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AHLAM MUFTAH ABDUSSALAM INVESTIGATION OF THE FINITE-DIFFERENCE METHOD FOR APPROXIMATING THE SOLUTION AND ITS DERIVATIVES OF THE DIRICHLET PROBLEM FOR 2D AND 3D LAPLACE’S EQUATION NEU 2018 INVESTIGATION OF THE FINITE-DIFFERENCE METHOD FOR APPROXIMATING THE SOLUTION AND ITS DERIVATIVES OF THE DIRICHLET PROBLEM FOR 2D AND 3D LAPLACE’S EQUATION A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF APPLIED SCIENCES OF NEAR EAST UNIVERSITY By AHLAM MUFTAH ABDUSSALAM In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy of Science in Mathematics NICOSIA, 2018

INVESTIGATION OF THE FINITE-DIFFERENCE METHOD ...docs.neu.edu.tr/library/6718922727.pdfits derivatives of PDEs in a rectangle or in a rectangular parallelepiped are important not only

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    2018

    INVESTIGATION OF THE FINITE-DIFFERENCE

    METHOD FOR APPROXIMATING THE SOLUTION

    AND ITS DERIVATIVES OF THE DIRICHLET

    PROBLEM FOR 2D AND 3D LAPLACE’S

    EQUATION

    A THESIS SUBMITTED TO THE GRADUATE

    SCHOOL OF APPLIED SCIENCES

    OF

    NEAR EAST UNIVERSITY

    By

    AHLAM MUFTAH ABDUSSALAM

    In Partial Fulfillment of the Requirements for

    the Degree of Doctor of

    Philosophy of Science

    in

    Mathematics

    NICOSIA, 2018

  • INVESTIGATION OF THE FINITE-DIFFERENCE

    METHOD FOR APPROXIMATING THE SOLUTION

    AND ITS DERIVATIVES OF THE DIRICHLET

    PROBLEM FOR 2D AND 3D LAPLACE’S

    EQUATION

    A THESIS SUBMITTED TO THE GRADUATE

    SCHOOL OF APPLIED SCIENCES

    OF

    NEAR EAST UNIVERSITY

    By

    AHLAM MUFTAH ABDUSSALAM

    In Partial Fulfillment of the Requirements for

    the Degree of Doctor of

    Philosophy of Science

    in

    Mathematics

    NICOSIA, 2018

  • Ahlam Muftah ABDUSSALAM: INVESTIGATION OF THE FINITE-

    DIFFERENCE METHOD FOR APPROXIMATING THE SOLUTION AND ITS

    DERIVATIVES OF THE DIRICHLET PROBLEM FOR 2D AND 3D LAPLACE’S

    EQUATION

    Approval of Director of Graduate School of

    Applied Sciences

    Prof. Dr. Nadire Γ‡AVUŞ

    We certify this thesis is satisfactory for the award of the degree of Doctor of

    Philosophy of Science in Mathematics

    Examining Committee in Charge:

    Prof.Dr. Agamirza Bashirov Department of Mathematics, EMU

    Prof.Dr. AdΔ±gΓΌzel Dosiyev Supervisor, Department of Mathematics,

    NEU

    Prof.Dr. Evren Hınçal Head of the Department of Mathematics,

    NEU

    Assoc.Prof.Dr. Suzan Cival Buranay Department of Mathematics, EMU

    Assoc.Prof.Dr. Murat Tezer Department of Mathematics, NEU

  • I hereby declare that all information in this document has been obtained and presented in

    accordance with academic rules and ethical conduct. I also declare that, as required by these

    rules and conduct, I have fully cited and referenced all material and results that are not

    original to this work.

    Name, Last name: AHLAM MUFTAH ABDUSSALAM

    Signature:

    Date:

  • iii

    ACKNOWLEDGEMENTS

    At the end of this thesis, I would like to thank all the people without whom this thesis

    would never have been possible. Although it is just my name on the cover, many people

    have contributed to the research in their own particular way and for that I want to give

    them special thanks.

    Foremost, I would like to express the deepest appreciation to my supervisor Prof. Dr.

    AdigΓΌzel Dosiyev. He supported me through time of study and research with his patient

    and immense knowledge. This thesis would have never been accomplished without his

    guidance and persistent help.

    I owe many thanks to my colleagues who helped me during whole academic journey and

    all staff in mathematics department.

    To my wonderful children, thank you for bearing with me and my mood swings and

    being my greatest supporters. To my husband and my mother thank you for not letting

    me give up and giving me all the encouragement, I needed to continue.

    Last but not least this dissertation is dedicated to my late father who has been my

    constant source of inspiration.

  • iv

    To my parents and family…

  • v

    ABSTRACT

    The Dirichlet problem for the Laplace equation is carefully weighed in a rectangle and a

    rectangular parallelepiped. In the case of a rectangle domain, the boundary functions of the

    Dirichlet problem are supposed to have seventh derivatives satisfying the HΓΆlder condition on

    the sides of the rectangle Ξ  . Moreover, it is assumed that on the vertices the continuity

    conditions as well as compatibility conditions, which result from Laplace equation, for even

    order derivatives up to sixth order are satisfied. Under these conditions the error 𝑒 βˆ’ π‘’β„Ž of the

    9-point solution π‘’β„Ž at each grid point (π‘₯, 𝑦) a pointwise estimation 𝑂(πœŒβ„Ž6) is obtained, where

    𝜌 = 𝜌(π‘₯, 𝑦) is the distance from the current grid point to the boundary of rectangle Ξ , 𝑒 is the

    exact solution and β„Ž is the grid step. The solution of difference problems constructed for the

    approximate values of the first derivatives converge with orders 𝑂(β„Ž6) and for pure second

    derivatives converge with orders 𝑂(β„Ž5+πœ†), 0 < πœ† < 1. In a rectangular parallelepiped domain,

    the seventh derivatives for the boundary functions of the Dirichlet problem on the faces of the

    parallelepiped 𝑅 are supposed to satisfy the HΓΆlder condition. While, their even order

    derivatives up to sixth satisfy the compatibility conditions on the edges. For the error 𝑒 βˆ’ π‘’β„Ž of

    the 27-point solution π‘’β„Ž at each grid point (π‘₯1, π‘₯2, π‘₯3) , a pointwise estimation 𝑂(πœŒβ„Ž6) is

    obtained, where 𝜌 = 𝜌(π‘₯1, π‘₯2, π‘₯3) is the distance from the current grid point to the boundary of

    the parallelepiped 𝑅. The solution of the constructed 27- point difference problems for the

    approximate values of the first converge with orders 𝑂(β„Ž6 ln β„Ž) and for pure second derivatives

    converge with orders 𝑂(β„Ž5+πœ†). In the constructed three-stage difference method for solving

    Dirichlet problem for Laplace's equation on a rectangular parallelepiped under some smoothness

    conditions for the boundary functions the difference solution obtained by 15+7+7- scheme

    converges uniformly as 𝑂(β„Ž6), as the 27-point scheme.

    Keywords: Approximation of the derivatives; pointwise error estimations; finite difference

    method; uniform error estimations; 2D and 3D Laplace's equation; numerical solution of the

    Laplace equation

  • vi

    Γ–ZET

    Bir dikdârtgen içindeki Laplace denklemi ve dikdârtgenler prizması için Dirichlet problem

    düşünülmüştür. Tanım bâlgesinin dikdârtgen olduğu durumda dikdârtgenin Π kenarlarında

    verilen sınır fonksiyonlarının yedinci türevlerinin Hâlder şartını sağladığı Kabul edildi.

    Kâşelerde süreklilik şartının yanında Laplace denkleminden sonuçlanan kâşelerin komşu

    kenarlarında verilen sınır değer fonksiyonlarının ikinci, dârdüncüve altıncı türevleri için

    uyumluluk şartları da sağlandı. Bu şartlar altında Dirichlet probleminin kare ızgara üzerinde

    çâzΓΌmΓΌ iΓ§in 𝑒 βˆ’ π‘’β„Ž yaklaşımΔ± 𝑂(πœŒβ„Ž6) olarak dΓΌzgΓΌn yakΔ±nsadığı bulunmuştur, burada π‘’β„Ž 9-

    nokta yaklaşımΔ± kullanΔ±ldığında elde elinen yaklaşık çâzΓΌm, π‘π‘Ÿπ‘œπ‘π‘™π‘’π‘š sağlayan kesin çâzΓΌm,

    𝜌 = 𝜌(π‘₯, 𝑦) , dikdΓΆrtgen sΔ±nΔ±rΔ±nΔ± işaret eden mevcut Δ±zgara uzunlığu ve h, Δ±zgara adΔ±mΔ±dΔ±r.

    ÇâzΓΌmΓΌn birinci ve pΓΌr tΓΌrevleri iΓ§in oluşturulan fark problemlerinin sΔ±rasΔ±yla çâzΓΌmleri, 𝑂(β„Ž6)

    ve 𝑂(β„Ž5+πœ†) , 0 < πœ† < 1 mertebesi ile yakΔ±nsar. TanΔ±m bΓΆlgesinin dikdΓΆrtgenler prizmasΔ±

    olduğu durumda prizmanın yüzeylerinde verilensınır fonksiyonlarının yedinci türevlerinin

    Hâlder şartını sağladığı Kabul edildi. Kâşelerde süreklilik şartının yanında Laplace

    denkleminden sonuçlanan kenarlarının komşu kenarlarında verilen sınır değer fonksiyonlarının

    ikinci, dârdüncü ve altıncı türevleri için uyumluluk şartlarınıda sağlar. Bu şartlar altında

    Dirichlet probleminin kΓΌp Δ±zgaralar ΓΌzerindeki çâzΓΌmΓΌ iΓ§in 𝑒 βˆ’ π‘’β„Ž yaklaşımΔ± 𝑂(πœŒβ„Ž6) olarak

    bulunmuştur, burada π‘’β„Ž 27-nokta yaklaşımΔ± kullanΔ±ldığında elde edilen yaklaşık çâzΓΌm,

    problem sağlayan kesin çâzΓΌm ve 𝜌 = 𝜌(π‘₯1, π‘₯2, π‘₯3) prizmanΔ±n sΔ±nΔ±rΔ±nΔ± işaret edenmevcut Δ±zgara

    ve h, Δ±zgara adΔ±mΔ±dΔ±r. ÇâzΓΌmΓΌn birinci ve pΓΌr tΓΌrevleri iΓ§in oluşturulan fark problemlerinin

    sΔ±rasΔ±yla çâzΓΌmleri 𝑂(β„Ž6 ln β„Ž) , ve 𝑂(β„Ž6) , 0 < πœ† < 1 mertebeleri ile yakΔ±nsar. Laplace

    denkleminin sınır fonksiyonları için bazı pürüzsüzlük koşulları altında Dirichlet problemi

    çâzmek için inşa edilmiş üç aşamalı fark yântemi, 15 + 7 + 7 - şeması ile elde edilen fark çâzümü

    𝑂(β„Ž6) olarak yakΔ±nsar 27 noktalΔ± şema yΓΆntemiyle elde edilen sonuΓ§ gibi.

    Anahtar Kelimeler: Sonlu fark metodu; noktasal hata tahminleri; türev yaklaşımları; düzgün

    hata tahminleri; 2D ve 3D Laplace denklemleri; Laplace denkleminin sayısal çâzümü

  • vii

    TABLE OF CONTENTS

    ACKNOWLEDGMENTS………………………………….…………………………..… iii

    ABSTRACT………………………………………………………………………............... v

    Γ–ZET…………………………………………..……………………………………......... vi

    LIST OF TABLES………………………………………………………........................... xi

    LIST OF FIGURES……………..……………………………………………………..… xii

    CHAPTER 1: INTRODUCTION ……………………………………...………………… 1

    CHAPTER 2: ON THE HIGH ORDER CONVERGENCE OF THE DIFFERENCE

    SOLUTION OF LAPLACE’S EQUATION IN A RECTANGLE

    2.1 The Dirichlet Problem for Laplace’s Equation on Rectangle and Some Differential

    Properties of its Solution………………………..…………………….………………... 6

    2.2 Difference Equations for the Dirichlet Problem and a Pointwise Estimation………..… 8

    2.3 Approximation of the First Derivatives …………….……………………………..……. 22

    2.4 Approximation of the Pure Second Derivatives ……………………………………...… 27

    CHAPTER 3: ON THE HIGH ORDER CONVERGENCE OF THE DIFFERENCE

    SOLUTION OF LAPLACE’S EQUATION IN A RECTANGULAR PARALLELIPIPED

    3.1 The Dirichlet Problem in a Rectangular Parallelepiped ……………………................… 34

  • viii

    3.2 27-Point Finite Difference Method for the Dirichlet Problem ……………...…………... 40

    3.3 Approximation of the First Derivatives…………...…………………………………..… 53

    3.4 Approximation of the Pure Second Derivatives...…………………………………….… 58

    CHAPTER 4: THE THREE STAGE DIFFERENCE METHOD FOR SOLVING

    DIRICHLET PROBLEM FOR LAPLACE’S EQUATION

    4.1 The Dirichlet Problem on Rectangular Parallelepiped…………………………….......... 64

    4.2 A Sixth Order Accurate Approximate Solution……………………………………….... 73

    4.3 The First Stage………………………………………………………………………..… 74

    4.4 The Second Stage……………………………………………..………………...…….… 75

    4.5 The Third Stage……………………………………………………………………….… 76

    CHAPTER 5: NUMERICAL EXAMPLES

    5.1 Domain in the Shape of a Rectangle………………………….………………………… 80

    5.2 Domain in the Shape of a Rectangular Parallelepiped………………………………….. 82

    CHAPTER 6: CONCLUSION……..…………………………………………………….. 86

    REFERENCES …………………………………………………………………………… 87

  • ix

    LIST OF TABLES

    Table 5.1: The approximate of solution in problem (5.1) ………….………………......… 81

    Table 5.2: First derivative approximation results with the sixth-order accurate formulae

    for problem (5.1) ………………………………………………………...….... 82

    Table 5.3: The approximate of solution in problem (5.3) …………….………………….. 83

    Table 5.4: First derivative approximation results with the sixth-order accurate formulae

    for problem (5.3) ………………………………………………………...…… 84

    Table 5.5: The approximate results for the pure second derivative ….………………...… 84

    Table 5.6: The approximate results by using a three stage difference method ….……..… 85

  • x

    LIST OF FIGURES

    Figure 2.1: The selected region in Ξ  is 𝜌 = π‘˜β„Ž ………………………………………...... 11

    Figure 2.2: The selected region in Ξ  is 𝜌 > π‘˜β„Ž ………………………………………….. 12

    Figure 2.3: The selected region in Ξ  is 𝜌 < π‘˜β„Ž ………………………………………….. 13

    Figure 2.4: The selected region in Ξ  is 𝜌 < π‘˜β„Ž ………………………………………..… 14

    Figure 3.1: 26 points around center point using operator β„œ ……………………………... 41

    Figure 3.2: The selected region in 𝑅 is 𝜌 = π‘˜β„Ž ………………………………………..… 44

    Figure 3.3: The selected region in 𝑅 is 𝜌 > π‘˜β„Ž ………………………………………….. 45

    Figure 3.4: The selected region in 𝑅 is 𝜌 < π‘˜β„Ž ………………………………………….. 46

  • 1

    CHAPTER 1

    INTRODUCTION

    The partial differential equations are highly used in many topics of applied sciences in order to

    solve equilibrium or steady state problem. Laplace equation is one of the most important elliptic

    equations, which has been used, to model many problems in real life situations.

    Further it can be used in the formulation of problems relevant to theory of electrostatics,

    gravitation and problems arising in the field of interest to mathematical physics. In addition, it

    is applied in engineering, when dealing with many problems such as analysis of steady heat

    condition in solid bodies, the irrotational flow of incompressible fluid, and so on.

    In many applied problems not only the calculation of the solution of the differential equation

    but also the calculation of the derivatives are very important to provide information about some

    physical phenomenas. For example, by the theory of Saint-Venant, the problem of the torsion

    of any prismatic body whose section is the region 𝐷 bounded by the contour 𝐿 reduces to the

    following boundary- value problem: to find, the solution of the Poisson equation

    Δ𝑒 = βˆ’2,

    that reduces to zero on the contour 𝐿:

    𝑒 = 0 on 𝐿.

    Here the basis quantities required from the calculation are expressed in terms of the function 𝑒

    the components of the tangential stress

    πœπ‘§π‘₯ = GΟ‘πœ•π‘’

    πœ•π‘¦, πœπ‘§π‘¦ = βˆ’GΟ‘

    πœ•π‘’

    πœ•π‘₯,

    and the torsional moment

    𝑀 = GΟ‘βˆ¬π‘’π‘‘π‘₯𝑑𝑦.

    𝐷

    Here Ο‘ is the angle of twist per unit length, and 𝐺 is the modulus of shear.

  • 2

    The construction and justification of highly accurate approximate methods for the solution and

    its derivatives of PDEs in a rectangle or in a rectangular parallelepiped are important not only

    for the development of theory of these methods, but also to improve some version of domain

    decomposition methods for more complicated domains, (Smith et al., 2004; Kantorovich and

    Krylov, 1958; Volkov, 1976; Volkov, 1979; Volkov, 2003; Volkov, 2006).

    Since the operation of differentiation is ill-conditioned, to find a highly accurate approximation

    for the derivatives of the solution of a differential equation becomes problematic, especially

    when smoothness is restricted.

    It is obvious that the accuracy of the approximate derivatives depends on the accuracy of the

    approximate solution. As is proved in (Lebedev, 1960), the high order difference derivatives

    uniformly converge to the corresponding derivatives of the solution for the 2D Laplace equation

    in any strictly interior subdomain with the same order β„Ž, (β„Ž is the grid step), with which the

    difference solution converges on the given domain. In (Volkov, 1999), 𝑂(β„Ž2) order difference

    derivatives uniform convergence of the solution of the difference equation, and its first and pure

    second difference derivatives over the whole grid domain to the solution, and corresponding

    derivatives of solution for the 2D Laplace equation was proved. In (Dosiyev and Sadeghi, 2015)

    three difference schemes were constructed to approximate the solution and its first and pure

    second derivatives of 2D Laplace’s equation with order of 𝑂(β„Ž4), when the sixth derivatives of

    the boundary functions on the sides of a rectangle satisfy the HΓΆlder condition, and on the

    vertices their second and fourth derivatives satisfy the compatibility condition that is implied by

    the Laplace equation.

    In (Volkov, 2004), for the 3D Laplace equation in a rectangular parallelepiped the constructed

    difference schemes converge with order of 𝑂(β„Ž2) to the first and pure second derivatives of the

    exact solution of the Dirichlet problem. It is assumed that the fourth derivatives of the boundary

    functions on the faces of a parallelepiped satisfy the HΓΆlder condition, and on the edges their

    second derivatives satisfy the compatibility condition that is implied by the Laplace equation.

    Whereas in (Volkov, 2005), the convergence with order 𝑂(β„Ž2) of the difference derivatives to

    the corresponding first order derivatives was proved, when the third derivatives of the boundary

    functions on the faces satisfy the HΓΆlder condition. Further, in (Dosiyev and Sadeghi, 2016) by

  • 3

    assuming that the boundary functions on the faces have the sixth order derivatives satisfying the

    HΓΆlder condition, and the second and fourth derivatives satisfy the compatibility conditions on

    the edges, for the uniform error of the approximate solution 𝑂(β„Ž6|ln β„Ž|) order, and for the first

    and pure second derivatives 𝑂(β„Ž4) order was obtained.

    We mention one more problem when we use the high order accurate finite- difference schemes

    for the approximation of the solution and its derivatives Since in the finite- difference

    approximations the obtained system of difference equations, in general, are banded matrices. To

    get a highly accurate results in the most of approximations, difference operators with the high

    number of pattern are used which increase the number of bandwidth of the difference equations.

    It is obvious that the complexity of the realization methods for the difference equations increases

    depending on the number of bandwidth of the matrices of these equations. As it was shown in

    (Tarjan, 1976) that in case of Gaussian elimination method the bandwidth elimination for 𝑛 Γ— 𝑛

    matrices with the bandwidth 𝑏 the computational cost is of order 𝑂(𝑏2𝑛) . Therefore, the

    construction of multistage finite difference methods with the use of low number of bandwidth

    matrix in each of stages becomes important.

    In (Volkov, 2009) a new two-stage difference method for solving the Dirichlet problem for

    Laplace's equation on a rectangular parallelepiped was proposed. It was assumed that the given

    boundary values are six times differentiable at the faces of the parallelepiped, those derivatives

    satisfy a HΓΆlder condition, and the boundary values are continuous at the edges and their second

    derivatives satisfy a compatibility condition implied by the Laplace equation. Under these

    conditions it was proved that by using the 7-point scheme on a cubic grid in each stage the order

    of uniform error is improved from 𝑂(β„Ž2) up to 𝑂(β„Ž4 ln β„Žβˆ’1), where β„Ž is the mesh size. It is

    known that, to get 𝑂(β„Ž4) order of accurate results by the existing one-stage methods for the

    approximation 3𝐷 Laplace's equation we have to use at least 15-point scheme, (Volkov, 2010).

    In this thesis, a highly accurate schemes for the solution and its the first and pure second

    derivatives of the Laplace equation on a rectangle and on a rectangular parallelepiped are

    constructed and justified. Two-dimensional case (Chapter 2) consider the classical 9-point, and

    in three-dimensional case (Chapter 3) the 27-point finite- difference approximation of Laplace

    equation are used. In Chapter 4, in the three-stage difference method at the first stage, the

  • 4

    difference equations are formulated using the 14-point averaging operator, and the difference

    equations at the second and third stages are formulated using the simplest six point averaging

    operator.

    The numerical experiments to justify the obtained theoretical results are presented in Chapter 5.

    Now, we formulated all results more explicitly.

    In Chapter 2, we consider the Dirichlet problem for the Laplace equation on a rectangle, when

    the boundary values belong to 𝐢7,πœ†, 0 < πœ† < 1, on the sides of the rectangle, and as whole are

    continuous on the vertices. Also, the 2π‘ž, π‘ž = 1,2,3, order derivatives satisfy the compatibility

    conditions on the vertices which result from the Laplace equation. Under these conditions, we

    present and justify difference schemes on a square grid for obtaining the solution of the Dirichlet

    problem, its first and pure second derivatives. For the approximate solution a pointwise

    estimation for the error of order 𝑂(πœŒβ„Ž6), where 𝜌 = 𝜌(π‘₯, 𝑦) is the distance from the current grid

    point (π‘₯, 𝑦) to the boundary of the rectangle, is obtained. This estimation is used to approximate

    the first derivatives with uniform error of order 𝑂(β„Ž6). The approximation of the pure second

    derivatives are obtained with uniform accuracy 𝑂(β„Ž5+πœ†), 0 < πœ† < 1.

    In Chapter 3, we consider the Dirichlet problem for the Laplace equation on a rectangular

    parallelepiped. The boundary functions on the faces of a parallelepiped are supposed to have the

    seventh order derivatives satisfying the HΓΆlder condition, and on the edges the second, fourth

    and sixth order derivatives satisfy the compatibility conditions. We present and justify

    difference schemes on a cubic grid for obtaining the solution of the Dirichlet problem, its first

    and pure second derivatives. For the approximate solution a pointwise estimation for the error

    of order 𝑂(πœŒβ„Ž6) with the weight function 𝜌, where 𝜌 = 𝜌(π‘₯1, π‘₯2, π‘₯3) is the distance from the

    current grid point (π‘₯1, π‘₯2, π‘₯3) to the boundary of the parallelepiped, is obtained. This estimation

    gives an additional accuracy of the finite difference solution near the boundary of the

    parallelepiped, which is used to approximate the first derivatives with uniform error of order

    𝑂(β„Ž6 ln β„Ž) . The approximation of the pure second derivatives are obtained with uniform

    accuracy 𝑂(β„Ž5+πœ†), 0 < πœ† < 1.

  • 5

    In Chapter 4, A three-stage difference method is proposed for solving the Dirichlet problem for

    the Laplace equation on a rectangular parallelepiped, at the first stage, approximate values of

    the sum of the pure fourth derivatives of the desired solution are sought on a cubic grid. At the

    second stage, approximate values of the sum of the pure sixth derivatives of the desired solution

    are sought on a cubic grid. At the third stage, the system of difference equations approximating

    the Dirichlet problem corrected by introducing the quantities determined at the first and second

    stages. The difference equations at the first stage is formulated using the 14-point averaging

    operator, and the difference equations at the second and third stages are formulated using the

    simplest six-point averaging operator. Under the assumptions that the given boundary functions

    on the faces of a parallelepiped have the eighth derivatives satisfying the HΓΆlder condition, and

    on the edges the second, fourth, and sixth order derivatives satisfy the compatibility conditions,

    it is proved that the difference solution to the Dirichlet problem converges uniformly as 𝑂(β„Ž6).

    In Chapter 5, the numerical experiments to justify the theoretical results obtained in each

    Chapters are demonstrated.

  • 6

    CHAPTER 2

    ON THE HIGH ORDER CONVERGENCE OF THE DIFFERENCE SOLUTION OF

    LAPLACE’S EQUATION IN A RECTANGLE

    In this Chapter we consider the Dirichlet problem for the Laplace equation on a rectangle, when

    the boundary values on the sides of the rectangle are supposed to have the seventh derivatives

    satisfying the HΓΆlder condition. On the vertices besides the continuity condition, the

    compatibility conditions, which result from the Laplace equation for the second, fourth and sixth

    derivatives of the boundary values, given on the adjacent sides are also satisfied. Under these

    conditions, we present and justify difference schemes on a square grid for obtaining the solution

    of the Dirichlet problem, its first and pure second derivatives. For the approximate solution a

    pointwise estimation for the error of order 𝑂(πœŒβ„Ž6) with the weight function 𝜌 , where 𝜌 =

    𝜌(π‘₯, 𝑦) is the distance from the current grid point (π‘₯, 𝑦) to the boundary of the rectangle is

    obtained. This estimation gives an additional accuracy of the finite difference solution near the

    boundary of the rectangle, which is used to approximate the first derivatives with uniform error

    of order 𝑂(β„Ž6). The approximation of the pure second derivatives are obtained with uniform

    accuracy 𝑂(β„Ž5+πœ†), 0 < πœ† < 1.

    2.1 The Dirichlet Problem for Laplace’s Equation on Rectangle and Some Differential

    Properties of Its Solution

    Let Ξ  = {(π‘₯, 𝑦): 0 < π‘₯ < π‘Ž , 0 < 𝑦 < 𝑏} be an open rectangle and π‘Ž 𝑏⁄ is a rational number.

    The sides are denoted by 𝛾𝑗 , 𝑗 = 1,2,3,4, including the ends. These sides are enumerated

    counterclockwise where 𝛾1 is the left side of Ξ , (𝛾0 ≑ 𝛾4 , 𝛾5 ≑ 𝛾1). Also let the boundary of Ξ 

    be defined by 𝛾 = ⋃ 𝛾𝑗4𝑗=1 .

    The arclength along 𝛾 is denoted by 𝑠, and 𝑠𝑗 is the value of 𝑠 at the beginning of 𝛾𝑗. We denote

    by 𝑓 ∈ πΆπ‘˜,πœ†(𝐷) if 𝑓 has π‘˜ βˆ’ π‘‘β„Ž deravatives on 𝐷 satisfying HΓΆlder condition, where exponent

    πœ† ∈ (0,1).

  • 7

    We consider the following boundary value problem

    βˆ†π‘’ = 0 on Ξ , 𝑒 = πœ‘π‘—(𝑠) on 𝛾𝑗, 𝑗 = 1,2,3,4 (2.1)

    where βˆ†β‰‘ πœ•2 πœ•π‘₯2 + πœ•2 πœ•π‘¦2⁄⁄ , πœ‘π‘— are given functions of 𝑠.

    Assume that

    πœ‘π‘— ∈ 𝐢7,πœ†(𝛾𝑗), 0 < πœ† < 1, 𝑗 = 1,2,3,4, (2.2)

    πœ‘π‘—(2π‘ž)(𝑠𝑗) = (βˆ’1)

    π‘žπœ‘π‘—βˆ’1(2π‘ž)(𝑠𝑗), π‘ž = 0,1,2,3. (2.3)

    Lemma 2.1 The solution 𝑒 of problem (2.1) is from 𝐢7,πœ†(𝛱).

    The proof of Lemma 2.1 follows from Theorem 3.1 by (Volkov, 1969).

    Lemma 2.2 Let 𝜌(π‘₯, 𝑦) be the distance from the current point of open rectangle 𝛱 to its

    boundary and let πœ• πœ•π‘™β„ ≑ 𝛼 πœ• πœ•π‘₯⁄ + 𝛽 πœ• πœ•π‘¦β„ , 𝛼2 + 𝛽2 = 1. Then the next inequality holds

    |πœ•8𝑒(π‘₯, 𝑦)

    πœ•π‘™8| ≀ π‘πœŒπœ†βˆ’1(π‘₯, 𝑦), (π‘₯, 𝑦) ∈ 𝛱, (2.4)

    where 𝑐 is a constant independent of the direction of differentiation πœ• πœ•π‘™β„ , and 𝑒 is a solution of

    problem (2.1).

    Proof. We choose an arbitrary point (π‘₯0, 𝑦0) ∈ Ξ . Let 𝜌0 = 𝜌(π‘₯0, 𝑦0), and 𝜎0 βŠ‚ Ξ Μ… be the closed

    circle of radius 𝜌0 centered at (π‘₯0, 𝑦0). Consider the harmonic function on Ξ 

    𝑣(π‘₯, 𝑦) =πœ•7𝑒(π‘₯, 𝑦)

    πœ•π‘™7βˆ’

    πœ•7𝑒(π‘₯0, 𝑦0)

    πœ•π‘™7. (2.5)

  • 8

    By Lemma 2.1, 𝑒 ∈ 𝐢7,πœ†(Ξ Μ…), for 0 < πœ† < 1. Then for the function (2.5) we have

    max(π‘₯,𝑦)βˆˆοΏ½Μ…οΏ½0

    |𝑣(π‘₯, 𝑦)| ≀ 𝑐0𝜌0πœ†, (2.6)

    where 𝑐0 is a constant independent of the point (π‘₯0, 𝑦0) ∈ Ξ  or the direction of πœ• πœ•π‘™β„ . Since 𝑒 is

    harmonic in Ξ , by using estimation (2.6) and applying Lemma 3 from (Mikhailov, 1978) we

    have

    |πœ•

    πœ•π‘™(πœ•7𝑒(π‘₯, 𝑦)

    πœ•π‘™7βˆ’

    πœ•7𝑒(π‘₯0, 𝑦0)

    πœ•π‘™7)| ≀ 𝑐1

    𝜌0πœ†

    𝜌0.

    or

    |πœ•8𝑒(π‘₯, 𝑦)

    πœ•π‘™8| ≀ 𝑐1𝜌0

    πœ†βˆ’1(π‘₯0, 𝑦0),

    where 𝑐1is a constant independent of the point (π‘₯0, 𝑦0) ∈ Ξ  or the direction of πœ• πœ•π‘™β„ . Since the

    point (π‘₯0, 𝑦0) ∈ Ξ  is arbitrary, inequality (2.4) holds true. ∎

    2.2 Difference Equations for the Dirichlet Problem and a Pointwise Estimation

    Let β„Ž > 0, and π‘šπ‘–π‘›{π‘Ž β„Žβ„ , 𝑏 β„Žβ„ } β‰₯ 6 where π‘Ž β„Žβ„ and 𝑏 β„Žβ„ are integers. A square net on Ξ  is

    assigned by Ξ β„Ž, with step β„Ž, obtained by the lines π‘₯, 𝑦 = 0, β„Ž, 2β„Ž,… .

    The set of nodes on 𝛾𝑗is denoted by π›Ύπ‘—β„Ž, and let

    π›Ύβ„Ž = ⋃ π›Ύπ‘—β„Ž4

    𝑗=1 , Ξ Μ…β„Ž = Ξ β„Ž βˆͺ π›Ύβ„Ž.

    Let the averaging operator 𝐡 be defined as following

  • 9

    𝐡𝑒(π‘₯, 𝑦) =1

    20[4(𝑒(π‘₯ + β„Ž, 𝑦) + 𝑒(π‘₯ βˆ’ β„Ž, 𝑦) + 𝑒(π‘₯, 𝑦 + β„Ž)

    +𝑒(π‘₯, 𝑦 βˆ’ β„Ž)) + 𝑒(π‘₯ + β„Ž, 𝑦 + β„Ž) + 𝑒(π‘₯ + β„Ž, 𝑦 βˆ’ β„Ž)

    +𝑒(π‘₯ βˆ’ β„Ž, 𝑦 + β„Ž) + 𝑒(π‘₯ βˆ’ β„Ž, 𝑦 βˆ’ β„Ž)]. (2.7)

    Let 𝑐, 𝑐0, 𝑐1, … be constants which are independent of β„Ž and the nearest factor, and for simplicity

    identical notation will be used for various constants.

    Consider the finite difference approximation of problem (2.1) as follows:

    π‘’β„Ž = π΅π‘’β„Ž on Ξ β„Ž, π‘’β„Ž = πœ‘π‘— on 𝛾𝑗

    β„Ž, 𝑗 = 1,2,3,4. (2.8)

    By the maximum principle system (2.8) has a unique solution (Samarskii, 2001).

    Let Ξ π‘˜β„Ž be the set of nodes of grid Ξ β„Ž whose distance from 𝛾 is π‘˜β„Ž. It is obvious that 1 ≀ π‘˜ ≀

    𝑁(β„Ž), where

    𝑁(β„Ž) = [1

    2β„Žπ‘šπ‘–π‘›{π‘Ž, 𝑏}],

    (2.9)

    [𝑑] is the integer part of 𝑑.

    We define for 1 ≀ π‘˜ ≀ 𝑁(β„Ž) the function

    π‘“β„Žπ‘˜ = {

    1, 𝜌(π‘₯, 𝑦) = π‘˜β„Ž,

    0, 𝜌(π‘₯, 𝑦) β‰  π‘˜β„Ž.

    (2.10)

    Consider the following systems

    π‘žβ„Ž = π΅π‘žβ„Ž + π‘”β„Ž on Ξ β„Ž, π‘žβ„Ž = 0 on 𝛾

    β„Ž, (2.11)

    οΏ½Μ…οΏ½β„Ž = π΅οΏ½Μ…οΏ½β„Ž + οΏ½Μ…οΏ½β„Ž on Ξ β„Ž, οΏ½Μ…οΏ½β„Ž = 0 on 𝛾

    β„Ž, (2.12)

    where π‘”β„Ž and οΏ½Μ…οΏ½β„Ž are given function, and |π‘”β„Ž| ≀ οΏ½Μ…οΏ½β„Ž on Ξ β„Ž.

  • 10

    Lemma 2.3 The solution π‘žβ„Ž and οΏ½Μ…οΏ½β„Ž of systems (2.11) and (2.12) satisfy the inequality

    |π‘žβ„Ž| ≀ οΏ½Μ…οΏ½β„Ž on Ξ Μ…β„Ž .

    The proof of Lemma 2.3 follows from comparison Theorem see Chapter 4 (Samarskii, 2001).

    Lemma 2.4 The solution of the system

    π‘£β„Žπ‘˜ = π΅π‘£β„Ž

    π‘˜ + π‘“β„Žπ‘˜ on Ξ β„Ž, π‘£β„Ž

    π‘˜ = 0 on π›Ύβ„Ž (2.13)

    satisfies the inequality

    π‘£β„Žπ‘˜(π‘₯, 𝑦) ≀ π‘„β„Ž

    π‘˜, 1 ≀ π‘˜ ≀ 𝑁(β„Ž), (2.14)

    where π‘„β„Žπ‘˜ is defined as follows

    π‘„β„Žπ‘˜ = π‘„β„Ž

    π‘˜(π‘₯, 𝑦) = {

    6𝜌

    β„Ž, 0 ≀ 𝜌(π‘₯, 𝑦) ≀ π‘˜β„Ž ,

    6π‘˜, 𝜌(π‘₯, 𝑦) > π‘˜β„Ž . (2.15)

    Proof. By virtue of (2.7) and (2.15) and in consider of Fig. (2.1), we have for 0 ≀ 𝜌 = π‘˜β„Ž,

    π΅π‘„β„Žπ‘˜ =

    1

    20[4(6π‘˜ + 6(π‘˜ βˆ’ 1) + 6(π‘˜ βˆ’ 1) + 6π‘˜)

    +6(π‘˜ βˆ’ 1) + 6π‘˜ + 6(π‘˜ βˆ’ 1) + 6(π‘˜ βˆ’ 1)]

    = 6π‘˜ βˆ’66

    20,

    which leads to

    π‘„β„Žπ‘˜ βˆ’ π΅π‘„β„Ž

    π‘˜ =66

    20> 1 = π‘“β„Ž

    π‘˜.

  • 11

    Figure 2.1: The selected region in Ξ  is 𝜌 = π‘˜β„Ž

    In consider of Fig. (2.2) for 𝜌 > π‘˜β„Ž, then

    π΅π‘„β„Žπ‘˜ =

    1

    20[4(6π‘˜ + 6π‘˜ + 6π‘˜ + 6π‘˜) + 6π‘˜

    +6π‘˜ + 6π‘˜ + 6π‘˜ = 6π‘˜,

    which leads to

    π΅π‘„β„Žπ‘˜ = π‘„β„Ž

    π‘˜ .

  • 12

    Figure 2.2: The selected region in Ξ  is 𝜌 > π‘˜β„Ž

    In consider of Fig. (2.3) for 𝜌 < π‘˜β„Ž, then

    π΅π‘„β„Žπ‘˜ =

    1

    20[4(6π‘˜ + 6(π‘˜ βˆ’ 2) + 6(π‘˜ βˆ’ 1) + 6(π‘˜ βˆ’ 1))

    +6(π‘˜ βˆ’ 1) + 6π‘˜ + 6(π‘˜ βˆ’ 2) + 6(π‘˜ βˆ’ 2)]

    = 6π‘˜ βˆ’126

    20,

    which leads to

    π‘„β„Žπ‘˜ βˆ’ π΅π‘„β„Ž

    π‘˜ =126

    20> 1 = π‘“β„Ž

    π‘˜.

  • 13

    Figure 2.3: The selected region in Ξ  is 𝜌 < π‘˜β„Ž

    In consider of Fig. (2.1) for 𝜌 < π‘˜β„Ž, then

    π΅π‘„β„Žπ‘˜ =

    1

    20= [4(6(π‘˜ βˆ’ 1) + 6(π‘˜ βˆ’ 1) + 6(π‘˜ βˆ’ 2) + 6π‘˜)

    +6π‘˜ + 6π‘˜ + 6(π‘˜ βˆ’ 2) + 6(π‘˜ βˆ’ 2)]

    = 6π‘˜ βˆ’ 6 = 6(π‘˜ βˆ’ 1),

    which leads to

    π‘„β„Žπ‘˜ = π΅π‘„β„Ž

    π‘˜.

  • 14

    Figure 2.4: The selected region in Ξ  is 𝜌 < π‘˜β„Ž

    From the above calculations we have

    π‘„β„Žπ‘˜ = π΅π‘„β„Ž

    π‘˜ + π‘žβ„Žπ‘˜ on Ξ β„Ž, π‘„β„Ž

    π‘˜ = 0 on π›Ύβ„Ž, π‘˜ = 1,… ,𝑁(β„Ž), (2.16)

    where |π‘žβ„Žπ‘˜| β‰₯ 1. On the basis of (2.10), (2.13), (2.16) and the Comparison Theorem from

    (Samarskii, 2001), we obtain

    |π‘£β„Žπ‘˜| ≀ π‘„β„Ž

    π‘˜ for all π‘˜, 1 ≀ π‘˜ ≀ 𝑁(β„Ž). ∎

    Lemma 2.5 The following equality is true

    𝐡𝑝7(π‘₯0, 𝑦0) = 𝑒(π‘₯0, 𝑦0)

    where 𝑝7 is the seventh order Taylor’s polynomial at (π‘₯0, 𝑦0), 𝑒 is a harmonic function.

  • 15

    Proof. The seventh order Taylor’s polynomial at (π‘₯0, 𝑦0) has the form

    𝑝7(π‘₯, 𝑦) = 𝑒(π‘₯, 𝑦) + β„Ž (πœ•π‘’

    πœ•π‘₯+

    πœ•π‘’

    πœ•π‘¦) +

    β„Ž2

    2!(πœ•2𝑒

    πœ•π‘₯2+ 2

    πœ•2𝑒

    πœ•π‘₯πœ•π‘¦+

    πœ•2𝑒

    πœ•π‘¦2) +

    β„Ž3

    3!(πœ•3𝑒

    πœ•π‘₯3

    +3πœ•3𝑒

    πœ•π‘₯2πœ•π‘¦+ 3

    πœ•3𝑒

    πœ•π‘₯πœ•π‘¦2+

    πœ•3𝑒

    πœ•π‘¦3) +

    β„Ž4

    4!(πœ•4𝑒

    πœ•π‘₯4+ 4

    πœ•4𝑒

    πœ•π‘₯3πœ•π‘¦+ 6

    πœ•4𝑒

    πœ•π‘₯2πœ•π‘¦2

    +4πœ•4𝑒

    πœ•π‘₯πœ•π‘¦3+

    πœ•4𝑒

    πœ•π‘¦4) +

    β„Ž5

    5!(πœ•5𝑒

    πœ•π‘₯5+ 5

    πœ•5𝑒

    πœ•π‘₯4πœ•π‘¦+ 20

    πœ•5𝑒

    πœ•π‘₯3πœ•π‘¦2+ 20

    πœ•5𝑒

    πœ•π‘₯2πœ•π‘¦3

    +5πœ•5𝑒

    πœ•π‘₯πœ•π‘¦4+

    πœ•5𝑒

    πœ•π‘¦5) +

    β„Ž6

    6!(πœ•6𝑒

    πœ•π‘₯6+ 6

    πœ•6𝑒

    πœ•π‘₯5πœ•π‘¦+ 15

    πœ•6𝑒

    πœ•π‘₯4πœ•π‘¦2+ 20

    πœ•6𝑒

    πœ•π‘₯3πœ•π‘¦3

    +15πœ•6𝑒

    πœ•π‘₯2πœ•π‘¦4+ 6

    πœ•6𝑒

    πœ•π‘₯πœ•π‘¦5+

    πœ•6𝑒

    πœ•π‘¦6) +

    β„Ž7

    7!(πœ•7𝑒

    πœ•π‘₯7+ 7

    πœ•7𝑒

    πœ•π‘₯6πœ•π‘¦+ 21

    πœ•7𝑒

    πœ•π‘₯5πœ•π‘¦2

    +35πœ•7𝑒

    πœ•π‘₯4πœ•π‘¦3+ 35

    πœ•7𝑒

    πœ•π‘₯3πœ•π‘¦4+ 21

    πœ•7𝑒

    πœ•π‘₯2πœ•π‘¦5+ 7

    πœ•7𝑒

    πœ•π‘₯πœ•π‘¦6+

    πœ•7𝑒

    πœ•π‘¦7). (2.17)

    Then according to (2.7) and (2.17) we have

    𝐡𝑝7(π‘₯0, 𝑦0) =1

    20[4(𝑝7(π‘₯0 + β„Ž, 𝑦0) + 𝑝7(π‘₯0 βˆ’ β„Ž, 𝑦0) + 𝑝7(π‘₯0, 𝑦0 + β„Ž)

    +𝑝7(π‘₯0, 𝑦0 βˆ’ β„Ž)) + 𝑝7(π‘₯0 + β„Ž, 𝑦0 + β„Ž) + 𝑝7(π‘₯0 + β„Ž, 𝑦0 βˆ’ β„Ž)

    +𝑝7(π‘₯0 βˆ’ β„Ž, 𝑦0 + β„Ž) + 𝑝7(π‘₯0 βˆ’ β„Ž, 𝑦0 βˆ’ β„Ž)]

    = 𝑒(π‘₯0, 𝑦0) +β„Ž4

    40

    πœ•2

    πœ•π‘₯2(πœ•2𝑒(π‘₯0, 𝑦0)

    πœ•π‘₯2+

    πœ•2𝑒(π‘₯0, 𝑦0)

    πœ•π‘¦2)

    +β„Ž4

    40

    πœ•2

    πœ•π‘¦2(πœ•2𝑒(π‘₯0, 𝑦0)

    πœ•π‘₯2+

    πœ•2𝑒(π‘₯0, 𝑦0)

    πœ•π‘¦2) +

    3β„Ž6

    5 Γ— 6!

    πœ•4

    πœ•π‘₯4(πœ•2𝑒(π‘₯0, 𝑦0)

    πœ•π‘₯2+

    πœ•2𝑒(π‘₯0, 𝑦0)

    πœ•π‘¦2)

    +3β„Ž6

    5 Γ— 6!

    πœ•4

    πœ•π‘¦4(πœ•2𝑒(π‘₯0, 𝑦0)

    πœ•π‘₯2+

    πœ•2𝑒(π‘₯0, 𝑦0)

    πœ•π‘¦2) +

    2β„Ž6

    5 Γ— 5!

    πœ•4

    πœ•π‘₯2πœ•π‘¦2(πœ•2𝑒(π‘₯0, 𝑦0)

    πœ•π‘₯2+

    πœ•2𝑒(π‘₯0, 𝑦0)

    πœ•π‘¦2).

    Since 𝑒 is harmonic, we obtain

  • 16

    𝐡𝑃7(π‘₯0, 𝑦0) = 𝑒(π‘₯0, 𝑦0) ∎

    Lemma 2.6 The inequality holds

    max(π‘₯,𝑦)βˆˆΞ π‘˜β„Ž

    |𝐡𝑒 βˆ’ 𝑒| ≀ π‘β„Ž7+πœ†

    π‘˜1βˆ’πœ†, π‘˜ = 1,2, … ,𝑁(β„Ž),

    (2.18)

    where 𝑒 is a solution of problem (2.1).

    Proof. Let (π‘₯0, 𝑦0) be a point of Ξ 1β„Ž, and let

    Ξ 0 = {(π‘₯, 𝑦): |π‘₯ βˆ’ π‘₯0| < β„Ž, |𝑦 βˆ’ 𝑦0| < β„Ž }, (2.19)

    be an elementary square, some sides of which lie on the boundary of the rectangle Ξ . On the

    vertices of Ξ 0, and on the mid points of its sides lie the nodes of which the function values are

    used to evaluate 𝐡𝑒(π‘₯0, 𝑦0). We represent a solution of problem (2.1) in some neighborhood of

    (π‘₯0, 𝑦0) ∈ Ξ 1β„Ž by Taylor’s formula

    𝑒(π‘₯, 𝑦) = 𝑝7(π‘₯, 𝑦) + π‘Ÿ8(π‘₯, 𝑦), (2.20)

    where 𝑝7(π‘₯, 𝑦) is the seventh order Taylor’s polynomial, π‘Ÿ8(π‘₯, 𝑦) is the remainder term, by

    Lemma 2.5 we have

    𝐡𝑝7(π‘₯0, 𝑦0) = 𝑒(π‘₯0, 𝑦0). (2.21)

    Now, we estimate π‘Ÿ8 at the nodes of the operator 𝐡. We take node (π‘₯0 + β„Ž, 𝑦0 + β„Ž) which is

    one of the eight nodes of 𝐡, and consider the function

  • 17

    οΏ½ΜƒοΏ½(𝑠) = 𝑒 (π‘₯0 +𝑠

    √2, 𝑦0 +

    𝑠

    √2) , βˆ’ √2β„Ž ≀ 𝑠 ≀ √2β„Ž (2.22)

    of one variable 𝑠. By virtue of Lemma 2.2, we have

    |πœ•8οΏ½ΜƒοΏ½(𝑠)

    πœ•π‘ 8| ≀ 𝑐2(√2β„Ž βˆ’ 𝑠)

    πœ†βˆ’1, 0 ≀ 𝑠 ≀ √2β„Ž . (2.23)

    We represent function (2.22) around the point 𝑠 = 0 by Taylor’s formula

    οΏ½ΜƒοΏ½(𝑠) = 𝑝7(𝑠) + οΏ½ΜƒοΏ½8(𝑠), (2.24)

    where

    𝑝7(𝑠) = 𝑝7 (π‘₯0 +𝑠

    √2, 𝑦0 +

    𝑠

    √2) (2.25)

    is the seventh order Taylor’s polynomial of the variable 𝑠, and

    οΏ½ΜƒοΏ½8(𝑠) = π‘Ÿ8 (π‘₯0 +𝑠

    √2, 𝑦0 +

    𝑠

    √2) , 0 ≀ |𝑠| ≀ √2β„Ž (2.26)

    is the remainder term. On the basis of continuity of οΏ½ΜƒοΏ½8(𝑠) on the interval [βˆ’βˆš2β„Ž, √2β„Ž], it follows

    from (2.26) that

    π‘Ÿ8 (π‘₯0 +𝑠

    √2, 𝑦0 +

    𝑠

    √2) = lim

    πœ–β†’+0οΏ½ΜƒοΏ½8(√2β„Ž βˆ’ πœ–). (2.27)

    Applying an integral representation for οΏ½ΜƒοΏ½8 we have

  • 18

    οΏ½ΜƒοΏ½8(√2β„Ž βˆ’ πœ–) =1

    7!∫ (√2β„Ž βˆ’ πœ– βˆ’ 𝑑)

    7οΏ½ΜƒοΏ½8(𝑑)𝑑𝑑

    √2β„Žβˆ’πœ–

    0

    , 0 < πœ– β‰€β„Ž

    √2 .

    Using estimation (2.23), we have

    |οΏ½ΜƒοΏ½8(√2β„Ž βˆ’ πœ–)| ≀ 𝑐3 ∫ (√2β„Ž βˆ’ πœ– βˆ’ 𝑑)7(√2β„Ž βˆ’ 𝑑)

    πœ†βˆ’1𝑑𝑑

    √2β„Žβˆ’πœ–

    0

    ≀ 𝑐41

    7!∫ (√2β„Ž βˆ’ 𝑑)

    6+πœ†π‘‘π‘‘

    √2β„Žβˆ’πœ–

    0

    ≀ π‘β„Žπœ†+7, 0 < πœ– β‰€β„Ž

    √2 . (2.28)

    From (2.26)-(2.28) yields

    |π‘Ÿ8(π‘₯0 + β„Ž, 𝑦0 + β„Ž| ≀ 𝑐1β„Žπœ†+7, (2.29)

    where 𝑐1 is a constant independent of the taken point (π‘₯0, 𝑦0) on Ξ 1β„Ž. Proceeding in a similar

    manner, we can find the same estimates of π‘Ÿ8 at the other vertices of square (2.19) and at the

    centers of its sides. Since the norm of 𝐡 in the uniform metric is equal to unity, we have

    |π΅π‘Ÿ8(π‘₯0, 𝑦0)| ≀ 𝑐5β„Žπœ†+7. (2.30)

    where 𝑐5 is a constant independent of the taken point (π‘₯0, 𝑦0) on Ξ 1β„Ž . From (2.20), (2.21),

    (2.30) and linearity of the operator 𝐡, we obtain

    |𝐡𝑒(π‘₯0, 𝑦0) βˆ’ 𝑒(π‘₯0, 𝑦0)| ≀ π‘β„Žπœ†+7, (2.31)

    for any (π‘₯0, 𝑦0) ∈ Ξ 1β„Ž.

  • 19

    Now let (π‘₯0, 𝑦0) ∈ Ξ π‘˜β„Ž, 2 ≀ π‘˜ ≀ 𝑁(β„Ž) and π‘Ÿ8(π‘₯, 𝑦) be the Lagrange remainder corresponding

    to this point in Taylor’s formula (2.20). Then π΅π‘Ÿ8(π‘₯0, 𝑦0) can be expressed linearly in terms of

    a fixed number of eighth derivatives of 𝑒 at some point of the open square Ξ 0 , which is a

    distance of π‘˜β„Ž 2⁄ away from the boundary of Ξ . The sum of the coefficients multiplying the

    eighth derivatives does not exceed π‘β„Ž8, which is independent of π‘˜ (2 ≀ π‘˜ ≀ 𝑁(β„Ž)). By using

    Lemma 2.2, we have

    |π΅π‘Ÿ8(π‘₯0, 𝑦0)| ≀ π‘β„Ž8

    (π‘˜β„Ž)1βˆ’πœ†= 𝑐

    β„Žπœ†+7

    π‘˜1βˆ’πœ†, (2.32)

    where 𝑐 is a constant independent of π‘˜ (2 ≀ π‘˜ ≀ 𝑁(β„Ž)). On the basis of (2.20), (2.21), (2.31),

    and (2.32) follows estimation (2.18) at any point (π‘₯0, 𝑦0) ∈ Ξ π‘˜β„Ž, 1 ≀ π‘˜ ≀ 𝑁(β„Ž). ∎

    Theorem 2.1 The next estimation holds

    |π‘’β„Ž βˆ’ 𝑒| ≀ π‘πœŒβ„Ž6,

    where 𝑐 is a constant independent of 𝜌 and β„Ž, 𝑒 is the exact solution of problem (2.1), π‘’β„Ž is the

    solution of the finite difference problem (2.8) and 𝜌 = 𝜌(π‘₯, 𝑦) is the distance from the current

    point (π‘₯, 𝑦) ∈ π›±β„Ž to the boundary of rectangle 𝛱.

    Proof. Let

    πœ–β„Ž(π‘₯, 𝑦) = π‘’β„Ž(π‘₯, 𝑦) βˆ’ 𝑒(π‘₯, 𝑦), (π‘₯, 𝑦) ∈ Ξ Μ…β„Ž. (2.33)

    Putting π‘’β„Ž = πœ–β„Ž + 𝑒 into (2.8), we have

    πœ–β„Ž = π΅πœ–β„Ž + (𝐡𝑒 βˆ’ 𝑒) on Ξ β„Ž, πœ–β„Ž = 0 on 𝛾

    β„Ž. (2.34)

    We represent a solution of system (2.34) as follows

  • 20

    πœ–β„Ž = βˆ‘ πœ–β„Žπ‘˜

    𝑁(β„Ž)

    π‘˜=1

    , 𝑁(β„Ž) = [1

    2β„Žmin{π‘Ž, 𝑏}], (2.35)

    where πœ–β„Žπ‘˜ is a solution of the system

    πœ–β„Žπ‘˜ = π΅πœ–β„Ž

    π‘˜ + πœŽβ„Žπ‘˜ on Ξ β„Ž, πœ–β„Ž = 0 on 𝛾

    β„Ž, π‘˜ = 1,2, … , 𝑁(β„Ž); (2.36)

    πœŽβ„Žπ‘˜ = {

    𝐡𝑒 βˆ’ 𝑒 on Ξ β„Žπ‘˜ ,

    0 on Ξ β„Ž Ξ β„Žπ‘˜β„ .

    (2.37)

    By virtue of (2.36), (2.37) and Lemma 2.4, for each π‘˜ , 1 ≀ π‘˜ ≀ 𝑁(β„Ž), follows the

    inequality

    |πœ–β„Žπ‘˜(π‘₯, 𝑦)| ≀ π‘„β„Ž

    π‘˜(π‘₯, 𝑦) max(π‘₯,𝑦)βˆˆΞ π‘˜β„Ž

    |(𝐡𝑒 βˆ’ 𝑒)| on Ξ Μ…β„Ž . (2.38)

    On the basis of (2.33), (2.35) and (2.38) we have

    |πœ–β„Ž| ≀ βˆ‘|πœ–β„Žπ‘˜|

    𝑁(β„Ž)

    π‘˜=1

    ≀ βˆ‘ π‘„β„Žπ‘˜(π‘₯, 𝑦) max

    (π‘₯,𝑦)βˆˆΞ π‘˜β„Ž|(𝐡𝑒 βˆ’ 𝑒)|

    𝑁(β„Ž)

    π‘˜=1

    = βˆ‘ π‘„β„Žπ‘˜(π‘₯, 𝑦) max

    (π‘₯,𝑦)βˆˆΞ π‘˜β„Ž|(𝐡𝑒 βˆ’ 𝑒)|

    𝜌

    β„Žβˆ’1

    π‘˜=1

    + βˆ‘ π‘„β„Žπ‘˜(π‘₯, 𝑦) max

    (π‘₯,𝑦)βˆˆΞ π‘˜β„Ž|(𝐡𝑒 βˆ’ 𝑒)|,

    𝑁(β„Ž)

    π‘˜=𝜌

    β„Ž

    (π‘₯, 𝑦) ∈ Ξ π‘˜β„Ž .

    By definition (2.15) of the function π‘„β„Žπ‘˜, we have

  • 21

    βˆ‘ π‘„β„Žπ‘˜(π‘₯, 𝑦) max

    (π‘₯,𝑦)βˆˆΞ π‘˜β„Ž|(𝐡𝑒 βˆ’ 𝑒)| ≀ 6π‘β„Ž8 βˆ‘

    π‘˜

    (π‘˜β„Ž)1βˆ’πœ†

    𝜌

    β„Žβˆ’1

    π‘˜=1

    𝜌

    β„Žβˆ’1

    π‘˜=1

    , (2.39)

    βˆ‘ π‘„β„Žπ‘˜(π‘₯, 𝑦) max

    (π‘₯,𝑦)βˆˆΞ π‘˜β„Ž|(𝐡𝑒 βˆ’ 𝑒)|6π‘β„Ž8 βˆ‘

    π‘˜

    (π‘˜β„Ž)1βˆ’πœ†

    𝑁(β„Ž)

    π‘˜=𝜌

    β„Ž

    .

    𝑁(β„Ž)

    π‘˜=𝜌

    β„Ž

    (2.40)

    Then from (2.39)-(2.40) we have

    |πœ–β„Ž(π‘₯, 𝑦)| ≀ 6π‘β„Ž7+πœ† βˆ‘ π‘˜πœ†

    𝜌

    β„Žβˆ’1

    π‘˜=1

    + 6π‘β„Ž6+πœ†πœŒ βˆ‘1

    π‘˜1βˆ’πœ†

    𝑁(β„Ž)

    π‘˜=𝜌

    β„Ž

    ≀ 6π‘β„Ž7+πœ†

    (

    1 + ∫ π‘₯πœ†

    𝜌

    β„Žβˆ’1

    1

    𝑑π‘₯

    )

    + 6π‘β„Ž6+πœ†πœŒ ((𝜌

    β„Ž)

    πœ†βˆ’1

    + ∫ π‘₯πœ†βˆ’1

    𝑁(β„Ž)

    𝜌

    β„Ž

    𝑑π‘₯)

    ≀ π‘β„Ž7+πœ† + π‘β„Ž7+πœ† ((𝜌

    β„Žβˆ’ 1)

    πœ†+1

    πœ† + 1βˆ’

    1

    πœ† + 1) + π‘β„Ž7πœŒπœ† + π‘β„Ž6+πœ†πœŒ (

    (π‘Ž

    β„Ž)πœ†

    πœ†βˆ’

    𝜌

    β„Ž

    πœ†)

    ≀ π‘β„Ž7+πœ† +𝑐

    πœ† + 1β„Ž7+πœ† (

    𝜌

    β„Žβˆ’ 1)

    πœ†+1

    βˆ’π‘

    πœ† + 1β„Ž7+πœ† + π‘β„Ž7πœŒπœ†

    +𝑐

    πœ†β„Ž6+πœ†πœŒ

    (π‘Ž

    β„Ž)

    πœ†

    πœ†βˆ’

    𝑐

    πœ†(𝜌

    β„Ž)

    πœ†

    β„Ž6+πœ†πœŒ

    ≀ π‘β„Ž7+πœ† + 𝑐1β„Ž7+πœ† (

    𝜌

    β„Žβˆ’ 1)

    πœ†+1

    βˆ’ 𝑐1β„Ž7+πœ† + π‘β„Ž7πœŒπœ† + 𝑐2β„Ž

    6πœŒπ‘Žπœ† βˆ’ 𝑐2β„Ž6πœŒπœ†+1

    = π‘β„Ž6𝜌(π‘₯, 𝑦), (π‘₯, 𝑦) ∈ Ξ Μ…β„Ž.

    Theorem 2.1 is proved. ∎

  • 22

    2.3 Approximation of the First Derivatives

    Let 𝑒 be a solution of the boundary value problem (2.1). We put 𝜐 =πœ•π‘’

    πœ•π‘₯. It is obvious that the

    function 𝜐 is a solution of boundary value problem

    βˆ†πœ = 0 on Ξ , 𝜐 = πœ“π‘— on 𝛾𝑗, 𝑗 = 1,2,3,4, (2.41)

    where πœ“π‘— =πœ•π‘’

    πœ•π‘₯ on 𝛾𝑗, 𝑗 = 1,2,3,4.

    Let π‘’β„Ž be a solution of finite difference problem (2.8). We define the following operators πœ“π‘£β„Ž,

    𝑣 = 1,2,3,4,

    πœ“1β„Ž(π‘’β„Ž) =1

    60β„Ž[βˆ’147πœ‘1(𝑦) + 360π‘’β„Ž(β„Ž, 𝑦) βˆ’ 450π‘’β„Ž(2β„Ž, 𝑦)

    +400π‘’β„Ž(3β„Ž, 𝑦) βˆ’ 225π‘’β„Ž(4β„Ž, 𝑦) + 72π‘’β„Ž(5β„Ž, 𝑦)

    βˆ’10π‘’β„Ž(6β„Ž, 𝑦)] on 𝛾1β„Ž, (2.42)

    πœ“3β„Ž(π‘’β„Ž) =1

    60β„Ž[147πœ‘3(𝑦) βˆ’ 360π‘’β„Ž(π‘Ž βˆ’ β„Ž, 𝑦)

    +450π‘’β„Ž(π‘Ž βˆ’ 2β„Ž, 𝑦) βˆ’ 400π‘’β„Ž(π‘Ž βˆ’ 3β„Ž, 𝑦) + 225π‘’β„Ž(π‘Ž βˆ’ 4β„Ž, 𝑦)

    βˆ’72π‘’β„Ž(π‘Ž βˆ’ 5β„Ž, 𝑦) + 10π‘’β„Ž(π‘Ž βˆ’ 6β„Ž, 𝑦)] on 𝛾3β„Ž, (2.43)

    πœ“π‘β„Ž(π‘’β„Ž) =πœ•πœ‘π‘

    πœ•π‘₯ on 𝛾𝑝

    β„Ž, 𝑝 = 2,4. (2.44)

    Lemma 2.7 The next inequality holds

    |πœ“π‘˜β„Ž(π‘’β„Ž) βˆ’ πœ“π‘˜β„Ž(𝑒)| ≀ π‘β„Ž6, π‘˜ = 1,3,

    where π‘’β„Ž is the solution of problem (2.8), and 𝑒 is the solution of problem (2.1).

  • 23

    Proof. It is obvious that πœ“π‘β„Ž(π‘’β„Ž) βˆ’ πœ“π‘β„Ž(𝑒) = 0 for 𝑝 = 2,4.

    For π‘˜ = 1, by (2.42) and Theorem 2.1 we have

    |πœ“1β„Ž(π‘’β„Ž) βˆ’ πœ“1β„Ž(𝑒)| = |1

    60β„Ž[(βˆ’147πœ‘1(𝑦) + 360π‘’β„Ž(β„Ž, 𝑦)

    βˆ’450π‘’β„Ž(2β„Ž, 𝑦) + 400π‘’β„Ž(3β„Ž, 𝑦) βˆ’ 225π‘’β„Ž(4β„Ž, 𝑦) + 72π‘’β„Ž(5β„Ž, 𝑦)

    βˆ’10π‘’β„Ž(6β„Ž, 𝑦)) βˆ’ (βˆ’147πœ‘1(𝑦) + 360𝑒(β„Ž, 𝑦) βˆ’ 450𝑒(2β„Ž, 𝑦)

    +400𝑒(3β„Ž, 𝑦) βˆ’ 225𝑒(4β„Ž, 𝑦) + 72𝑒(5β„Ž, 𝑦)10𝑒(6β„Ž, 𝑦))]|

    ≀1

    60β„Ž[|π‘’β„Ž(β„Ž, 𝑦) βˆ’ 𝑒(β„Ž, 𝑦)| + 450|π‘’β„Ž(2β„Ž, 𝑦) βˆ’ 𝑒(2β„Ž, 𝑦)|

    +400|π‘’β„Ž(3β„Ž, 𝑦) βˆ’ 𝑒(3β„Ž, 𝑦)| + 225|π‘’β„Ž(4β„Ž, 𝑦) βˆ’ 𝑒(4β„Ž, 𝑦)|

    +72|π‘’β„Ž(5β„Ž, 𝑦) βˆ’ 𝑒(5β„Ž, 𝑦)| + 10|π‘’β„Ž(6β„Ž, 𝑦) βˆ’ 𝑒(6β„Ž, 𝑦)|]

    ≀1

    60β„Ž[360(π‘β„Ž)β„Ž6 + 450(2π‘β„Ž)β„Ž6 + 400(3π‘β„Ž)β„Ž6 + 225(4π‘β„Ž)β„Ž6

    +72(5π‘β„Ž)β„Ž6 + 10(6π‘β„Ž)β„Ž6].

    ≀ 𝑐7β„Ž6.

    The same inequality is true for π‘˜ = 3. ∎

    Lemma 2.8 The inequality holds

    max(π‘₯,𝑦)βˆˆπ›Ύπ‘˜

    β„Ž|πœ“π‘˜β„Ž(π‘’β„Ž) βˆ’ πœ“π‘˜| ≀ 𝑐8β„Ž

    6 , π‘˜ = 1,3,

    where πœ“π‘˜β„Ž, π‘˜ = 1,3 are the functions defined by (2.42) and (2.43), πœ“π‘˜ =πœ•π‘’

    πœ•π‘₯ on π›Ύπ‘˜,

    π‘˜ = 1,3.

    Proof. From Lemma 2.1 follows that 𝑒 ∈ 𝐢7,0(Ξ Μ…). Then at the end points (0, πœβ„Ž) ∈ 𝛾1β„Ž and

    (π‘Ž, πœβ„Ž) ∈ 𝛾3β„Ž of each line segment {(π‘₯, 𝑦): 0 ≀ π‘₯ ≀ π‘Ž , 0 < 𝑦 = πœβ„Ž < 𝑏} expressions (2.42) and

    (2.43) give the sixth order approximation of πœ•π‘’

    πœ•π‘₯ respectively.

    From the truncation error formulae follows that

  • 24

    max(π‘₯,𝑦)βˆˆπ›Ύπ‘˜

    β„Ž|πœ“π‘˜β„Ž(𝑒) βˆ’ πœ“π‘˜| ≀ max

    (π‘₯,𝑦)βˆˆΞ Μ…

    1

    7!|πœ•7𝑒

    πœ•π‘₯7| β„Ž(2β„Ž)(3β„Ž)(4β„Ž)(5β„Ž)(6β„Ž)

    ≀ 𝑐9β„Ž6, k = 1,3. (2.45)

    On the basis of Lemma 2.7 and estimation (2.45)

    max(π‘₯,𝑦)βˆˆπ›Ύπ‘˜

    β„Ž|πœ“π‘˜β„Ž(π‘’β„Ž) βˆ’ πœ“π‘˜| ≀ max

    (π‘₯,𝑦)βˆˆπ›Ύπ‘˜β„Ž|πœ“π‘˜β„Ž(π‘’β„Ž) βˆ’ πœ“π‘˜β„Ž(𝑒)| + max

    (π‘₯,𝑦)βˆˆπ›Ύπ‘˜β„Ž|πœ“π‘˜β„Ž(𝑒) βˆ’ πœ“π‘˜|

    ≀ 𝑐7β„Ž6 + 𝑐9β„Ž

    6 = 𝑐8β„Ž6, π‘˜ = 1,3.∎

    Consider the finite difference problem

    π‘£β„Ž = π΅π‘£β„Ž on Ξ β„Ž, 𝑣 = πœ“π‘—β„Ž on 𝛾𝑗

    β„Ž, 𝑗 = 1,2,3,4, (2.46)

    where πœ“π‘—β„Ž , 𝑗 = 1,2,3,4 are defined by the formulas (2.42)-(2.44).

    Since the boundary values πœ“π‘—β„Ž for 𝑗 = 1,3 are defined by the solution of finite-difference

    problem (2.46) which assumed to be known and πœ“π‘—β„Ž =πœ•πœ‘π‘—

    πœ•π‘₯, 𝑗 = 2,4 are calculated by the

    boundary functions πœ‘π‘— , 𝑗 = 2,4 the existence and uniqueness follows from the discrete

    maximum principle.

    To estimate the convergence order of problem (2.41) we consider the problem

    βˆ†π‘‰ = 0 on Ξ , 𝑉 = Φ𝑗 on 𝛾𝑗, 𝑗 = 1,2,3,4,

    where Φ𝑗 in the given function, which satisfy the following conditions

    Φ𝑗 ∈ 𝐢6,πœ†(𝛾𝑗), 0 < πœ† < 1, (2.47)

  • 25

    Φ𝑗2π‘ž(𝑠𝑗) = (βˆ’1)

    π‘žΞ¦π‘—βˆ’12π‘ž (𝑠𝑗), π‘ž = 0,1,2. (2.48)

    Let π‘‰β„Ž be a solution of the finite–difference problem

    π‘‰β„Ž = π΅π‘‰β„Ž on Ξ β„Ž, π‘‰β„Ž = Ξ¦π‘—β„Ž on 𝛾𝑗

    β„Ž , 𝑗 = 1,2,3,4, (2.49)

    where πœ“π‘—β„Ž is the value of πœ“π‘— on π›Ύπ‘—β„Ž, 𝑗 = 1,2,3,4. It is clear that the error function πœ–β„Ž = π‘‰β„Ž βˆ’ 𝑉

    is a solution of the boundary value problem

    πœ–β„Ž = π΅πœ–β„Ž + (π΅π‘‰β„Ž βˆ’ 𝑉) on Ξ β„Ž, πœ–β„Ž = 0 on 𝛾𝑗

    β„Ž , 𝑗 = 1,2,3,4. (2.50)

    As follows from Theorem 12 in (Dosiyev, 2003) the following estimation

    max(π‘₯,𝑦)βˆˆΞ Μ…β„Ž

    |π‘‰β„Ž βˆ’ 𝑉| ≀ π‘β„Ž6, (2.51)

    is true.

    Theorem 2.2 The following estimation holds

    max(π‘₯,𝑦)βˆˆΞ Μ…

    |π‘£β„Ž βˆ’πœ•π‘’

    πœ•π‘₯| ≀ 𝑐0β„Ž

    6,

    where 𝑒 is the solution of problem (2.1), and π‘£β„Ž is the solution of finite difference problem

    (2.46).

    Proof. Let

    πœ–β„Ž = π‘£β„Ž βˆ’ 𝑣 on Ξ Μ…β„Ž, (2.52)

  • 26

    where 𝑣 =πœ•π‘’

    πœ•π‘₯. From (2.46) and (2.52), we have

    πœ–β„Ž = π΅πœ–β„Ž + (𝐡𝑣 βˆ’ 𝑣) on Ξ β„Ž, πœ–β„Ž = πœ“π‘˜β„Ž(π‘’β„Ž) βˆ’ 𝑣 on π›Ύπ‘˜

    β„Ž,

    π‘˜ = 1,3, πœ–β„Ž = 0 on π›Ύπ‘β„Ž, 𝑝 = 2,4. (2.53)

    We represent

    πœ–β„Ž = πœ–β„Ž1 + πœ–β„Ž

    2 (2.54)

    where

    πœ–β„Ž1 = π΅πœ–β„Ž

    1 on Ξ β„Ž, πœ–β„Ž1 = πœ“π‘˜β„Ž(π‘’β„Ž) βˆ’ 𝑣 on π›Ύπ‘˜

    β„Ž, π‘˜ = 1,3,

    πœ–β„Ž1 = 0 on 𝛾𝑝

    β„Ž, 𝑝 = 2,4; (2.55)

    πœ–β„Ž2 = π΅πœ–β„Ž

    2 + (𝐡𝑣 βˆ’ 𝑣) on Ξ β„Ž, πœ–β„Ž2 = 0 on 𝛾𝑗

    β„Ž, 𝑗 = 1,2,3,4. (2.56)

    By Lemma 2.8 and by maximum principle, for the solution of system (2.55), we have

    max(π‘₯,𝑦)βˆˆΞ β„Ž

    |πœ–β„Ž1| ≀ max

    π‘ž=1,3max

    (π‘₯,𝑦)βˆˆπ›Ύπ‘žβ„Ž|πœ“π‘žβ„Ž(π‘’β„Ž) βˆ’ 𝑣| ≀ 𝑐1β„Ž

    6. (2.57)

    From (2.50) follows that πœ–β„Ž2 in (2.56) is a solution of problem (2.50) with the boundary

    value of 𝑣 satisfy the equations (2.47), (2.48). By the estimation (2.51), we obtain

    max(π‘₯,𝑦)βˆˆΞ Μ…β„Ž

    |πœ–β„Ž2| ≀ 𝑐2β„Ž

    6 (2.58)

    By virtue of (2.54), (2.57) and (2.58) the proof is completed. ∎

  • 27

    2.4 Approximation of the Pure Second Derivatives

    We denote πœ” =πœ•2𝑒

    πœ•π‘₯2. The function πœ” is harmonic on Ξ  , on the basis of Lemma 2.1 πœ” is

    continuous on Ξ Μ…, and it is a solution of the following Dirichlet problem

    βˆ†πœ” = 0 on Ξ , πœ” = πœ’π‘— on 𝛾𝑗 , 𝑗 = 1,2,3,4 (2.59)

    where

    πœ’πœ =πœ•2πœ‘πœπœ•π‘₯2

    , 𝜏 = 2,4, (2.60)

    πœ’πœˆ = βˆ’πœ•2Ο†πœˆπœ•π‘¦2

    , 𝜈 = 1,3. (2.61)

    From the continuity of the function πœ” on Ξ Μ…, and from (2.2), (2.3) and (2.60), (2.61) it follows

    that

    πœ’π‘— ∈ 𝐢5,πœ†(𝛾𝑗), 0 < πœ† < 1, 𝑗 = 1,2,3,4. (2.62)

    πœ’π‘—(2π‘ž)

    (𝑠𝑗) = (βˆ’1)π‘žπœ’π‘—βˆ’1

    2π‘ž (𝑠𝑗), π‘ž = 0,1,2 , 𝑗 = 1,2,3,4. (2.63)

    Let πœ”β„Ž be a solution of the finite difference problem

    πœ”β„Ž = π΅πœ”β„Ž on Ξ β„Ž, πœ”β„Ž = πœ’π‘— on 𝛾𝑗

    β„Ž, 𝑗 = 1,2,3,4, (2.64)

    where πœ’π‘— are functions determined by (2.60) and (2.61).

  • 28

    Lemma 2.9 The next inequality holds true

    |πœ•8πœ”(π‘₯, 𝑦)

    πœ•π‘™8| ≀ 𝑐1𝜌

    πœ†βˆ’3(π‘₯, 𝑦), (π‘₯, 𝑦) ∈ Ξ  (2.65)

    where 𝑐1is a constant independent of the direction of differentiation πœ• πœ•π‘™β„ .

    Proof. We choose an arbitrary point (π‘₯0, 𝑦0) ∈ Ξ . Let 𝜌0 = 𝜌(π‘₯0, 𝑦0) and 𝜎 βŠ‚ Ξ Μ… be the closed

    circle of radius 𝜌0 centered at (π‘₯0, 𝑦0), consider the harmonic function on

    𝑣(π‘₯, 𝑦) =πœ•5πœ”(π‘₯, 𝑦)

    πœ•π‘™5βˆ’

    πœ•5πœ”(π‘₯0, 𝑦0)

    πœ•π‘™5. (2.66)

    By (2.62), πœ” =πœ•2𝑒

    πœ•π‘₯12 ∈ 𝐢

    5,πœ†(Ξ Μ…), for, 0 < πœ† < 1. Then for the function (2.66) we have

    max(π‘₯,𝑦)βˆˆοΏ½Μ…οΏ½0

    |𝑣(π‘₯, 𝑦)| ≀ 𝑐0𝜌0πœ†, (2.67)

    where 𝑐0 is a constant independent of the point (π‘₯0, 𝑦0) ∈ Ξ  or the direction of πœ• πœ•π‘™β„ . Since 𝑒

    is harmonic in Ξ , by using estimation (2.67) and applying Lemma 3 from (Mikhailov, 1978)

    we have

    |πœ•3

    πœ•π‘™3(πœ•5πœ”(π‘₯, 𝑦)

    πœ•π‘™5βˆ’

    πœ•5πœ”(π‘₯0, 𝑦0)

    πœ•π‘™5)| ≀ 𝑐

    𝜌0πœ†

    𝜌03 ,

    or

    |πœ•8πœ”(π‘₯, 𝑦)

    πœ•π‘™8| ≀ π‘πœŒ0

    πœ†βˆ’3(π‘₯0, 𝑦0),

    where 𝑐 is a constant independent of the point (π‘₯0, 𝑦0) ∈ Ξ  or the direction of πœ• πœ•π‘™β„ . Since the

    point (π‘₯0, 𝑦0) ∈ Ξ  is arbitrary, inequality (2.65) holds true. ∎

  • 29

    Lemma 2.10 The inequality holds

    max(π‘₯,𝑦)βˆˆΞ π‘˜β„Ž

    |π΅πœ” βˆ’ πœ”| ≀ π‘β„Ž5+πœ†

    π‘˜3βˆ’πœ†, π‘˜ = 1,2, … ,𝑁(β„Ž), (2.68)

    where 𝑒 is a solution of problem (2.1).

    Proof. Let (π‘₯0, 𝑦0) be a point of Ξ 1β„Ž βŠ‚ Ξ β„Ž, and let

    Ξ 0 = {(π‘₯, 𝑦): |π‘₯ βˆ’ π‘₯0| < β„Ž , |𝑦 βˆ’ 𝑦0| < β„Ž }, (2.69)

    be an elementary square, some sides of which lie on the boundary of the rectangle Ξ . On the

    vertices of Ξ 0, and on the mid points of its sides lie the nodes of which the function values are

    used to evaluate π΅πœ”(π‘₯0, 𝑦0). We represent a solution of problem (2.59) in some neighborhood

    of (π‘₯0, 𝑦0) ∈ Ξ 1β„Ž by Taylor’s formula

    πœ”(π‘₯, 𝑦) = 𝑝7(π‘₯, 𝑦) + π‘Ÿ8(π‘₯, 𝑦), (2.70)

    where 𝑝7(π‘₯, 𝑦) is the seventh order Taylor’s polynomial, π‘Ÿ8(π‘₯, 𝑦) is the remainder term, and

    𝐡𝑝7(π‘₯0, 𝑦0) = πœ”(π‘₯0, 𝑦0). (2.71)

    Now, we estimate π‘Ÿ8 at the nodes of the operator 𝐡. We take node (π‘₯0 + β„Ž, 𝑦0 + β„Ž) which is

    one of the eight nodes of 𝐡, and consider the function

    αΏΆ(𝑠) = πœ” (π‘₯0 +𝑠

    √2, 𝑦0 +

    𝑠

    √2) , βˆ’ √2β„Ž ≀ 𝑠 ≀ √2β„Ž (2.72)

    of one variable 𝑠 . By virtue of Lemma 2.9, we have

  • 30

    |πœ•8αΏΆ(𝑠)

    πœ•π‘ 8| ≀ 𝑐2(√2β„Ž βˆ’ 𝑠)

    πœ†βˆ’3, 0 ≀ 𝑠 ≀ √2β„Ž (2.73)

    we represent function (2.72) around the point 𝑠 = 0 by Taylor’s formula as

    αΏΆ(𝑠) = 𝑝7(𝑠) + οΏ½ΜƒοΏ½8(𝑠), (2.74)

    where

    𝑝7(𝑠) = 𝑝7 (π‘₯0 +𝑠

    √2, 𝑦0 +

    𝑠

    √2) (2.75)

    is the seventh order Taylor’s polynomial of the variable 𝑠, and

    οΏ½ΜƒοΏ½8(𝑠) = π‘Ÿ8 (π‘₯0 +𝑠

    √2, 𝑦0 +

    𝑠

    √2) , 0 ≀ |𝑠| ≀ √2β„Ž, (2.76)

    is the remainder term. On the basis of continuity of οΏ½ΜƒοΏ½8(𝑠) on the interval [βˆ’βˆš2β„Ž, √2β„Ž], it follows

    from (2.76) that

    π‘Ÿ8 (π‘₯0 +𝑠

    √2, 𝑦0 +

    𝑠

    √2) = lim

    πœ–β†’+0οΏ½ΜƒοΏ½8(√2β„Ž βˆ’ πœ–). (2.77)

    Applying an integral representation for οΏ½ΜƒοΏ½8 we have

    οΏ½ΜƒοΏ½8(√2β„Ž βˆ’ πœ–) =1

    7!∫ (√2β„Ž βˆ’ πœ– βˆ’ 𝑑)

    7οΏ½ΜƒοΏ½8(𝑑)𝑑𝑑

    √2β„Žβˆ’πœ–

    0

    , 0 < πœ– β‰€β„Ž

    √2. (2.78)

    Using estimation (2.73), we have

  • 31

    |οΏ½ΜƒοΏ½8(√2β„Ž βˆ’ πœ–)| ≀ 𝑐3 ∫ (√2β„Ž βˆ’ πœ– βˆ’ 𝑑)7(√2β„Ž βˆ’ 𝑑)

    πœ†βˆ’3𝑑𝑑

    √2β„Žβˆ’πœ–

    0

    ≀ 𝑐41

    7!∫ (√2β„Ž βˆ’ 𝑑)

    4+πœ†π‘‘π‘‘

    √2β„Žβˆ’πœ–

    0

    ≀ π‘β„Žπœ†+5, 0 < πœ– β‰€β„Ž

    √2. (2.79)

    From (2.76)-(2.79) yields

    |π‘Ÿ8(π‘₯0 + β„Ž, 𝑦0 + β„Ž| ≀ 𝑐1β„Žπœ†+5,

    where 𝑐1 is a constant independent of the taken point (π‘₯0, 𝑦0) on Ξ 1β„Ž. Proceeding in a similar

    manner, we can find the same estimates of π‘Ÿ8 at the other vertices of square (2.69) and at the

    centers of its sides. Since the norm of 𝐡 in the uniform metric is equal to unity, we have

    |π΅π‘Ÿ8(π‘₯0, 𝑦0)| ≀ 𝑐5β„Žπœ†+5, (2.80)

    where 𝑐5 is a constant independent of the taken point (π‘₯0, 𝑦0) on Ξ 1β„Ž . From (2.70), (2.71),

    (2.80) and linearity of the operator 𝐡, we obtain

    |π΅πœ”(π‘₯0, 𝑦0) βˆ’ πœ”(π‘₯0, 𝑦0)| ≀ π‘β„Žπœ†+5, (2.81)

    for any (π‘₯0, 𝑦0) ∈ Ξ 1β„Ž.

    Now let (π‘₯0, 𝑦0) ∈ Ξ π‘˜β„Ž, 2 ≀ π‘˜ ≀ 𝑁(β„Ž) and π‘Ÿ8(π‘₯, 𝑦) be the Lagrange remainder corresponding

    to this point in Taylor’s formula (2.70). Then π΅π‘Ÿ8(π‘₯0, 𝑦0) can be expressed linearly in terms of

    a fixed number of eighth derivatives of𝑒 at some point of the open square Ξ 0, which is a distance

    of π‘˜β„Ž 2⁄ away from the boundary of Ξ . The sum of the coefficients multiplying the eighth

  • 32

    derivatives does not exceed π‘β„Ž8, which is independent of π‘˜, (2 ≀ π‘˜ ≀ 𝑁(β„Ž)). By Lemma 2.9,

    we have

    |π΅π‘Ÿ8(π‘₯0, 𝑦0)| ≀ π‘β„Ž8

    (π‘˜β„Ž)3βˆ’πœ†= 𝑐

    β„Žπœ†+5

    π‘˜3βˆ’πœ†, (2.82)

    where 𝑐 is a constant independent of π‘˜, (2 ≀ π‘˜ ≀ 𝑁(β„Ž)). On the basis of (2.70), (2.71), (2.81)

    and (2.82) follows estimation (2.68) at any point (π‘₯0, 𝑦0) ∈ Ξ π‘˜β„Ž, 1 ≀ π‘˜ ≀ 𝑁(β„Ž). ∎

    Theorem 2.3 The estimation holds

    max(π‘₯0,𝑦0)βˆˆΞ Μ…β„Ž

    |πœ”β„Ž βˆ’ πœ”| ≀ 𝑐12 β„Žπœ†+5, (2.83)

    where πœ” =πœ•2𝑒

    πœ•π‘₯2, 𝑒 is the solution of problem (2.1), and πœ”β„Ž is the solution of the finite

    difference problem (2.64).

    Proof. Let

    πœ–β„Ž = πœ”β„Ž βˆ’ πœ”, (2.84)

    where πœ”β„Ž, and πœ” is a solution of problem (2.64) and (2.59) respectively. Then for πœ–β„Ž, we have

    πœ–β„Ž = π΅πœ–β„Ž + (π΅πœ” βˆ’ πœ”) on Ξ β„Ž, πœ–β„Ž = 0 on 𝛾

    β„Ž. (2.85)

    We represent a solution of system (2.85) as follows

    πœ–β„Ž = βˆ‘ πœ–β„Ž, π‘˜ 𝑁(β„Ž) = [

    1

    2β„Žmin{π‘Ž, 𝑏}] ,

    𝑁(β„Ž)

    π‘˜=1

    (2.86)

  • 33

    where πœ–β„Žπ‘˜ is a solution of the system

    πœ–β„Žπ‘˜ = π΅πœ–β„Ž

    π‘˜ + π‘“β„Žπ‘˜ on Ξ β„Ž, πœ–β„Ž

    π‘˜ = 0 on Ξ³β„Ž, π‘˜ = 1,2, … ,𝑁(β„Ž) (2.87)

    π‘“β„Žπ‘˜ = {

    π΅πœ” βˆ’ πœ” on Ξ β„Žπ‘˜ ,

    0 on Ξ β„Ž Ξ β„Žπ‘˜β„ .

    (2.88)

    By virtue of (2.87), (2.88) and Lemma 2.4 for each π‘˜ , 1 ≀ π‘˜ ≀ 𝑁(β„Ž), follows the inequality

    |πœ–β„Žπ‘˜(π‘₯, 𝑦)| ≀ π‘„β„Ž

    π‘˜(π‘₯, 𝑦) max(π‘₯,𝑦)βˆˆΞ β„Ž

    |(π΅πœ” βˆ’ πœ”)|, on Ξ Μ…β„Ž. (2.89)

    On the basis of (2.84), (2.86) and (2.89) we have

    max(π‘₯,𝑦)βˆˆΞ β„Ž

    |πœ–β„Ž| ≀ βˆ‘ 6π‘˜ max(π‘₯,𝑦)βˆˆΞ β„Ž

    |(π΅πœ” βˆ’ πœ”)|

    𝑁(β„Ž)

    π‘˜=1

    ≀ βˆ‘ 6π‘˜π‘13β„Ž5+πœ†

    π‘˜3βˆ’πœ†

    𝑁(β„Ž)

    π‘˜=1

    ≀ 6π‘β„Ž5+πœ† [1 + ∫ π‘₯πœ†βˆ’2

    𝑁(β„Ž)

    1

    𝑑π‘₯]

    ≀ 6𝑐14β„Ž5+πœ† [1 +

    π‘₯πœ†βˆ’1

    πœ† βˆ’ 1|1

    π‘Ž

    β„Ž

    ]

    = 6𝑐14β„Ž5+πœ† [1 + (

    π‘Ž

    β„Ž)

    πœ†βˆ’1 1

    πœ† βˆ’ 1βˆ’

    1

    πœ† βˆ’ 1]

    = 6𝑐14β„Ž5+πœ† + 6𝑐1β„Ž

    6π‘Žπœ†βˆ’1 βˆ’ 6𝑐1β„Ž5+πœ†

    ≀ 𝑐4β„Ž5+πœ†.

    Theorem 2.3 is proved. ∎

  • 34

    CHAPTER 3

    ON THE HIGH ORDER CONVERGENCE OF THE DIFFERENCE SOLUTION OF

    LAPLACE’S EQUATION IN A RECTANGULAR PARALLELEPIPED

    In this Chapter, we consider the Dirichlet problem for the Laplace equation on a rectangular

    parallelepiped. The boundary functions on the faces of a parallelepiped are supposed to have the

    seventh order derivatives satisfying the HΓΆlder condition, and on the edges the second, fourth

    and sixth order derivatives satisfy the compatibility conditions. We present and justify

    difference schemes on a cubic grid for obtaining the solution of the Dirichlet problem, its first

    and pure second derivatives. For the approximate solution a pointwise estimation for the error

    of order 𝑂(πœŒβ„Ž6) with the weight function 𝜌, where 𝜌 = 𝜌(π‘₯1, π‘₯2, π‘₯3) is the distance from the

    current grid point (π‘₯1, π‘₯2, π‘₯3) to the boundary of the parallelepiped, is obtained. This estimation

    gives an additional accuracy of the finite difference solution near the boundary of the

    parallelepiped, which is used to approximate the first derivatives with uniform error of order

    𝑂(β„Ž6|ln β„Ž|) . The approximation of the pure second derivatives are obtained with uniform

    accuracy 𝑂(β„Ž5+πœ†), 0 < πœ† < 1.

    3.1 The Dirichlet Problem in a Rectangular Parallelepiped

    Let 𝑅 = {(π‘₯1, π‘₯2, π‘₯3): 0 < π‘₯𝑖 < π‘Žπ‘– , 𝑖 = 1,2,3} be an open rectangular parallelepiped; Γ𝑗 (𝑗 =

    1,2, … ,6) be its faces including the edges; Γ𝑗 for 𝑗 = 1,2,3 (for 𝑗 = 4,5,6) belongs to the plane

    π‘₯𝑗 = 0 (to the plane π‘₯π‘—βˆ’3 = π‘Žπ‘—βˆ’3), let Ξ“ = ⋃ Ξ“j6j=1 be the boundary of the parallelepiped, let 𝛾

    be the union of the edges of 𝑅 , and let Ξ“πœ‡πœˆ = Ξ“πœ‡ βˆͺ Ξ“πœˆ . We say that 𝑓 ∈ πΆπ‘˜,πœ†(𝐷) , if 𝑓 has

    continuous π‘˜ βˆ’ π‘‘β„Ž deravatives on 𝐷 satisfying HΓΆlder condition with exponent πœ† ∈ (0,1).

    We consider the following boundary value problem

    βˆ†π‘’ = 0 on 𝑅, 𝑒 = πœ‘π‘— on Γ𝑗, 𝑗 = 1,2, … ,6, (3.1)

  • 35

    where βˆ†β‰‘πœ•2

    πœ•π‘₯12 +

    πœ•2

    πœ•π‘₯22 +

    πœ•2

    πœ•π‘₯32, πœ‘π‘— are given functions.

    Assume that

    πœ‘π‘— ∈ 𝐢7,πœ†(Γ𝑗), 0 < πœ† < 1, 𝑗 = 1,2, … ,6, (3.2)

    πœ‘πœ‡ = πœ‘πœˆ on Ξ³πœ‡πœˆ , (3.3)

    πœ•2πœ‘πœ‡

    πœ•π‘‘πœ‡2+

    πœ•2πœ‘πœˆπœ•π‘‘πœˆ2

    +πœ•2πœ‘πœ‡

    πœ•π‘‘πœ‡πœˆ2= 0 on Ξ³πœ‡πœˆ , (3.4)

    πœ•4πœ‘πœ‡

    πœ•π‘‘πœ‡4+

    πœ•4πœ‘πœ‡

    πœ•π‘‘πœ‡2πœ•π‘‘πœ‡πœˆ2=

    πœ•4πœ‘πœˆπœ•π‘‘πœˆ4

    +πœ•4πœ‘πœ‡

    πœ•π‘‘πœˆ2πœ•π‘‘πœˆπœ‡2 on Ξ³πœ‡πœˆ , (3.5)

    πœ•6πœ‘πœ‡

    πœ•π‘‘πœ‡6

    +πœ•6πœ‘πœ‡

    πœ•π‘‘πœ‡4πœ•π‘‘πœ‡πœˆ2+

    πœ•6πœ‘πœ‡

    πœ•π‘‘πœ‡4πœ•π‘‘πœˆ2=

    πœ•6πœ‘πœ‡

    πœ•π‘‘πœ‡2πœ•π‘‘πœˆ4+

    πœ•6πœ‘πœˆ

    πœ•π‘‘πœˆ6 +

    πœ•6πœ‘πœ‡

    πœ•π‘‘πœˆ4πœ•π‘‘πœ‡πœˆ2 on Ξ³πœ‡πœˆ . (3.6)

    Where 1 ≀ πœ‡ < 𝜈 ≀ 6, 𝜈 βˆ’ πœ‡ β‰  3, tπœ‡πœˆ is an element in Ξ³πœ‡πœˆ , tπœ‡ and t𝜈 is an element of the

    normal to Ξ³πœ‡πœˆ on the face Ξ“πœ‡ and Ξ“πœˆ, respectively. The boundary function as hole are continuous

    on the edges and satisfy second, fourth, and sixth compatibility conditions which result from

    Laplace equations. Indeed, (3.3) is differentiated twice with respect to tπœ‡ . Then, it is

    differentiated twice with respect to t𝜈 . We have

    πœ•2πœ‘πœ‡

    πœ•π‘‘πœ‡2=

    πœ•2πœ‘πœˆπœ•π‘‘πœ‡2

    πœ•2

    πœ•π‘‘πœˆ2(πœ•2πœ‘πœ‡

    πœ•π‘‘πœ‡2) =

    πœ•2

    πœ•π‘‘πœˆ2(πœ•2πœ‘πœˆπœ•π‘‘πœ‡2

    )

  • 36

    πœ•4πœ‘πœ‡

    πœ•π‘‘πœ‡2πœ•π‘‘πœˆ2=

    πœ•4πœ‘πœˆπœ•π‘‘πœ‡2πœ•π‘‘πœˆ2

    . (3.7)

    (3.4) is differentiated twice with respect to tπœ‡

    πœ•2

    πœ•π‘‘πœ‡2(πœ•2πœ‘πœ‡

    πœ•π‘‘πœ‡2+

    πœ•2πœ‘πœˆπœ•π‘‘πœˆ2

    +πœ•2πœ‘πœ‡

    πœ•π‘‘πœ‡πœˆ2) = 0.

    We have

    πœ•4πœ‘πœ‡

    πœ•π‘‘πœ‡4+

    πœ•4πœ‘πœˆπœ•π‘‘πœ‡2πœ•π‘‘πœˆ2

    +πœ•4πœ‘πœ‡

    πœ•π‘‘πœ‡πœˆ2 πœ•π‘‘πœ‡2= 0. (3.8)

    (3.4) is differentiated twice with respect to t𝜈

    πœ•2

    πœ•π‘‘πœˆ2(πœ•2πœ‘πœ‡

    πœ•π‘‘πœ‡2+

    πœ•2πœ‘πœˆπœ•π‘‘πœˆ2

    +πœ•2πœ‘πœ‡

    πœ•π‘‘πœ‡πœˆ2) = 0.

    We have

    πœ•4πœ‘πœ‡

    πœ•π‘‘πœ‡2πœ•π‘‘πœˆ2+

    πœ•4πœ‘πœˆπœ•π‘‘πœˆ4

    +πœ•4πœ‘πœ‡

    πœ•π‘‘πœ‡πœˆ2 πœ•π‘‘πœˆ2= 0. (3.9)

    From (3.7), (3.8) and (3.9) follows

    πœ•4πœ‘πœ‡

    πœ•π‘‘πœ‡4+

    πœ•4πœ‘πœ‡

    πœ•π‘‘πœ‡2πœ•π‘‘πœ‡πœˆ2=

    πœ•4πœ‘πœˆπœ•π‘‘πœˆ4

    +πœ•4πœ‘πœ‡

    πœ•π‘‘πœˆ2πœ•π‘‘πœˆπœ‡2.

    (3.7) is differentiated twice with respect to tπœ‡πœˆ

  • 37

    πœ•2

    πœ•π‘‘πœ‡πœˆ2(

    πœ•4πœ‘πœ‡

    πœ•π‘‘πœ‡2πœ•π‘‘πœ‡πœˆ2) =

    πœ•2

    πœ•π‘‘πœ‡πœˆ2(

    πœ•4πœ‘πœˆπœ•π‘‘πœ‡2πœ•π‘‘πœˆ2

    )

    πœ•6πœ‘πœ‡

    πœ•π‘‘πœ‡2πœ•π‘‘πœˆ2πœ•π‘‘πœ‡πœˆ2=

    πœ•6πœ‘πœˆπœ•π‘‘πœ‡2πœ•π‘‘πœˆ2πœ•π‘‘πœ‡πœˆ2

    . (3.10)

    (3.5) is differentiated twice with respect to tπœ‡ follows

    πœ•2

    πœ•π‘‘πœ‡2(πœ•4πœ‘πœ‡

    πœ•π‘‘πœ‡4+

    πœ•4πœ‘πœ‡

    πœ•π‘‘πœ‡2πœ•π‘‘πœ‡πœˆ2) =

    πœ•2

    πœ•π‘‘πœ‡2(πœ•4πœ‘πœˆπœ•π‘‘πœˆ4

    +πœ•4πœ‘πœ‡

    πœ•π‘‘πœˆ2πœ•π‘‘πœˆπœ‡2),

    πœ•6πœ‘πœ‡

    πœ•π‘‘πœ‡6

    +πœ•6πœ‘πœ‡

    πœ•π‘‘πœ‡4πœ•π‘‘πœˆπœ‡2=

    πœ•6πœ‘πœˆπœ•π‘‘πœ‡2πœ•π‘‘πœˆ4

    +πœ•6πœ‘πœ‡

    πœ•π‘‘πœ‡2πœ•π‘‘πœˆ2πœ•π‘‘πœˆπœ‡2. (3.11)

    (3.5) is differentiated twice with respect to t𝜈 follows

    πœ•2

    πœ•π‘‘πœˆ2(πœ•4πœ‘πœ‡

    πœ•π‘‘πœ‡4+

    πœ•4πœ‘πœ‡

    πœ•π‘‘πœ‡2πœ•π‘‘πœ‡πœˆ2) =

    πœ•2

    πœ•π‘‘πœˆ2(πœ•4πœ‘πœˆπœ•π‘‘πœˆ4

    +πœ•4πœ‘πœ‡

    πœ•π‘‘πœˆ2πœ•π‘‘πœˆπœ‡2)

    πœ•6πœ‘πœ‡

    πœ•π‘‘πœ‡4πœ•π‘‘πœˆ2+

    πœ•6πœ‘πœ‡

    πœ•π‘‘πœ‡2πœ•π‘‘πœˆ2πœ•π‘‘πœˆπœ‡2=

    πœ•6πœ‘πœˆ

    πœ•π‘‘πœˆ6 +

    πœ•6πœ‘πœ‡

    πœ•π‘‘πœˆ4πœ•π‘‘πœˆπœ‡2. (3.12)

    From (3.10), (3.11) and (3.12) follows

    πœ•6πœ‘πœ‡

    πœ•π‘‘πœ‡6

    +πœ•6πœ‘πœ‡

    πœ•π‘‘πœ‡4πœ•π‘‘πœ‡πœˆ2+

    πœ•6πœ‘πœ‡

    πœ•π‘‘πœ‡4πœ•π‘‘πœˆ2=

    πœ•6πœ‘πœ‡

    πœ•π‘‘πœ‡2πœ•π‘‘πœˆ4+

    πœ•6πœ‘πœˆ

    πœ•π‘‘πœˆ6 +

    πœ•6πœ‘πœ‡

    πœ•π‘‘πœˆ4πœ•π‘‘πœ‡πœˆ2.

    Lemma 3.1 The solution 𝑒 of the problem (3.1) is from 𝐢7,πœ†(οΏ½Μ…οΏ½),

    The proof of Lemma 3.1 follows from Theorem 2.1 (Volkov, 1969).

  • 38

    Lemma 3.2 Let 𝜌 = (π‘₯1, π‘₯2, π‘₯3) be the distance from the current point of the open

    parallelepiped 𝑅 to its boundary and let πœ• πœ•π‘™β„ ≑ 𝛼1πœ•

    πœ•π‘₯1+ 𝛼2

    πœ•

    πœ•π‘₯2+ 𝛼3

    πœ•

    πœ•π‘₯3, 𝛼1

    2 + 𝛼22 + 𝛼3

    2 = 1.

    Then the next inequality holds

    |πœ•8𝑒(π‘₯1, π‘₯2, π‘₯3)

    πœ•π‘™8| ≀ π‘πœŒπœ†βˆ’1(π‘₯1, π‘₯2, π‘₯3), (π‘₯1, π‘₯2, π‘₯3) ∈ 𝑅, (3.13)

    where 𝑐 is a constant independent of the direction of differentiation πœ• πœ•π‘™β„ , 𝑒 is a solution of

    problem (3.1).

    Proof. We choose an arbitrary point (π‘₯10, π‘₯20, π‘₯30) ∈ R. Let 𝜌0 = 𝜌(π‘₯10, π‘₯20, π‘₯30), and 𝜎0 βŠ‚ RΜ…

    be the closed ball of radius 𝜌0 centred at (π‘₯10, π‘₯20, π‘₯30). Consider the harmonic function on 𝑅

    𝑣(π‘₯1, π‘₯2, π‘₯3) =πœ•7𝑒(π‘₯1, π‘₯2, π‘₯3)

    πœ•π‘™7βˆ’

    πœ•7𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘™7. (3.14)

    As it follows from Theorem 2.1 in (Volkov, 1969) the solution 𝑒 of problem (3.1) which

    satisfies the conditions (3.3)-(3.6) belongs to the class 𝐢7,πœ†(RΜ…), for 0 < πœ† < 1. Then for the

    function (3.14) we have

    max(π‘₯1,π‘₯2,π‘₯3)βˆˆοΏ½Μ…οΏ½0

    |𝑣(π‘₯1, π‘₯2, π‘₯3)| ≀ 𝑐0𝜌0πœ†, (3.15)

    where 𝑐0 is a constant independent of the point (π‘₯10, π‘₯20, π‘₯30) ∈ 𝑅 or the direction of πœ• πœ•π‘™β„ .

    By using estimation (3.15) and applying Lemma 3 from (Mikeladze, 1978) we have

    |πœ•

    πœ•π‘™(πœ•7𝑒(π‘₯1, π‘₯2, π‘₯3)

    πœ•π‘™7βˆ’

    πœ•7𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘™7)| ≀ 𝑐

    𝜌0πœ†

    𝜌0,

    or

  • 39

    |πœ•8𝑒(π‘₯1, π‘₯2, π‘₯3)

    πœ•π‘™8| ≀ π‘πœŒ0

    πœ†βˆ’1(π‘₯1, π‘₯2, π‘₯3),

    where 𝑐 is a constant independent of the point (π‘₯10, π‘₯20, π‘₯30) ∈ 𝑅 or the direction of πœ• πœ•π‘™β„ . Since

    the point (π‘₯10, π‘₯20, π‘₯30) ∈ 𝑅 is arbitrary, inequality (3.13) holds true. ∎

    Let 𝑣 be a solution of the problem

    βˆ†π‘£ = 0 on 𝑅, 𝑣 = Ψ𝑗 on Γ𝑗, 𝑗 = 1,2, … ,6, (3.16)

    where Ψ𝑗, 𝑗 = 1,2, … ,6 are given functions and

    Ψ𝑗 ∈ 𝐢5,πœ†(Γ𝑗), 0 < πœ† < 1, 𝑗 = 1,2, … ,6, Ξ¨πœ‡ = Ψ𝜐 on Ξ³πœ‡πœˆ, (3.17)

    πœ•2Ξ¨πœ‡

    πœ•π‘‘πœ‡2+

    πœ•2Ξ¨πœπœ•π‘‘πœ2

    +πœ•2Ξ¨πœ‡

    πœ•π‘‘πœ‡πœ2= 0 on Ξ³πœ‡πœ, (3.18)

    πœ•4Ξ¨πœ‡

    πœ•π‘‘πœ‡4+

    πœ•4Ξ¨πœ‡

    πœ•π‘‘πœ‡2πœ•π‘‘πœ‡πœ2=

    πœ•4Ξ¨πœπœ•π‘‘πœ4

    +πœ•4Ξ¨πœ‡

    πœ•π‘‘πœ2πœ•π‘‘π‘£πœ‡2 on Ξ³πœ‡πœˆ . (3.19)

    Lemma 3.3 The next inequality is true

    |πœ•8𝑣(π‘₯1, π‘₯2, π‘₯3)

    πœ•π‘™8| ≀ 𝑐1𝜌

    πœ†βˆ’3(π‘₯1, π‘₯2, π‘₯3), (π‘₯1, π‘₯2, π‘₯3) ∈ 𝑅, (3.20)

    where 𝑐1 is a constant independent of the direction of differentiation πœ• πœ•π‘™β„ .

    Proof. We choose an arbitrary point (π‘₯10, π‘₯20, π‘₯30) ∈ R. Let 𝜌0 = 𝜌(π‘₯10, π‘₯20, π‘₯30) and 𝜎 βŠ‚ RΜ…

    be the closed ball of radius 𝜌0 centred at (π‘₯10, π‘₯20, π‘₯30). Consider the harmonic function on 𝑅

  • 40

    𝜐(π‘₯1, π‘₯2, π‘₯3) =πœ•5𝑣(π‘₯1, π‘₯2, π‘₯3)

    πœ•π‘™5βˆ’

    πœ•5𝑣(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘™5. (3.21)

    As it follows from Theorem 2.1 in (Volkov, 1969) the solution 𝑒 of problem (3.1), which

    satisfies the conditions (3.17)-(3.19) belongs to the class 𝐢5,πœ†(RΜ…), 0 < πœ† < 1. Then for

    the function (3.21) we have

    max(π‘₯1,π‘₯2,π‘₯3)βˆˆοΏ½Μ…οΏ½0

    |𝜐(π‘₯1, π‘₯2, π‘₯3)| ≀ 𝑐0𝜌0πœ†, (3.22)

    where 𝑐0 is a constant independent of the point (π‘₯10, π‘₯20, π‘₯30) ∈ R or the direction of πœ• πœ•π‘™β„ .

    Since 𝑒 is harmonic in 𝑅, by using estimation (3.22) and applying Lemma 3 in (Mikeladze,

    1978) we have

    |πœ•3

    πœ•π‘™3(πœ•5𝑣(π‘₯1, π‘₯2, π‘₯3)

    πœ•π‘™5βˆ’

    πœ•5𝑣(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘™5)| ≀ 𝑐

    𝜌0πœ†

    𝜌03 ,

    or

    |πœ•8𝑣(π‘₯1, π‘₯2, π‘₯3)

    πœ•π‘™8| ≀ π‘πœŒ0

    πœ†βˆ’3(π‘₯1, π‘₯2, π‘₯3),

    where 𝑐 is a constant independent of the point (π‘₯10, π‘₯20, π‘₯30) ∈ R or the direction of πœ• πœ•π‘™β„ .

    Since the point (π‘₯10, π‘₯20, π‘₯30) ∈ R is arbitrary, inequality (3.20) holds true. ∎

    3.2 27-Point Finite Difference Method for the Dirichlet Problem

    Let β„Ž > 0, and π‘Žπ‘– β„Žβ„ β‰₯ 6 where 𝑖 = 1,2,3, … ,6 integers. We assign π‘…β„Ž a cubic grid on 𝑅, with

    step β„Ž, obtained by the planes π‘₯𝑖 = 0, β„Ž, 2β„Ž,… , 𝑖 = 1,2,3. Let π·β„Ž be a set of nodes of this grid,

    Rβ„Ž = 𝑅 ∩ π·β„Ž, Ξ“π‘—β„Ž = Γ𝑗 ∩ 𝐷

    β„Ž, and Ξ“β„Ž = Ξ“1β„Ž βˆͺ Ξ“2

    β„Ž βˆͺ …βˆͺ Ξ“6β„Ž.

    Let the operator β„œ be defined as follows by (Mikeladze, 1938).

  • 41

    β„œπ‘’(π‘₯1, π‘₯2, π‘₯3) =1

    128(14 βˆ‘ 𝑒𝑝 + 3 βˆ‘ π‘’π‘ž

    18

    π‘ž=7(2)

    6

    𝑝=1(1)

    + βˆ‘ π‘’π‘Ÿ

    26

    π‘Ÿ=19(3)

    ),

    (π‘₯1, π‘₯2, π‘₯3) ∈ 𝑅, (3.23)

    where the sum βˆ‘(π‘˜) is taken over the grid nodes that are at a distance of βˆšπ‘˜β„Ž from the point

    (π‘₯1, π‘₯2, π‘₯3), and 𝑒𝑝, π‘’π‘ž and π‘’π‘Ÿ are the values of 𝑒 at the corresponding grid points.

    The red points have distance β„Ž from the center point, the white points have

    distance √2β„Ž from the center point and the blue points have distance √3β„Ž

    from center point.

    Let 𝑐, 𝑐0, 𝑐1, … be constants which are independent of β„Ž and the nearest factor, and for simplicity

    identical notation will be used for various constants.

    We consider the finite difference approximations of problem (3.1):

    π‘’β„Ž = β„œπ‘’β„Ž on Rβ„Ž, π‘’β„Ž = πœ‘π‘— on Γ𝑗

    β„Ž, 𝑗 = 1,2, … ,6. (3.24)

    Figure 3.1: 26 points around center point using

    operator β„œ

  • 42

    By the maximum principle system (3.24) has a unique solution.

    Let π‘…π‘˜β„Ž be the set of the grid nodes Rβ„Ž whose distance from Ξ“ is π‘˜β„Ž. It is obvious that 1 ≀ π‘˜ ≀

    𝑁(β„Ž), where

    𝑁(β„Ž) = [1

    2β„Žπ‘šπ‘–π‘›{π‘Ž1, π‘Ž2, π‘Ž3}], (3.25)

    [𝑑] is the integer part of 𝑑.

    We define for 1 ≀ π‘˜ ≀ 𝑁(β„Ž) the function

    π‘“β„Žπ‘˜ = {

    1, 𝜌(π‘₯1, π‘₯2, π‘₯3) = π‘˜β„Ž,

    0, 𝜌(π‘₯1, π‘₯2, π‘₯3) β‰  π‘˜β„Ž. (3.26)

    Consider two systems of grid equations

    π‘£β„Ž = π΄π‘£β„Ž + π‘”β„Ž, on π‘…β„Ž, π‘£β„Ž = 0 on Ξ“β„Ž, (3.27)

    οΏ½Μ…οΏ½β„Ž = π΄οΏ½Μ…οΏ½β„Ž + οΏ½Μ…οΏ½β„Ž, on π‘…β„Ž, οΏ½Μ…οΏ½β„Ž = 0 on Ξ“β„Ž, (3.28)

    where π‘”β„Ž and οΏ½Μ…οΏ½β„Ž are given functions and |π‘”β„Ž| ≀ οΏ½Μ…οΏ½β„Ž on π‘…β„Ž.

    Lemma 3.4 The solution π‘£β„Ž and οΏ½Μ…οΏ½β„Ž to systems (3.23) and (3.31) satisfy the inequality

    |π‘£β„Ž| ≀ οΏ½Μ…οΏ½β„Ž on π‘…β„Ž.

    Proof. The proof of Lemma 3.4 follows from the comparison theorem (Samarskii, 2001).

  • 43

    Lemma 3.5 The solution of the system

    π‘£β„Žπ‘˜ = β„œπ‘£β„Ž

    π‘˜ + π‘“β„Žπ‘˜ on Rβ„Ž, π‘£β„Ž

    π‘˜ = 0 on Ξ“β„Ž (3.29)

    satisfies the inequality

    π‘£β„Žπ‘˜(π‘₯1, π‘₯2, π‘₯3) ≀ π‘„β„Ž

    π‘˜, 1 ≀ π‘˜ ≀ 𝑁(β„Ž), (3.30)

    where π‘„β„Žπ‘˜ is defined as follows

    π‘„β„Žπ‘˜ = π‘„β„Ž

    π‘˜(π‘₯1, π‘₯2, π‘₯3) = {6𝜌

    β„Ž, 0 ≀ 𝜌(π‘₯1, π‘₯2, π‘₯3) ≀ π‘˜β„Ž ,

    6π‘˜, 𝜌(π‘₯1, π‘₯2, π‘₯3) > π‘˜β„Ž . (3.31)

    Proof. By virtue of (3.23) and (3.31), and in consider of Fig 3.2, we have for 0 ≀ 𝜌 = π‘˜β„Ž

    β„œπ‘„β„Žπ‘˜ =

    1

    128[14(2 Γ— 6(π‘˜ βˆ’ 1) + 4 Γ— 6π‘˜) + 3(7 Γ— 6(π‘˜ βˆ’ 1) + 5 Γ— 6π‘˜)

    +6 Γ— 6(π‘˜ βˆ’ 1) + 2 Γ— 6π‘˜] =1

    128[504π‘˜ βˆ’ 168

    +216π‘˜ βˆ’ 126 + 48π‘˜ βˆ’ 36] =1

    128[768π‘˜ βˆ’ 330]

    = 6π‘˜ βˆ’330

    128,

    which leads to

    π‘„β„Žπ‘˜ βˆ’ β„œπ‘„β„Ž

    π‘˜ =330

    128> 1 = π‘“β„Ž

    π‘˜.

  • 44

    In consider of Fig 3.3 for 𝜌 > π‘˜β„Ž, then

    β„œπ‘„β„Žπ‘˜ =

    1

    128[14(6 Γ— 6π‘˜) + 3(12 Γ— 6π‘˜) + 8 Γ— 6π‘˜] =

    768

    128π‘˜ = 6π‘˜,

    which leads to

    π‘„β„Žπ‘˜ βˆ’ β„œπ‘„β„Ž

    π‘˜ = 0.

    Figure 3.2: The selected region in 𝑅 is 𝜌 = π‘˜β„Ž

  • 45

    In consider of Fig 3.4 for 𝜌 < π‘˜β„Ž, then

    β„œπ‘„β„Žπ‘˜ =

    1

    128[14(6(π‘˜ βˆ’ 2) + 4 Γ— 6(π‘˜ βˆ’ 1) + 6π‘˜) + 3(4 Γ— 6(π‘˜ βˆ’ 2)

    +4 Γ— 6(π‘˜ βˆ’ 1) + 4 Γ— 6π‘˜) + 4 Γ— 6(π‘˜ βˆ’ 2) + 4 Γ— 6π‘˜]

    = 504π‘˜ βˆ’ 504 + 216π‘˜ βˆ’ 216 + 48π‘˜ βˆ’ 48

    =768

    128π‘˜ βˆ’

    768

    128= 6π‘˜ βˆ’ 6 = 6(π‘˜ βˆ’ 1),

    which leads to

    π‘„β„Žπ‘˜ βˆ’ β„œπ‘„β„Ž

    π‘˜ = 0.

    Figure 3.3: The selected region in 𝑅 is 𝜌 > π‘˜β„Ž

  • 46

    From the above calculations we have

    π‘„β„Žπ‘˜ = β„œπ‘„β„Ž

    π‘˜ + π‘žβ„Žπ‘˜ on Rβ„Ž, π‘„β„Ž

    π‘˜ = 0 on Ξ“β„Ž, π‘˜ = 1,… ,𝑁(β„Ž), (3.32)

    where |π‘žβ„Žπ‘˜| β‰₯ 1.

    On the basis of (3.26), (3.31), (3.32) and Lemma 3.4, we obtain

    π‘£β„Žπ‘˜ ≀ π‘„β„Ž

    π‘˜ for all π‘˜, 1 ≀ π‘˜ ≀ 𝑁(β„Ž).

    ∎

    Introducing the notation π‘₯0 = (π‘₯10, π‘₯20, π‘₯30) , we used Taylor’s formulate to represent the

    solution of Dirichlet problem around some point π‘₯0 ∈ π‘…β„Ž

    𝑒(π‘₯1, π‘₯2, π‘₯3) = 𝑝7(π‘₯1, π‘₯2, π‘₯3; π‘₯0) + π‘Ÿ8(π‘₯1, π‘₯2, π‘₯3; π‘₯0), (3.33)

    where 𝑝7 is the seventh order Taylor’s polynomial and π‘Ÿ8(π‘₯1, π‘₯2, π‘₯3; π‘₯0) is the remainder term.

    Figure 3.4: The selected region in 𝑅 is 𝜌 < π‘˜β„Ž

  • 47

    Here,

    𝑝7(π‘₯10, π‘₯20, π‘₯30; π‘₯0) = 𝑒(π‘₯10, π‘₯20, π‘₯30; π‘₯0) + π‘Ÿ8(π‘₯10, π‘₯20, π‘₯30; π‘₯0) = 0.

    Lemma 3.6 It is true that

    β„œπ‘’(π‘₯10, π‘₯20, π‘₯30) = 𝑒(π‘₯10, π‘₯20, π‘₯30) + β„œπ‘Ÿ8(π‘₯10, π‘₯20, π‘₯30; π‘₯0), (π‘₯10, π‘₯20, π‘₯30) ∈

    π‘…β„Ž,

    Proof. Let 𝑝7(π‘₯10, π‘₯20, π‘₯30; π‘₯0) be a Taylor’s polynomial.

    By Direct calculations we have

    β„œπ‘7(π‘₯10, π‘₯20, π‘₯30) = 𝑒(π‘₯10, π‘₯20, π‘₯30) +1

    128[πœ•2

    πœ•π‘₯12 (

    πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯12

    +πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯22 +

    πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯32 ) +

    πœ•2

    πœ•π‘₯22 (

    πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯12

    +πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯22 +

    πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯32 ) +

    πœ•2

    πœ•π‘₯32 (

    πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯12

    +πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯22 +

    πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯32 )] +

    1

    1536[πœ•4

    πœ•π‘₯14 (

    πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯12

    +πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯22 +

    πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯32 ) +

    πœ•4

    πœ•π‘₯24 (

    πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯12

    +πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯22 +

    πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯32 ) +

    πœ•4

    πœ•π‘₯34 (

    πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯12

    +πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯22 +

    πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯32 ) + 4

    πœ•4

    πœ•π‘₯12πœ•π‘₯2

    2 (πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯12

    +πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯22 +

    πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯32 ) + 4

    πœ•4

    πœ•π‘₯12πœ•π‘₯3

    2 (πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯12

    +πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯22 +

    πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯32 ) + 4

    πœ•4

    πœ•π‘₯22πœ•π‘₯3

    2 (πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯12

    +πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯22 +

    πœ•2𝑒(π‘₯10, π‘₯20, π‘₯30)

    πœ•π‘₯32 )]

    = 𝑒(π‘₯10, π‘₯20, π‘₯30).

  • 48

    Since 𝑒 is harmonic function, all terms on the right-hand side of this equality vanished except

    the first term. Thus

    β„œπ‘7(π‘₯10, π‘₯20, π‘₯30) = 𝑒(π‘₯10, π‘₯20, π‘₯30).

    Combining this with (3.33) and recalling the linearity of β„œ, Lemma 3.6. is proved. ∎

    Lemma 3.7 Let 𝑒 be a solution of problem (3.1) The inequality holds

    max(π‘₯1,π‘₯2,π‘₯3)∈Rπ‘˜β„Ž

    |β„œπ‘’ βˆ’ 𝑒| ≀ π‘β„Ž7+πœ†

    π‘˜1βˆ’πœ†, π‘˜ = 1,… ,𝑁(β„Ž). (3.34)

    Proof. Let (π‘₯01, π‘₯02, π‘₯03) be a point of 𝑅1β„Ž, and let

    𝑅0 = {(π‘₯1, π‘₯2, π‘₯3): |π‘₯𝑖 βˆ’ π‘₯𝑖0| < β„Ž, 𝑖 = 1,2,3 } (3.35)

    be an elementary cube, some faces of which lie on the boundary of the rectangular parallelepiped

    𝑅.

    On the vertices of 𝑅0, and on the center of its faces and edges lie the nodes of which the function

    values are used to evaluate β„œ(π‘₯10, π‘₯20, π‘₯30). We represent a solution of problem (3.1) in some

    neighborhood of π‘₯0 = (π‘₯10, π‘₯20, π‘₯30) ∈ R1β„Ž by Taylor’s formula

    𝑒(π‘₯1, π‘₯2, π‘₯3) = 𝑝7(π‘₯1, π‘₯2, π‘₯3; π‘₯0) + π‘Ÿ8(π‘₯1, π‘₯2, π‘₯3; π‘₯0), (3.36)

    where 𝑝7(π‘₯1, π‘₯2, π‘₯3; π‘₯0) is the seventh order Taylor’s polynomial, π‘Ÿ8(π‘₯1, π‘₯2, π‘₯3; π‘₯0) is the

    remainder term. Taking into account the function 𝑒 is harmonic, hence by Lemma 3.6 we have

    β„œπ‘7(π‘₯10, π‘₯20, π‘₯30; π‘₯0) = 𝑒(π‘₯10, π‘₯20, π‘₯30). (3.37)

  • 49

    Now, we estimate π‘Ÿ8 at the nodes of the operator β„œ. We take node (π‘₯10 + β„Ž, π‘₯20, π‘₯3