuesof 120582 (proper values) are values to be determined such that acontinuous solution 119909(119904) equiv 0 exists

119909 (119904) minus int

1

0

119870 (119904 119905) 119909 (119905) 119889119905 = 119910 (119904) 0 le 119904 le 1

119909 (119904) minus 120582int

1

0

119870 (119904 119905) 119909 (119905) 119889119905 = 119910 (119904) 0 le 119904 le 1

(1)

In his book Pachpatte [3] stated a few number ofmonographsthat he accepted as an excellent account of integral equationsmay be found in the following Burton [8] Corduneanu [9ndash11] Gripenberg et al [12] Krasnoselskii [13] Miller [14] andTricomi [15]

12 On Adaptive Refinement In 1988 Babuska [16] publishedhis work on the advances in the 119901 and ℎ-119901 versions of thefinite elementmethod and in this study he distinguished threeversions of the finite elementmethod (FEM) as follows the ℎ-version the 119901-version and the ℎ-119901 version The main idea inthe ℎ-version is to refine the size of the meshes while degreesof the polynomials used for approximation are kept fixed(usually 119901 = 1 2) in the 119901-version it is the opposite size ofthemeshes kept fixed but degrees of the polynomials used forapproximation are increased In the ℎ-119901 version both changesare done simultaneously size of the meshes are refined andthe degrees of the polynomials used for approximation areincreased In [16] while the ℎ-version of FEM is introducedto be the standard one the other versions are said to bedeveloped later and the first theoretical papers about the 119901-version and the ℎ-119901 version which appeared in 1981 are givenwith [17] and [18] respectively

In Demkowiczrsquos book [19] which is about computationwith ℎ119901-adaptive finite elements he studied one- and two-dimensional elliptic and Maxwell problems and he men-tioned two major components of the one-dimensional ver-sion of their ℎ119901-algorithm as fine grid solution and optimalmesh selection For the first component a given (coarse)mesh is refined in both ℎ and119901 to obtain a corresponding finemesh and then the problem is solved on this finemesh to findthe fine mesh solution In the latter component he used thisfine mesh solution to determine optimal mesh refinement ofthe coarse mesh by minimizing the projection based inter-polation error solving the following discrete optimizationproblem where 119906 prod

ℎ119901119906 prodℎ119901opt119906 119873ℎ119901opt

and 119873ℎ119901

denote

respectively the solution on the fine mesh the interpolant ofthe fine grid solution on the original mesh interpolant of thefine grid solution on the new optimal mesh to be determinedthe corresponding number of degrees of freedom on thenew optimal mesh to be determined and the correspondingnumber of degrees of freedom on the original mesh

10038171003817100381710038171003817119906 minus prod

ℎ11990111990610038171003817100381710038171003817

2

1198671(0119897)minus100381710038171003817100381710038171003817119906 minus prod

ℎ119901opt119906100381710038171003817100381710038171003817

2

1198671(0119897)

119873ℎ119901opt

minus 119873ℎ119901

997888rarr min (2)

Here the aim of the optimization problem is said tomaximizethe rate of decrease of the interpolation error Asadzadehand Eriksson [20] gave a few number of references [21ndash25]on solving integral equations with FEM in their paper inwhich they have chosen to work on the single layer potentialproblem for Laplacersquos equation with Neumann boundaryconditions in order to be concrete In their paper the studieson solving integral equations with adaptive FEM in thatperiod are givenwith [25ndash28] Adaptive FEMare usually usedto solve partial differential equations but in literature thesemethods are also seen to be used for solving different typeof problems in various branches of science such as hydro-dynamics [29] optimal design [30] elliptic stochastic equa-tions [31] parabolic problems [32] parabolic systems [33]elliptic problems [34] elliptic partial differential equations[35] elliptic boundary value problems [36 37] electrostatics[38] electromagnetic problems [39] biological flows [40]and Laplace eigenvalue problem [41]

2 Refining a Finer Mesh from a Coarse Meshand Construction of Basis Functions

21 Refining a Finer Mesh from a Coarse Mesh Let 119886 = 1199051lt

1199052lt sdot sdot sdot lt 119905

119899= 119887 be the node points of a given (finite element)

mesh which is accepted as the coarsemesh and denote the listof these node points as follows

119871119888= [1199051 1199052 sdot sdot sdot 119905119899] (3)

For each 119894 isin 1 2 119899 minus 1 the interval of the form 119868119894=

[119905119894 119905119894+1] is called an element (or finite element) having two

parameters element length ℎ119894= 119905119894+1minus 119905119894and element local

polynomial of order of approximation 119901119894(119901119894ge 2) What is

meant by element local polynomial of order of approximationcan be explained as follows ldquoIf an element has element localpolynomial of order of approximation of order119901 it means thenonlinear base functions on that element are the polynomialsof degree starting from 2 to 119901rdquo Let 119863119888 be the list of 119899 minus 1numbers of element local polynomial order of approximationof the coarse mesh elements

119863119888= [1199011 1199012 sdot sdot sdot 119901119899minus1] (4)

Firstly dividing each element of thismesh from themiddle (ℎ-refinement) and then increasing the element local polynomialorder of approximation by 1 for each new element (119901-refine-ment) we obtain a finer mesh having new lists of node pointsand the element local polynomial orders of approximationgiven belowwhich are of length 2119899minus1 and 2119899minus2 respectively


119871119891= [1199051

(1199051+ 1199052)


(119905119899minus1+ 119905119899)

2119905119899] (5)



1205931(119905) =

(1199052minus 119905)

(1199052minus 1199051) if 119905 isin [119905

1 1199052]

0 otherwise

120593119894(119905) =

(119905 minus 119905119894)

(119905119894+1minus 119905119894) if 119905 isin [119905

119894minus1 119905119894]

(119905119894+1minus 119905)

(119905119894+1minus 119905119894) if 119905 isin [119905

119894 119905119894+1]

0 otherwise

119894 = 2 119899 minus 1

120593119899(119905) =

(119905 minus 119905119899minus1)


119899minus1 119905119899]

0 otherwise

(7)



120595 [119905119894 119905119894+1] 997888rarr [minus1 1] 120595 (119905) =

2119905 minus (119905119894+1+ 119905119894)

119905119894+1minus 119905119894

(8)



120593119894119901(119905) =

119871119901∘ 120595 (119905) if 119905 isin [119905

119894minus1 119905119894]

0 otherwise119894 = 2 119899 119901 ge 2

(9)





120582119909 (119905) minus int

119887

119886

119870 (119905 119904) 119909 (119904) 119889119904 = 119891 (119905) 119886 le 119905 le 119887 120582 = 0

(10)


1+1199012+sdot sdot sdot+119901

119899minus1)


119906 (119905) =

119889119891

sum

119895=1

119888119895120601119895(119905) 119905 isin [119886 119887] (11)



119891


119903 (119905) =

119889119891

sum

119895=1

119888119895120582120601119895(119905) minus int

119863

119870 (119905 119904) 120601119895(119905) 119889119904 minus 119910 (119905) 119905 isin 119863

(12)


⟨119903 120601119894⟩ = 0 119894 = 1 119889

119891 (13)


119889119891

sum

119895=1

119888119895[120582int

119887

119886

120601119895(119905) 120601119894(119905) 119889119905 minus int

119887

119886

int

119887

119886

119870 (119905 119904) 120601119895(119904) 120601119894(119905) 119889119904 119889119905]

= int

119887

119886

119891 (119905) 120601119894(119905) 119889119905

(14)



119864 =

[[[[[

[

11986411

sdot sdot sdot 1198641198891198911

11986421

sdot sdot sdot 1198641198891198912

1198641198891198911sdot sdot sdot 119864119889119891119889119891

]]]]]

]

119864119894119895= 120582int

119887

119886

120601119895(119905) 120601119894(119905) 119889119905

119870 =

[[[[[

[

11987011

sdot sdot sdot 1198701198891198911

11987021

sdot sdot sdot 1198701198891198912

1198701198891198911sdot sdot sdot 119870

119889119891119889119891

]]]]]

]

119870119894119895= int

119887

119886

int

119887

119886

119870 (119905 119904) 120601119895(119904) 120601119894(119905) 119889119904 119889119905

119865 = [int

119887

119886

119891(119905)1206011(119905)119889119905 int

119887

119886


119887

119886

119891(119905)120601119889119891

(119905)119889119905]

119879

119862 = [1198881 1198882 sdot sdot sdot 119888119889119891]119879

(15)


(119864 minus 119870) sdot 119862 = 119865 (16)


119862 = (119864 minus 119870)minus1sdot 119865 (17)



10038171003817100381710038171003817119906 minus Π

ℎ11990111990610038171003817100381710038171003817

2

minus100381710038171003817100381710038171003817119906 minus Π

ℎ119901opt119906100381710038171003817100381710038171003817

2

119873ℎ119901opt

minus 119873ℎ119901

997888rarr min (18)



119871119891= [119886

119886 + 119887

2119887] 119863

119891= [119901 + 1 119901 + 1] (19)


119871opt= [119886 119887] 119863

opt= [119901 + 1] (20)

or 119871opt= [119886

119886 + 119887

2119887] 119863

opt= [119901 119901] (21)

or 119871opt= [119886

119886 + 119887

2119887] 119863

opt= [119901 119901 + 1] (22)

or 119871opt= [119886

119886 + 119887

2119887] 119863

opt= [119901 + 1 119901] (23)





119895

(119895 = 1 2 119889119888) and 120585opt

119895 (119895 = 1 2 119889opt) be the basis


Πℎ119901119906 =

119889119888

sum

119895=1

120585119888

119895120601119888

119895 (24)

Πℎ119901opt119906 =

119889opt

sum

119895=1

120585opt119895120601opt119895 (25)

Let 120585119888 = [120585119888

1120585119888


119889119888

]119879 120585opt = [120585

opt1

120585opt2






119885119888=

[[[[[

[

119885119888


119888

1119889119891

119885119888


119888

2119889119891

119885119888

1198891198881sdot sdot sdot 119885

119888

119889119888119889119891

]]]]]

]

119885119888

119894119895= int

119887

119886

120601119888

119894(119904) 120601119891

119895(119904) 119889119904

119872119888=

[[[[

[

119872119888


119888

1119889119888

119872119888


119888

2119889119888

119872119888

1198891198881sdot sdot sdot 119872

119888

119889119888119889119888

]]]]

]

119872119888

119894119895= int

119887

119886

120601119888

119894(119904) 120601119888

119895(119904) 119889119904

(26)


120585119888= (119872119888)minus1

sdot 119885119888sdot 119862 (27)





119885opt=

[[[[[[

[

119885opt11


119885opt21


119885opt119889opt1


]]]]]]

]

119885opt119894119895= int

119887

119886

120601opt119894(119904) 120601119891

119895(119904) 119889119904

119872opt=

[[[[[[

[

119872opt11


119872opt21


119872opt119889opt1


]]]]]]

]

119872opt119894119895=int

119887

119886

120601opt119894(119904) 120601119888

119895(119904) 119889119904

(28)


120585opt= (119872

opt)minus1

sdot 119885optsdot 119862 (29)


ℎ119901opt119906


(119862119879sdot [(119885

opt)119879


)

119879

sdot 119885optminus (119885119888)119879

sdot ((119872119888)minus1

)119879

sdot 119885119888] sdot 119862)


(30)





ℎ119901119906 If we



Πℎ119901119906 (119886) = 119906 (119886)

997904rArr 120585119888

1sdot 120601119888

1(119886) + 0 = 119888

1sdot 120601119891

1(119886) + 0

997904rArr 120585119888

1= 1198881

Πℎ119901119906 (119887) = 119906 (119887)

997904rArr 120585119888

2sdot 120601119888

2(119887) + 0 = 119888

3sdot 120601119891

3(119887) + 0

997904rArr 120585119888

2= 1198883

(31)


1and 120585119888



= 1198881and 120585opt



ℎ119901opt


Πℎ119901opt119906 (119886) = 119906 (119886)

997904rArr 120585opt1sdot 120601

opt1(119886) + 0 = 119888

1sdot 120601119891

1(119886) + 0

997904rArr 120585opt1= 1198881

Πℎ119901opt119906 (119887) = 119906 (119887)

997904rArr 120585opt3sdot 120601

opt3(119887) + 0 = 119888

3sdot 120601119891

3(119887) + 0

997904rArr 120585opt3= 1198883

(32)



opt1



ℎ119901opt



119909 =1

120582(119891 + 119911) (33)

where 119911 = K119909 = int119887

119886119870(119905 119904)119909(119904)119889119904 119905 isin [119886 119887] Atkinson [44]



119909119899=1

120582(119891 +K119909

119899) (34)




119909 (119905) =1

120582[int

119887

119886

119870 (119905 119904) 119909 (119904) 119889119904 + 119891 (119905)] 119905 isin [119886 119887] (35)


119909 (119905) =1

120582

[

[

119891 (119905) +

119889

sum

119895=1

119888119895int

119887

119886

119870 (119905 119904) 120601119895(119904) 119889119904]

]

119905 isin [119886 119887]

(36)




119885119888=

[[[[[

[

119885119888


119888

1198891198911

119885119888


119888

1198891198912

119885119888

1119889119888


119889119891119889119888

]]]]]

]

119885119888= int

119887

119886

int

119887

119886

119870 (119905 119904) 120601119891

119895(119904) 120601119888

119894(119905) 119889119904 119889119905

119865119888= [int

119887

119886

119891 (119905) 120601119888

1(119905) 119889119905 int

119887

119886

119891 (119905) 120601119888


119887

119886

119891 (119905) 120601119888

119889119888

(119905) 119889119905]

119879

(37)


120585119888=1

120582(119872119888)minus1

sdot (119865119888+ 119885119888sdot 119862) (38)





119885opt=

[[[[[[[[

[

119885opt11


119885opt12


119885opt1119889opt


]]]]]]]]

]

119885opt119894119895= int

119887

119886

int

119887

119886

119870 (119905 119904) 120601119891

119895(119904) 120601

opt119894(119905) 119889119904 119889119905

119865opt= [int

119887

119886

119891 (119905) 120601opt1(119905) 119889119905 int

119887

119886


119887

119886

119891 (119905) 120601opt119889opt(119905) 119889119905]

119879

(39)


120585opt=1

120582(119872

opt)minus1

sdot (119865opt+ 119885

optsdot 119862) (40)






119899 1198661198712 119866max 119878

1198712 119878max

I 3807778247544568119890 + 00 3962758499440351119890 + 00 6142245995845852119890 minus 02 2691129516577462119890 minus 02






119873119901 1198661198712 119866max 119878

1198712 119878max






1

1205822(((119865

opt)119879

+ 119862119879sdot (119885

opt)119879


)

119879

sdot (119865opt+ 119885


119888)119879

+ 119862119879sdot (119885119888)119879

)

sdot ((119872119888)minus1

)119879

sdot (119865119888+ 119885119888sdot 119862))

(41)


5 Some Applications







119909 (119904) +1

120587int

120587

minus120587

03

1 minus 064cos2 ((119904 + 119905) 2)119909 (119905) 119889119905


(42)








100

100

10minus1

10minus2

10minus3

10minus4

100

10minus1

10minus2

10minus3

10minus410minus5

101 102

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof)100 101 102

log (dof)


100

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

101 102 100 101 102

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof) log (dof)






119899 1198661198712 119866max 119878

1198712 119878max





[minus120587 120587] p = 3

[minus120587 0] p = 3 [0 120587] p = 4




[minus1205874 0] p = 5 [0 1205872] p = 6 [1205872 120587] p = 7

I Refinement

II Refinement

III Refinement

IV Refinement

V Refinement






15708119890 + 00 31416119890 + 00 ]

119863 = [5 4 6 4 5 6 7]

(43)



0 15708119890 + 00 23562119890 + 00 31416119890 + 00 ]

119863 = [5 5 4 5 5 4]

(44)


minus119909 (119904)

2minus int

1

0

01

001 + (119904 minus 119905)2119909 (119905) 119889119905 = 119891 (119904) 0 le 119904 le 1

(45)







119909 (119904) minus int

1

0

119896 (119905 119904) 119909 (119905) 119889119905 = (1 minus1

1205872) sin (120587119904) 0 le 119904 le 1

(46)

with

119896 (119905 119904) = 119904 (1 minus 119905) 119904 le 119905

119905 (1 minus 119904) 119905 le 119904(47)



log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof)

10minus4

10minus5

10minus6

10minus7

100 101 102

10minus4

10minus5

10minus6

10minus7

log (dof)100 101 102



119873119901 1198661198712 119866max 119878

1198712 119878max





119899 1198661198712 119866max 119878

1198712 119878max









119873119901 1198661198712 119866max 119878

1198712 119878max







6 Conclusions



10minus2 10minus3

10minus4

10minus5

10minus3

10minus4

10minus5 10minus6

100 102 104 100 102 104

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof) log (dof)






Acknowledgment


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119871119891= [1199051

(1199051+ 1199052)


(119905119899minus1+ 119905119899)

2119905119899] (5)



1205931(119905) =

(1199052minus 119905)

(1199052minus 1199051) if 119905 isin [119905

1 1199052]

0 otherwise

120593119894(119905) =

(119905 minus 119905119894)

(119905119894+1minus 119905119894) if 119905 isin [119905

119894minus1 119905119894]

(119905119894+1minus 119905)

(119905119894+1minus 119905119894) if 119905 isin [119905

119894 119905119894+1]

0 otherwise

119894 = 2 119899 minus 1

120593119899(119905) =

(119905 minus 119905119899minus1)


119899minus1 119905119899]

0 otherwise

(7)



120595 [119905119894 119905119894+1] 997888rarr [minus1 1] 120595 (119905) =

2119905 minus (119905119894+1+ 119905119894)

119905119894+1minus 119905119894

(8)



120593119894119901(119905) =

119871119901∘ 120595 (119905) if 119905 isin [119905

119894minus1 119905119894]

0 otherwise119894 = 2 119899 119901 ge 2

(9)





120582119909 (119905) minus int

119887

119886

119870 (119905 119904) 119909 (119904) 119889119904 = 119891 (119905) 119886 le 119905 le 119887 120582 = 0

(10)


1+1199012+sdot sdot sdot+119901

119899minus1)


119906 (119905) =

119889119891

sum

119895=1

119888119895120601119895(119905) 119905 isin [119886 119887] (11)



119891


119903 (119905) =

119889119891

sum

119895=1

119888119895120582120601119895(119905) minus int

119863

119870 (119905 119904) 120601119895(119905) 119889119904 minus 119910 (119905) 119905 isin 119863

(12)


⟨119903 120601119894⟩ = 0 119894 = 1 119889

119891 (13)


119889119891

sum

119895=1

119888119895[120582int

119887

119886

120601119895(119905) 120601119894(119905) 119889119905 minus int

119887

119886

int

119887

119886

119870 (119905 119904) 120601119895(119904) 120601119894(119905) 119889119904 119889119905]

= int

119887

119886

119891 (119905) 120601119894(119905) 119889119905

(14)



119864 =

[[[[[

[

11986411

sdot sdot sdot 1198641198891198911

11986421

sdot sdot sdot 1198641198891198912

1198641198891198911sdot sdot sdot 119864119889119891119889119891

]]]]]

]

119864119894119895= 120582int

119887

119886

120601119895(119905) 120601119894(119905) 119889119905

119870 =

[[[[[

[

11987011

sdot sdot sdot 1198701198891198911

11987021

sdot sdot sdot 1198701198891198912

1198701198891198911sdot sdot sdot 119870

119889119891119889119891

]]]]]

]

119870119894119895= int

119887

119886

int

119887

119886

119870 (119905 119904) 120601119895(119904) 120601119894(119905) 119889119904 119889119905

119865 = [int

119887

119886

119891(119905)1206011(119905)119889119905 int

119887

119886


119887

119886

119891(119905)120601119889119891

(119905)119889119905]

119879

119862 = [1198881 1198882 sdot sdot sdot 119888119889119891]119879

(15)


(119864 minus 119870) sdot 119862 = 119865 (16)


119862 = (119864 minus 119870)minus1sdot 119865 (17)



10038171003817100381710038171003817119906 minus Π

ℎ11990111990610038171003817100381710038171003817

2

minus100381710038171003817100381710038171003817119906 minus Π

ℎ119901opt119906100381710038171003817100381710038171003817

2

119873ℎ119901opt

minus 119873ℎ119901

997888rarr min (18)



119871119891= [119886

119886 + 119887

2119887] 119863

119891= [119901 + 1 119901 + 1] (19)


119871opt= [119886 119887] 119863

opt= [119901 + 1] (20)

or 119871opt= [119886

119886 + 119887

2119887] 119863

opt= [119901 119901] (21)

or 119871opt= [119886

119886 + 119887

2119887] 119863

opt= [119901 119901 + 1] (22)

or 119871opt= [119886

119886 + 119887

2119887] 119863

opt= [119901 + 1 119901] (23)





119895

(119895 = 1 2 119889119888) and 120585opt

119895 (119895 = 1 2 119889opt) be the basis


Πℎ119901119906 =

119889119888

sum

119895=1

120585119888

119895120601119888

119895 (24)

Πℎ119901opt119906 =

119889opt

sum

119895=1

120585opt119895120601opt119895 (25)

Let 120585119888 = [120585119888

1120585119888


119889119888

]119879 120585opt = [120585

opt1

120585opt2






119885119888=

[[[[[

[

119885119888


119888

1119889119891

119885119888


119888

2119889119891

119885119888

1198891198881sdot sdot sdot 119885

119888

119889119888119889119891

]]]]]

]

119885119888

119894119895= int

119887

119886

120601119888

119894(119904) 120601119891

119895(119904) 119889119904

119872119888=

[[[[

[

119872119888


119888

1119889119888

119872119888


119888

2119889119888

119872119888

1198891198881sdot sdot sdot 119872

119888

119889119888119889119888

]]]]

]

119872119888

119894119895= int

119887

119886

120601119888

119894(119904) 120601119888

119895(119904) 119889119904

(26)


120585119888= (119872119888)minus1

sdot 119885119888sdot 119862 (27)





119885opt=

[[[[[[

[

119885opt11


119885opt21


119885opt119889opt1


]]]]]]

]

119885opt119894119895= int

119887

119886

120601opt119894(119904) 120601119891

119895(119904) 119889119904

119872opt=

[[[[[[

[

119872opt11


119872opt21


119872opt119889opt1


]]]]]]

]

119872opt119894119895=int

119887

119886

120601opt119894(119904) 120601119888

119895(119904) 119889119904

(28)


120585opt= (119872

opt)minus1

sdot 119885optsdot 119862 (29)


ℎ119901opt119906


(119862119879sdot [(119885

opt)119879


)

119879

sdot 119885optminus (119885119888)119879

sdot ((119872119888)minus1

)119879

sdot 119885119888] sdot 119862)


(30)





ℎ119901119906 If we



Πℎ119901119906 (119886) = 119906 (119886)

997904rArr 120585119888

1sdot 120601119888

1(119886) + 0 = 119888

1sdot 120601119891

1(119886) + 0

997904rArr 120585119888

1= 1198881

Πℎ119901119906 (119887) = 119906 (119887)

997904rArr 120585119888

2sdot 120601119888

2(119887) + 0 = 119888

3sdot 120601119891

3(119887) + 0

997904rArr 120585119888

2= 1198883

(31)


1and 120585119888



= 1198881and 120585opt



ℎ119901opt


Πℎ119901opt119906 (119886) = 119906 (119886)

997904rArr 120585opt1sdot 120601

opt1(119886) + 0 = 119888

1sdot 120601119891

1(119886) + 0

997904rArr 120585opt1= 1198881

Πℎ119901opt119906 (119887) = 119906 (119887)

997904rArr 120585opt3sdot 120601

opt3(119887) + 0 = 119888

3sdot 120601119891

3(119887) + 0

997904rArr 120585opt3= 1198883

(32)



opt1



ℎ119901opt



119909 =1

120582(119891 + 119911) (33)

where 119911 = K119909 = int119887

119886119870(119905 119904)119909(119904)119889119904 119905 isin [119886 119887] Atkinson [44]



119909119899=1

120582(119891 +K119909

119899) (34)




119909 (119905) =1

120582[int

119887

119886

119870 (119905 119904) 119909 (119904) 119889119904 + 119891 (119905)] 119905 isin [119886 119887] (35)


119909 (119905) =1

120582

[

[

119891 (119905) +

119889

sum

119895=1

119888119895int

119887

119886

119870 (119905 119904) 120601119895(119904) 119889119904]

]

119905 isin [119886 119887]

(36)




119885119888=

[[[[[

[

119885119888


119888

1198891198911

119885119888


119888

1198891198912

119885119888

1119889119888


119889119891119889119888

]]]]]

]

119885119888= int

119887

119886

int

119887

119886

119870 (119905 119904) 120601119891

119895(119904) 120601119888

119894(119905) 119889119904 119889119905

119865119888= [int

119887

119886

119891 (119905) 120601119888

1(119905) 119889119905 int

119887

119886

119891 (119905) 120601119888


119887

119886

119891 (119905) 120601119888

119889119888

(119905) 119889119905]

119879

(37)


120585119888=1

120582(119872119888)minus1

sdot (119865119888+ 119885119888sdot 119862) (38)





119885opt=

[[[[[[[[

[

119885opt11


119885opt12


119885opt1119889opt


]]]]]]]]

]

119885opt119894119895= int

119887

119886

int

119887

119886

119870 (119905 119904) 120601119891

119895(119904) 120601

opt119894(119905) 119889119904 119889119905

119865opt= [int

119887

119886

119891 (119905) 120601opt1(119905) 119889119905 int

119887

119886


119887

119886

119891 (119905) 120601opt119889opt(119905) 119889119905]

119879

(39)


120585opt=1

120582(119872

opt)minus1

sdot (119865opt+ 119885

optsdot 119862) (40)






119899 1198661198712 119866max 119878

1198712 119878max

I 3807778247544568119890 + 00 3962758499440351119890 + 00 6142245995845852119890 minus 02 2691129516577462119890 minus 02






119873119901 1198661198712 119866max 119878

1198712 119878max






1

1205822(((119865

opt)119879

+ 119862119879sdot (119885

opt)119879


)

119879

sdot (119865opt+ 119885


119888)119879

+ 119862119879sdot (119885119888)119879

)

sdot ((119872119888)minus1

)119879

sdot (119865119888+ 119885119888sdot 119862))

(41)


5 Some Applications







119909 (119904) +1

120587int

120587

minus120587

03

1 minus 064cos2 ((119904 + 119905) 2)119909 (119905) 119889119905


(42)








100

100

10minus1

10minus2

10minus3

10minus4

100

10minus1

10minus2

10minus3

10minus410minus5

101 102

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof)100 101 102

log (dof)


100

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

101 102 100 101 102

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof) log (dof)






119899 1198661198712 119866max 119878

1198712 119878max





[minus120587 120587] p = 3

[minus120587 0] p = 3 [0 120587] p = 4




[minus1205874 0] p = 5 [0 1205872] p = 6 [1205872 120587] p = 7

I Refinement

II Refinement

III Refinement

IV Refinement

V Refinement






15708119890 + 00 31416119890 + 00 ]

119863 = [5 4 6 4 5 6 7]

(43)



0 15708119890 + 00 23562119890 + 00 31416119890 + 00 ]

119863 = [5 5 4 5 5 4]

(44)


minus119909 (119904)

2minus int

1

0

01

001 + (119904 minus 119905)2119909 (119905) 119889119905 = 119891 (119904) 0 le 119904 le 1

(45)







119909 (119904) minus int

1

0

119896 (119905 119904) 119909 (119905) 119889119905 = (1 minus1

1205872) sin (120587119904) 0 le 119904 le 1

(46)

with

119896 (119905 119904) = 119904 (1 minus 119905) 119904 le 119905

119905 (1 minus 119904) 119905 le 119904(47)



log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof)

10minus4

10minus5

10minus6

10minus7

100 101 102

10minus4

10minus5

10minus6

10minus7

log (dof)100 101 102



119873119901 1198661198712 119866max 119878

1198712 119878max





119899 1198661198712 119866max 119878

1198712 119878max









119873119901 1198661198712 119866max 119878

1198712 119878max







6 Conclusions



10minus2 10minus3

10minus4

10minus5

10minus3

10minus4

10minus5 10minus6

100 102 104 100 102 104

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof) log (dof)






Acknowledgment


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119864 =

[[[[[

[

11986411

sdot sdot sdot 1198641198891198911

11986421

sdot sdot sdot 1198641198891198912

1198641198891198911sdot sdot sdot 119864119889119891119889119891

]]]]]

]

119864119894119895= 120582int

119887

119886

120601119895(119905) 120601119894(119905) 119889119905

119870 =

[[[[[

[

11987011

sdot sdot sdot 1198701198891198911

11987021

sdot sdot sdot 1198701198891198912

1198701198891198911sdot sdot sdot 119870

119889119891119889119891

]]]]]

]

119870119894119895= int

119887

119886

int

119887

119886

119870 (119905 119904) 120601119895(119904) 120601119894(119905) 119889119904 119889119905

119865 = [int

119887

119886

119891(119905)1206011(119905)119889119905 int

119887

119886


119887

119886

119891(119905)120601119889119891

(119905)119889119905]

119879

119862 = [1198881 1198882 sdot sdot sdot 119888119889119891]119879

(15)


(119864 minus 119870) sdot 119862 = 119865 (16)


119862 = (119864 minus 119870)minus1sdot 119865 (17)



10038171003817100381710038171003817119906 minus Π

ℎ11990111990610038171003817100381710038171003817

2

minus100381710038171003817100381710038171003817119906 minus Π

ℎ119901opt119906100381710038171003817100381710038171003817

2

119873ℎ119901opt

minus 119873ℎ119901

997888rarr min (18)



119871119891= [119886

119886 + 119887

2119887] 119863

119891= [119901 + 1 119901 + 1] (19)


119871opt= [119886 119887] 119863

opt= [119901 + 1] (20)

or 119871opt= [119886

119886 + 119887

2119887] 119863

opt= [119901 119901] (21)

or 119871opt= [119886

119886 + 119887

2119887] 119863

opt= [119901 119901 + 1] (22)

or 119871opt= [119886

119886 + 119887

2119887] 119863

opt= [119901 + 1 119901] (23)





119895

(119895 = 1 2 119889119888) and 120585opt

119895 (119895 = 1 2 119889opt) be the basis


Πℎ119901119906 =

119889119888

sum

119895=1

120585119888

119895120601119888

119895 (24)

Πℎ119901opt119906 =

119889opt

sum

119895=1

120585opt119895120601opt119895 (25)

Let 120585119888 = [120585119888

1120585119888


119889119888

]119879 120585opt = [120585

opt1

120585opt2






119885119888=

[[[[[

[

119885119888


119888

1119889119891

119885119888


119888

2119889119891

119885119888

1198891198881sdot sdot sdot 119885

119888

119889119888119889119891

]]]]]

]

119885119888

119894119895= int

119887

119886

120601119888

119894(119904) 120601119891

119895(119904) 119889119904

119872119888=

[[[[

[

119872119888


119888

1119889119888

119872119888


119888

2119889119888

119872119888

1198891198881sdot sdot sdot 119872

119888

119889119888119889119888

]]]]

]

119872119888

119894119895= int

119887

119886

120601119888

119894(119904) 120601119888

119895(119904) 119889119904

(26)


120585119888= (119872119888)minus1

sdot 119885119888sdot 119862 (27)





119885opt=

[[[[[[

[

119885opt11


119885opt21


119885opt119889opt1


]]]]]]

]

119885opt119894119895= int

119887

119886

120601opt119894(119904) 120601119891

119895(119904) 119889119904

119872opt=

[[[[[[

[

119872opt11


119872opt21


119872opt119889opt1


]]]]]]

]

119872opt119894119895=int

119887

119886

120601opt119894(119904) 120601119888

119895(119904) 119889119904

(28)


120585opt= (119872

opt)minus1

sdot 119885optsdot 119862 (29)


ℎ119901opt119906


(119862119879sdot [(119885

opt)119879


)

119879

sdot 119885optminus (119885119888)119879

sdot ((119872119888)minus1

)119879

sdot 119885119888] sdot 119862)


(30)





ℎ119901119906 If we



Πℎ119901119906 (119886) = 119906 (119886)

997904rArr 120585119888

1sdot 120601119888

1(119886) + 0 = 119888

1sdot 120601119891

1(119886) + 0

997904rArr 120585119888

1= 1198881

Πℎ119901119906 (119887) = 119906 (119887)

997904rArr 120585119888

2sdot 120601119888

2(119887) + 0 = 119888

3sdot 120601119891

3(119887) + 0

997904rArr 120585119888

2= 1198883

(31)


1and 120585119888



= 1198881and 120585opt



ℎ119901opt


Πℎ119901opt119906 (119886) = 119906 (119886)

997904rArr 120585opt1sdot 120601

opt1(119886) + 0 = 119888

1sdot 120601119891

1(119886) + 0

997904rArr 120585opt1= 1198881

Πℎ119901opt119906 (119887) = 119906 (119887)

997904rArr 120585opt3sdot 120601

opt3(119887) + 0 = 119888

3sdot 120601119891

3(119887) + 0

997904rArr 120585opt3= 1198883

(32)



opt1



ℎ119901opt



119909 =1

120582(119891 + 119911) (33)

where 119911 = K119909 = int119887

119886119870(119905 119904)119909(119904)119889119904 119905 isin [119886 119887] Atkinson [44]



119909119899=1

120582(119891 +K119909

119899) (34)




119909 (119905) =1

120582[int

119887

119886

119870 (119905 119904) 119909 (119904) 119889119904 + 119891 (119905)] 119905 isin [119886 119887] (35)


119909 (119905) =1

120582

[

[

119891 (119905) +

119889

sum

119895=1

119888119895int

119887

119886

119870 (119905 119904) 120601119895(119904) 119889119904]

]

119905 isin [119886 119887]

(36)




119885119888=

[[[[[

[

119885119888


119888

1198891198911

119885119888


119888

1198891198912

119885119888

1119889119888


119889119891119889119888

]]]]]

]

119885119888= int

119887

119886

int

119887

119886

119870 (119905 119904) 120601119891

119895(119904) 120601119888

119894(119905) 119889119904 119889119905

119865119888= [int

119887

119886

119891 (119905) 120601119888

1(119905) 119889119905 int

119887

119886

119891 (119905) 120601119888


119887

119886

119891 (119905) 120601119888

119889119888

(119905) 119889119905]

119879

(37)


120585119888=1

120582(119872119888)minus1

sdot (119865119888+ 119885119888sdot 119862) (38)





119885opt=

[[[[[[[[

[

119885opt11


119885opt12


119885opt1119889opt


]]]]]]]]

]

119885opt119894119895= int

119887

119886

int

119887

119886

119870 (119905 119904) 120601119891

119895(119904) 120601

opt119894(119905) 119889119904 119889119905

119865opt= [int

119887

119886

119891 (119905) 120601opt1(119905) 119889119905 int

119887

119886


119887

119886

119891 (119905) 120601opt119889opt(119905) 119889119905]

119879

(39)


120585opt=1

120582(119872

opt)minus1

sdot (119865opt+ 119885

optsdot 119862) (40)






119899 1198661198712 119866max 119878

1198712 119878max

I 3807778247544568119890 + 00 3962758499440351119890 + 00 6142245995845852119890 minus 02 2691129516577462119890 minus 02






119873119901 1198661198712 119866max 119878

1198712 119878max






1

1205822(((119865

opt)119879

+ 119862119879sdot (119885

opt)119879


)

119879

sdot (119865opt+ 119885


119888)119879

+ 119862119879sdot (119885119888)119879

)

sdot ((119872119888)minus1

)119879

sdot (119865119888+ 119885119888sdot 119862))

(41)


5 Some Applications







119909 (119904) +1

120587int

120587

minus120587

03

1 minus 064cos2 ((119904 + 119905) 2)119909 (119905) 119889119905


(42)








100

100

10minus1

10minus2

10minus3

10minus4

100

10minus1

10minus2

10minus3

10minus410minus5

101 102

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof)100 101 102

log (dof)


100

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

101 102 100 101 102

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof) log (dof)






119899 1198661198712 119866max 119878

1198712 119878max





[minus120587 120587] p = 3

[minus120587 0] p = 3 [0 120587] p = 4




[minus1205874 0] p = 5 [0 1205872] p = 6 [1205872 120587] p = 7

I Refinement

II Refinement

III Refinement

IV Refinement

V Refinement






15708119890 + 00 31416119890 + 00 ]

119863 = [5 4 6 4 5 6 7]

(43)



0 15708119890 + 00 23562119890 + 00 31416119890 + 00 ]

119863 = [5 5 4 5 5 4]

(44)


minus119909 (119904)

2minus int

1

0

01

001 + (119904 minus 119905)2119909 (119905) 119889119905 = 119891 (119904) 0 le 119904 le 1

(45)







119909 (119904) minus int

1

0

119896 (119905 119904) 119909 (119905) 119889119905 = (1 minus1

1205872) sin (120587119904) 0 le 119904 le 1

(46)

with

119896 (119905 119904) = 119904 (1 minus 119905) 119904 le 119905

119905 (1 minus 119904) 119905 le 119904(47)



log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof)

10minus4

10minus5

10minus6

10minus7

100 101 102

10minus4

10minus5

10minus6

10minus7

log (dof)100 101 102



119873119901 1198661198712 119866max 119878

1198712 119878max





119899 1198661198712 119866max 119878

1198712 119878max









119873119901 1198661198712 119866max 119878

1198712 119878max







6 Conclusions



10minus2 10minus3

10minus4

10minus5

10minus3

10minus4

10minus5 10minus6

100 102 104 100 102 104

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof) log (dof)






Acknowledgment


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119885119888=

[[[[[

[

119885119888


119888

1119889119891

119885119888


119888

2119889119891

119885119888

1198891198881sdot sdot sdot 119885

119888

119889119888119889119891

]]]]]

]

119885119888

119894119895= int

119887

119886

120601119888

119894(119904) 120601119891

119895(119904) 119889119904

119872119888=

[[[[

[

119872119888


119888

1119889119888

119872119888


119888

2119889119888

119872119888

1198891198881sdot sdot sdot 119872

119888

119889119888119889119888

]]]]

]

119872119888

119894119895= int

119887

119886

120601119888

119894(119904) 120601119888

119895(119904) 119889119904

(26)


120585119888= (119872119888)minus1

sdot 119885119888sdot 119862 (27)





119885opt=

[[[[[[

[

119885opt11


119885opt21


119885opt119889opt1


]]]]]]

]

119885opt119894119895= int

119887

119886

120601opt119894(119904) 120601119891

119895(119904) 119889119904

119872opt=

[[[[[[

[

119872opt11


119872opt21


119872opt119889opt1


]]]]]]

]

119872opt119894119895=int

119887

119886

120601opt119894(119904) 120601119888

119895(119904) 119889119904

(28)


120585opt= (119872

opt)minus1

sdot 119885optsdot 119862 (29)


ℎ119901opt119906


(119862119879sdot [(119885

opt)119879


)

119879

sdot 119885optminus (119885119888)119879

sdot ((119872119888)minus1

)119879

sdot 119885119888] sdot 119862)


(30)





ℎ119901119906 If we



Πℎ119901119906 (119886) = 119906 (119886)

997904rArr 120585119888

1sdot 120601119888

1(119886) + 0 = 119888

1sdot 120601119891

1(119886) + 0

997904rArr 120585119888

1= 1198881

Πℎ119901119906 (119887) = 119906 (119887)

997904rArr 120585119888

2sdot 120601119888

2(119887) + 0 = 119888

3sdot 120601119891

3(119887) + 0

997904rArr 120585119888

2= 1198883

(31)


1and 120585119888



= 1198881and 120585opt



ℎ119901opt


Πℎ119901opt119906 (119886) = 119906 (119886)

997904rArr 120585opt1sdot 120601

opt1(119886) + 0 = 119888

1sdot 120601119891

1(119886) + 0

997904rArr 120585opt1= 1198881

Πℎ119901opt119906 (119887) = 119906 (119887)

997904rArr 120585opt3sdot 120601

opt3(119887) + 0 = 119888

3sdot 120601119891

3(119887) + 0

997904rArr 120585opt3= 1198883

(32)



opt1



ℎ119901opt



119909 =1

120582(119891 + 119911) (33)

where 119911 = K119909 = int119887

119886119870(119905 119904)119909(119904)119889119904 119905 isin [119886 119887] Atkinson [44]



119909119899=1

120582(119891 +K119909

119899) (34)




119909 (119905) =1

120582[int

119887

119886

119870 (119905 119904) 119909 (119904) 119889119904 + 119891 (119905)] 119905 isin [119886 119887] (35)


119909 (119905) =1

120582

[

[

119891 (119905) +

119889

sum

119895=1

119888119895int

119887

119886

119870 (119905 119904) 120601119895(119904) 119889119904]

]

119905 isin [119886 119887]

(36)




119885119888=

[[[[[

[

119885119888


119888

1198891198911

119885119888


119888

1198891198912

119885119888

1119889119888


119889119891119889119888

]]]]]

]

119885119888= int

119887

119886

int

119887

119886

119870 (119905 119904) 120601119891

119895(119904) 120601119888

119894(119905) 119889119904 119889119905

119865119888= [int

119887

119886

119891 (119905) 120601119888

1(119905) 119889119905 int

119887

119886

119891 (119905) 120601119888


119887

119886

119891 (119905) 120601119888

119889119888

(119905) 119889119905]

119879

(37)


120585119888=1

120582(119872119888)minus1

sdot (119865119888+ 119885119888sdot 119862) (38)





119885opt=

[[[[[[[[

[

119885opt11


119885opt12


119885opt1119889opt


]]]]]]]]

]

119885opt119894119895= int

119887

119886

int

119887

119886

119870 (119905 119904) 120601119891

119895(119904) 120601

opt119894(119905) 119889119904 119889119905

119865opt= [int

119887

119886

119891 (119905) 120601opt1(119905) 119889119905 int

119887

119886


119887

119886

119891 (119905) 120601opt119889opt(119905) 119889119905]

119879

(39)


120585opt=1

120582(119872

opt)minus1

sdot (119865opt+ 119885

optsdot 119862) (40)






119899 1198661198712 119866max 119878

1198712 119878max

I 3807778247544568119890 + 00 3962758499440351119890 + 00 6142245995845852119890 minus 02 2691129516577462119890 minus 02






119873119901 1198661198712 119866max 119878

1198712 119878max






1

1205822(((119865

opt)119879

+ 119862119879sdot (119885

opt)119879


)

119879

sdot (119865opt+ 119885


119888)119879

+ 119862119879sdot (119885119888)119879

)

sdot ((119872119888)minus1

)119879

sdot (119865119888+ 119885119888sdot 119862))

(41)


5 Some Applications







119909 (119904) +1

120587int

120587

minus120587

03

1 minus 064cos2 ((119904 + 119905) 2)119909 (119905) 119889119905


(42)








100

100

10minus1

10minus2

10minus3

10minus4

100

10minus1

10minus2

10minus3

10minus410minus5

101 102

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof)100 101 102

log (dof)


100

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

101 102 100 101 102

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof) log (dof)






119899 1198661198712 119866max 119878

1198712 119878max





[minus120587 120587] p = 3

[minus120587 0] p = 3 [0 120587] p = 4




[minus1205874 0] p = 5 [0 1205872] p = 6 [1205872 120587] p = 7

I Refinement

II Refinement

III Refinement

IV Refinement

V Refinement






15708119890 + 00 31416119890 + 00 ]

119863 = [5 4 6 4 5 6 7]

(43)



0 15708119890 + 00 23562119890 + 00 31416119890 + 00 ]

119863 = [5 5 4 5 5 4]

(44)


minus119909 (119904)

2minus int

1

0

01

001 + (119904 minus 119905)2119909 (119905) 119889119905 = 119891 (119904) 0 le 119904 le 1

(45)







119909 (119904) minus int

1

0

119896 (119905 119904) 119909 (119905) 119889119905 = (1 minus1

1205872) sin (120587119904) 0 le 119904 le 1

(46)

with

119896 (119905 119904) = 119904 (1 minus 119905) 119904 le 119905

119905 (1 minus 119904) 119905 le 119904(47)



log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof)

10minus4

10minus5

10minus6

10minus7

100 101 102

10minus4

10minus5

10minus6

10minus7

log (dof)100 101 102



119873119901 1198661198712 119866max 119878

1198712 119878max





119899 1198661198712 119866max 119878

1198712 119878max









119873119901 1198661198712 119866max 119878

1198712 119878max







6 Conclusions



10minus2 10minus3

10minus4

10minus5

10minus3

10minus4

10minus5 10minus6

100 102 104 100 102 104

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof) log (dof)






Acknowledgment


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











opt1



ℎ119901opt



119909 =1

120582(119891 + 119911) (33)

where 119911 = K119909 = int119887

119886119870(119905 119904)119909(119904)119889119904 119905 isin [119886 119887] Atkinson [44]



119909119899=1

120582(119891 +K119909

119899) (34)




119909 (119905) =1

120582[int

119887

119886

119870 (119905 119904) 119909 (119904) 119889119904 + 119891 (119905)] 119905 isin [119886 119887] (35)


119909 (119905) =1

120582

[

[

119891 (119905) +

119889

sum

119895=1

119888119895int

119887

119886

119870 (119905 119904) 120601119895(119904) 119889119904]

]

119905 isin [119886 119887]

(36)




119885119888=

[[[[[

[

119885119888


119888

1198891198911

119885119888


119888

1198891198912

119885119888

1119889119888


119889119891119889119888

]]]]]

]

119885119888= int

119887

119886

int

119887

119886

119870 (119905 119904) 120601119891

119895(119904) 120601119888

119894(119905) 119889119904 119889119905

119865119888= [int

119887

119886

119891 (119905) 120601119888

1(119905) 119889119905 int

119887

119886

119891 (119905) 120601119888


119887

119886

119891 (119905) 120601119888

119889119888

(119905) 119889119905]

119879

(37)


120585119888=1

120582(119872119888)minus1

sdot (119865119888+ 119885119888sdot 119862) (38)





119885opt=

[[[[[[[[

[

119885opt11


119885opt12


119885opt1119889opt


]]]]]]]]

]

119885opt119894119895= int

119887

119886

int

119887

119886

119870 (119905 119904) 120601119891

119895(119904) 120601

opt119894(119905) 119889119904 119889119905

119865opt= [int

119887

119886

119891 (119905) 120601opt1(119905) 119889119905 int

119887

119886


119887

119886

119891 (119905) 120601opt119889opt(119905) 119889119905]

119879

(39)


120585opt=1

120582(119872

opt)minus1

sdot (119865opt+ 119885

optsdot 119862) (40)






119899 1198661198712 119866max 119878

1198712 119878max

I 3807778247544568119890 + 00 3962758499440351119890 + 00 6142245995845852119890 minus 02 2691129516577462119890 minus 02






119873119901 1198661198712 119866max 119878

1198712 119878max






1

1205822(((119865

opt)119879

+ 119862119879sdot (119885

opt)119879


)

119879

sdot (119865opt+ 119885


119888)119879

+ 119862119879sdot (119885119888)119879

)

sdot ((119872119888)minus1

)119879

sdot (119865119888+ 119885119888sdot 119862))

(41)


5 Some Applications







119909 (119904) +1

120587int

120587

minus120587

03

1 minus 064cos2 ((119904 + 119905) 2)119909 (119905) 119889119905


(42)








100

100

10minus1

10minus2

10minus3

10minus4

100

10minus1

10minus2

10minus3

10minus410minus5

101 102

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof)100 101 102

log (dof)


100

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

101 102 100 101 102

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof) log (dof)






119899 1198661198712 119866max 119878

1198712 119878max





[minus120587 120587] p = 3

[minus120587 0] p = 3 [0 120587] p = 4




[minus1205874 0] p = 5 [0 1205872] p = 6 [1205872 120587] p = 7

I Refinement

II Refinement

III Refinement

IV Refinement

V Refinement






15708119890 + 00 31416119890 + 00 ]

119863 = [5 4 6 4 5 6 7]

(43)



0 15708119890 + 00 23562119890 + 00 31416119890 + 00 ]

119863 = [5 5 4 5 5 4]

(44)


minus119909 (119904)

2minus int

1

0

01

001 + (119904 minus 119905)2119909 (119905) 119889119905 = 119891 (119904) 0 le 119904 le 1

(45)







119909 (119904) minus int

1

0

119896 (119905 119904) 119909 (119905) 119889119905 = (1 minus1

1205872) sin (120587119904) 0 le 119904 le 1

(46)

with

119896 (119905 119904) = 119904 (1 minus 119905) 119904 le 119905

119905 (1 minus 119904) 119905 le 119904(47)



log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof)

10minus4

10minus5

10minus6

10minus7

100 101 102

10minus4

10minus5

10minus6

10minus7

log (dof)100 101 102



119873119901 1198661198712 119866max 119878

1198712 119878max





119899 1198661198712 119866max 119878

1198712 119878max









119873119901 1198661198712 119866max 119878

1198712 119878max







6 Conclusions



10minus2 10minus3

10minus4

10minus5

10minus3

10minus4

10minus5 10minus6

100 102 104 100 102 104

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof) log (dof)






Acknowledgment


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119899 1198661198712 119866max 119878

1198712 119878max

I 3807778247544568119890 + 00 3962758499440351119890 + 00 6142245995845852119890 minus 02 2691129516577462119890 minus 02






119873119901 1198661198712 119866max 119878

1198712 119878max






1

1205822(((119865

opt)119879

+ 119862119879sdot (119885

opt)119879


)

119879

sdot (119865opt+ 119885


119888)119879

+ 119862119879sdot (119885119888)119879

)

sdot ((119872119888)minus1

)119879

sdot (119865119888+ 119885119888sdot 119862))

(41)


5 Some Applications







119909 (119904) +1

120587int

120587

minus120587

03

1 minus 064cos2 ((119904 + 119905) 2)119909 (119905) 119889119905


(42)








100

100

10minus1

10minus2

10minus3

10minus4

100

10minus1

10minus2

10minus3

10minus410minus5

101 102

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof)100 101 102

log (dof)


100

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

101 102 100 101 102

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof) log (dof)






119899 1198661198712 119866max 119878

1198712 119878max





[minus120587 120587] p = 3

[minus120587 0] p = 3 [0 120587] p = 4




[minus1205874 0] p = 5 [0 1205872] p = 6 [1205872 120587] p = 7

I Refinement

II Refinement

III Refinement

IV Refinement

V Refinement






15708119890 + 00 31416119890 + 00 ]

119863 = [5 4 6 4 5 6 7]

(43)



0 15708119890 + 00 23562119890 + 00 31416119890 + 00 ]

119863 = [5 5 4 5 5 4]

(44)


minus119909 (119904)

2minus int

1

0

01

001 + (119904 minus 119905)2119909 (119905) 119889119905 = 119891 (119904) 0 le 119904 le 1

(45)







119909 (119904) minus int

1

0

119896 (119905 119904) 119909 (119905) 119889119905 = (1 minus1

1205872) sin (120587119904) 0 le 119904 le 1

(46)

with

119896 (119905 119904) = 119904 (1 minus 119905) 119904 le 119905

119905 (1 minus 119904) 119905 le 119904(47)



log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof)

10minus4

10minus5

10minus6

10minus7

100 101 102

10minus4

10minus5

10minus6

10minus7

log (dof)100 101 102



119873119901 1198661198712 119866max 119878

1198712 119878max





119899 1198661198712 119866max 119878

1198712 119878max









119873119901 1198661198712 119866max 119878

1198712 119878max







6 Conclusions



10minus2 10minus3

10minus4

10minus5

10minus3

10minus4

10minus5 10minus6

100 102 104 100 102 104

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof) log (dof)






Acknowledgment


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










100

100

10minus1

10minus2

10minus3

10minus4

100

10minus1

10minus2

10minus3

10minus410minus5

101 102

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof)100 101 102

log (dof)


100

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

101 102 100 101 102

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof) log (dof)






119899 1198661198712 119866max 119878

1198712 119878max





[minus120587 120587] p = 3

[minus120587 0] p = 3 [0 120587] p = 4




[minus1205874 0] p = 5 [0 1205872] p = 6 [1205872 120587] p = 7

I Refinement

II Refinement

III Refinement

IV Refinement

V Refinement






15708119890 + 00 31416119890 + 00 ]

119863 = [5 4 6 4 5 6 7]

(43)



0 15708119890 + 00 23562119890 + 00 31416119890 + 00 ]

119863 = [5 5 4 5 5 4]

(44)


minus119909 (119904)

2minus int

1

0

01

001 + (119904 minus 119905)2119909 (119905) 119889119905 = 119891 (119904) 0 le 119904 le 1

(45)







119909 (119904) minus int

1

0

119896 (119905 119904) 119909 (119905) 119889119905 = (1 minus1

1205872) sin (120587119904) 0 le 119904 le 1

(46)

with

119896 (119905 119904) = 119904 (1 minus 119905) 119904 le 119905

119905 (1 minus 119904) 119905 le 119904(47)



log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof)

10minus4

10minus5

10minus6

10minus7

100 101 102

10minus4

10minus5

10minus6

10minus7

log (dof)100 101 102



119873119901 1198661198712 119866max 119878

1198712 119878max





119899 1198661198712 119866max 119878

1198712 119878max









119873119901 1198661198712 119866max 119878

1198712 119878max







6 Conclusions



10minus2 10minus3

10minus4

10minus5

10minus3

10minus4

10minus5 10minus6

100 102 104 100 102 104

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof) log (dof)






Acknowledgment


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119899 1198661198712 119866max 119878

1198712 119878max





[minus120587 120587] p = 3

[minus120587 0] p = 3 [0 120587] p = 4




[minus1205874 0] p = 5 [0 1205872] p = 6 [1205872 120587] p = 7

I Refinement

II Refinement

III Refinement

IV Refinement

V Refinement






15708119890 + 00 31416119890 + 00 ]

119863 = [5 4 6 4 5 6 7]

(43)



0 15708119890 + 00 23562119890 + 00 31416119890 + 00 ]

119863 = [5 5 4 5 5 4]

(44)


minus119909 (119904)

2minus int

1

0

01

001 + (119904 minus 119905)2119909 (119905) 119889119905 = 119891 (119904) 0 le 119904 le 1

(45)







119909 (119904) minus int

1

0

119896 (119905 119904) 119909 (119905) 119889119905 = (1 minus1

1205872) sin (120587119904) 0 le 119904 le 1

(46)

with

119896 (119905 119904) = 119904 (1 minus 119905) 119904 le 119905

119905 (1 minus 119904) 119905 le 119904(47)



log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof)

10minus4

10minus5

10minus6

10minus7

100 101 102

10minus4

10minus5

10minus6

10minus7

log (dof)100 101 102



119873119901 1198661198712 119866max 119878

1198712 119878max





119899 1198661198712 119866max 119878

1198712 119878max









119873119901 1198661198712 119866max 119878

1198712 119878max







6 Conclusions



10minus2 10minus3

10minus4

10minus5

10minus3

10minus4

10minus5 10minus6

100 102 104 100 102 104

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof) log (dof)






Acknowledgment


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof)

10minus4

10minus5

10minus6

10minus7

100 101 102

10minus4

10minus5

10minus6

10minus7

log (dof)100 101 102



119873119901 1198661198712 119866max 119878

1198712 119878max





119899 1198661198712 119866max 119878

1198712 119878max









119873119901 1198661198712 119866max 119878

1198712 119878max







6 Conclusions



10minus2 10minus3

10minus4

10minus5

10minus3

10minus4

10minus5 10minus6

100 102 104 100 102 104

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof) log (dof)






Acknowledgment


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10minus2 10minus3

10minus4

10minus5

10minus3

10minus4

10minus5 10minus6

100 102 104 100 102 104

log

(rel

ativ

e err

or in

L2-n

orm

)

log

(rel

ativ

e err

or in

max

imum

nor

m)

log (dof) log (dof)






Acknowledgment


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article An Adaptive Approach to Solutions...

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