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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 LAPLACEβS
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
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INVESTIGATION OF THE FINITE-DIFFERENCE
METHOD FOR APPROXIMATING THE SOLUTION
AND ITS DERIVATIVES OF THE DIRICHLET
PROBLEM FOR 2D AND 3D LAPLACEβS
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 LAPLACEβS
EQUATION
Approval of Director of Graduate School of
Applied Sciences
Prof. Dr. Nadire ΓAVUΕ
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
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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:
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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.
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To my parents and familyβ¦
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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 π’ β π’β of the
9-point solution π’β at each grid point (π₯, π¦) a pointwise estimation π(πβ6) is obtained, where
π = π(π₯, π¦) is the distance from the current grid point to the boundary of rectangle Ξ , π’ is the
exact solution and β is the grid step. The solution of difference problems constructed for the
approximate values of the first derivatives converge with orders π(β6) and for pure second
derivatives converge with orders π(β5+π), 0 < π < 1. In a rectangular parallelepiped domain,
the seventh derivatives for the boundary functions of the Dirichlet problem on the faces of the
parallelepiped π 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 π’ β π’β of
the 27-point solution π’β at each grid point (π₯1, π₯2, π₯3) , a pointwise estimation π(πβ6) is
obtained, where π = π(π₯1, π₯2, π₯3) is the distance from the current grid point to the boundary of
the parallelepiped π . The solution of the constructed 27- point difference problems for the
approximate values of the first converge with orders π(β6 ln β) and for pure second derivatives
converge with orders π(β5+π). 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 π(β6), 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
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ΓZET
Bir dikdârtgen içindeki Laplace denklemi ve dikdârtgenler prizması için Dirichlet problem
dΓΌΕΓΌnΓΌlmΓΌΕtΓΌr. TanΔ±m bΓΆlgesinin dikdΓΆrtgen olduΔu durumda dikdΓΆrtgenin Ξ kenarlarΔ±nda
verilen sΔ±nΔ±r fonksiyonlarΔ±nΔ±n yedinci tΓΌrevlerinin HΓΆlder ΕartΔ±nΔ± saΔladΔ±ΔΔ± Kabul edildi.
KΓΆΕelerde sΓΌreklilik ΕartΔ±nΔ±n yanΔ±nda Laplace denkleminden sonuΓ§lanan kΓΆΕelerin komΕu
kenarlarΔ±nda verilen sΔ±nΔ±r deΔer fonksiyonlarΔ±nΔ±n ikinci, dΓΆrdΓΌncΓΌve altΔ±ncΔ± tΓΌrevleri iΓ§in
uyumluluk ΕartlarΔ± da saΔlandΔ±. Bu Εartlar altΔ±nda Dirichlet probleminin kare Δ±zgara ΓΌzerinde
çâzΓΌmΓΌ iΓ§in π’ β π’β yaklaΕΔ±mΔ± π(πβ6) olarak dΓΌzgΓΌn yakΔ±nsadΔ±ΔΔ± bulunmuΕtur, burada π’β 9-
nokta yaklaΕΔ±mΔ± kullanΔ±ldΔ±ΔΔ±nda elde elinen yaklaΕΔ±k çâzΓΌm, πππππππ saΔlayan kesin çâzΓΌm,
π = π(π₯, π¦) , dikdΓΆrtgen sΔ±nΔ±rΔ±nΔ± iΕaret eden mevcut Δ±zgara uzunlΔ±Δu ve h, Δ±zgara adΔ±mΔ±dΔ±r.
ΓΓΆzΓΌmΓΌn birinci ve pΓΌr tΓΌrevleri iΓ§in oluΕturulan fark problemlerinin sΔ±rasΔ±yla çâzΓΌmleri, π(β6)
ve π(β5+π) , 0 < π < 1 mertebesi ile yakΔ±nsar. TanΔ±m bΓΆlgesinin dikdΓΆrtgenler prizmasΔ±
olduΔu durumda prizmanΔ±n yΓΌzeylerinde verilensΔ±nΔ±r fonksiyonlarΔ±nΔ±n yedinci tΓΌrevlerinin
HΓΆlder ΕartΔ±nΔ± saΔladΔ±ΔΔ± Kabul edildi. KΓΆΕelerde sΓΌreklilik ΕartΔ±nΔ±n yanΔ±nda Laplace
denkleminden sonuΓ§lanan kenarlarΔ±nΔ±n komΕu kenarlarΔ±nda verilen sΔ±nΔ±r deΔer fonksiyonlarΔ±nΔ±n
ikinci, dΓΆrdΓΌncΓΌ ve altΔ±ncΔ± tΓΌrevleri iΓ§in uyumluluk ΕartlarΔ±nΔ±da saΔlar. Bu Εartlar altΔ±nda
Dirichlet probleminin kΓΌp Δ±zgaralar ΓΌzerindeki çâzΓΌmΓΌ iΓ§in π’ β π’β yaklaΕΔ±mΔ± π(πβ6) olarak
bulunmuΕtur, burada π’β 27-nokta yaklaΕΔ±mΔ± kullanΔ±ldΔ±ΔΔ±nda elde edilen yaklaΕΔ±k çâzΓΌm,
problem saΔlayan kesin çâzΓΌm ve π = π(π₯1, π₯2, π₯3) prizmanΔ±n sΔ±nΔ±rΔ±nΔ± iΕaret edenmevcut Δ±zgara
ve h, Δ±zgara adΔ±mΔ±dΔ±r. ΓΓΆzΓΌmΓΌn birinci ve pΓΌr tΓΌrevleri iΓ§in oluΕturulan fark problemlerinin
sΔ±rasΔ±yla çâzΓΌmleri π(β6 ln β) , ve π(β6) , 0 < π < 1 mertebeleri ile yakΔ±nsar. Laplace
denkleminin sΔ±nΔ±r fonksiyonlarΔ± iΓ§in bazΔ± pΓΌrΓΌzsΓΌzlΓΌk koΕullarΔ± altΔ±nda Dirichlet problemi
çâzmek iΓ§in inΕa edilmiΕ ΓΌΓ§ aΕamalΔ± fark yΓΆntemi, 15 + 7 + 7 - ΕemasΔ± ile elde edilen fark çâzΓΌmΓΌ
π(β6) olarak yakΔ±nsar 27 noktalΔ± Εema yΓΆntemiyle elde edilen sonuΓ§ gibi.
Anahtar Kelimeler: Sonlu fark metodu; noktasal hata tahminleri; tΓΌrev yaklaΕΔ±mlarΔ±; dΓΌzgΓΌn
hata tahminleri; 2D ve 3D Laplace denklemleri; Laplace denkleminin sayısal çâzümü
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TABLE OF CONTENTS
ACKNOWLEDGMENTSβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦ iii
ABSTRACTβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦............... v
ΓZETβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦......... vi
LIST OF TABLESβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦........................... xi
LIST OF FIGURESβ¦β¦β¦β¦β¦..β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦ xii
CHAPTER 1: INTRODUCTION β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...β¦β¦β¦β¦β¦β¦β¦ 1
CHAPTER 2: ON THE HIGH ORDER CONVERGENCE OF THE DIFFERENCE
SOLUTION OF LAPLACEβS EQUATION IN A RECTANGLE
2.1 The Dirichlet Problem for Laplaceβs Equation on Rectangle and Some Differential
Properties of its Solutionβ¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦β¦β¦β¦... 6
2.2 Difference Equations for the Dirichlet Problem and a Pointwise Estimationβ¦β¦β¦..β¦ 8
2.3 Approximation of the First Derivatives β¦β¦β¦β¦β¦.β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦. 22
2.4 Approximation of the Pure Second Derivatives β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...β¦ 27
CHAPTER 3: ON THE HIGH ORDER CONVERGENCE OF THE DIFFERENCE
SOLUTION OF LAPLACEβS EQUATION IN A RECTANGULAR PARALLELIPIPED
3.1 The Dirichlet Problem in a Rectangular Parallelepiped β¦β¦β¦β¦β¦β¦β¦β¦................β¦ 34
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3.2 27-Point Finite Difference Method for the Dirichlet Problem β¦β¦β¦β¦β¦...β¦β¦β¦β¦... 40
3.3 Approximation of the First Derivativesβ¦β¦β¦β¦...β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦ 53
3.4 Approximation of the Pure Second Derivatives...β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦ 58
CHAPTER 4: THE THREE STAGE DIFFERENCE METHOD FOR SOLVING
DIRICHLET PROBLEM FOR LAPLACEβS EQUATION
4.1 The Dirichlet Problem on Rectangular Parallelepipedβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.......... 64
4.2 A Sixth Order Accurate Approximate Solutionβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.... 73
4.3 The First Stageβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦ 74
4.4 The Second Stageβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦β¦β¦β¦β¦...β¦β¦.β¦ 75
4.5 The Third Stageβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦ 76
CHAPTER 5: NUMERICAL EXAMPLES
5.1 Domain in the Shape of a Rectangleβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ 80
5.2 Domain in the Shape of a Rectangular Parallelepipedβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.. 82
CHAPTER 6: CONCLUSIONβ¦β¦..β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.. 86
REFERENCES β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ 87
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LIST OF TABLES
Table 5.1: The approximate of solution in problem (5.1) β¦β¦β¦β¦.β¦β¦β¦β¦β¦β¦......β¦ 81
Table 5.2: First derivative approximation results with the sixth-order accurate formulae
for problem (5.1) β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...β¦.... 82
Table 5.3: The approximate of solution in problem (5.3) β¦β¦β¦β¦β¦.β¦β¦β¦β¦β¦β¦β¦.. 83
Table 5.4: First derivative approximation results with the sixth-order accurate formulae
for problem (5.3) β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...β¦β¦ 84
Table 5.5: The approximate results for the pure second derivative β¦.β¦β¦β¦β¦β¦β¦...β¦ 84
Table 5.6: The approximate results by using a three stage difference method β¦.β¦β¦..β¦ 85
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LIST OF FIGURES
Figure 2.1: The selected region in Ξ is π = πβ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...... 11
Figure 2.2: The selected region in Ξ is π > πβ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.. 12
Figure 2.3: The selected region in Ξ is π < πβ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.. 13
Figure 2.4: The selected region in Ξ is π < πβ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦ 14
Figure 3.1: 26 points around center point using operator β β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦... 41
Figure 3.2: The selected region in π is π = πβ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦ 44
Figure 3.3: The selected region in π is π > πβ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.. 45
Figure 3.4: The selected region in π is π < πβ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.. 46
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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 π· bounded by the contour πΏ reduces to the
following boundary- value problem: to find, the solution of the Poisson equation
Ξπ’ = β2,
that reduces to zero on the contour πΏ:
π’ = 0 on πΏ.
Here the basis quantities required from the calculation are expressed in terms of the function π’
the components of the tangential stress
ππ§π₯ = GΟππ’
ππ¦, ππ§π¦ = βGΟ
ππ’
ππ₯,
and the torsional moment
π = GΟβ¬π’ππ₯ππ¦.
π·
Here Ο is the angle of twist per unit length, and πΊ is the modulus of shear.
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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, (β is the grid step), with which the
difference solution converges on the given domain. In (Volkov, 1999), π(β2) 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 Laplaceβs equation with order of π(β4), 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 π(β2) 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 π(β2) 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
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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 π(β6|ln β|) order, and for the first
and pure second derivatives π(β4) 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 π Γ π
matrices with the bandwidth π the computational cost is of order π(π2π) . 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 π(β2) up to π(β4 ln ββ1), where β is the mesh size. It is
known that, to get π(β4) order of accurate results by the existing one-stage methods for the
approximation 3π· 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
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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 πΆ7,π, 0 < π < 1, on the sides of the rectangle, and as whole are
continuous on the vertices. Also, the 2π, π = 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 π(πβ6), where π = π(π₯, π¦) is the distance from the current grid
point (π₯, π¦) to the boundary of the rectangle, is obtained. This estimation is used to approximate
the first derivatives with uniform error of order π(β6). The approximation of the pure second
derivatives are obtained with uniform accuracy π(β5+π), 0 < π < 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 π(πβ6) with the weight function π, where π = π(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
π(β6 ln β) . The approximation of the pure second derivatives are obtained with uniform
accuracy π(β5+π), 0 < π < 1.
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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 π(β6).
In Chapter 5, the numerical experiments to justify the theoretical results obtained in each
Chapters are demonstrated.
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CHAPTER 2
ON THE HIGH ORDER CONVERGENCE OF THE DIFFERENCE SOLUTION OF
LAPLACEβS 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 π(πβ6) with the weight function π , where π =
π(π₯, π¦) is the distance from the current grid point (π₯, π¦) 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 π(β6). The approximation of the pure second derivatives are obtained with uniform
accuracy π(β5+π), 0 < π < 1.
2.1 The Dirichlet Problem for Laplaceβs Equation on Rectangle and Some Differential
Properties of Its Solution
Let Ξ = {(π₯, π¦): 0 < π₯ < π , 0 < π¦ < π} be an open rectangle and π πβ is a rational number.
The sides are denoted by πΎπ , π = 1,2,3,4, including the ends. These sides are enumerated
counterclockwise where πΎ1 is the left side of Ξ , (πΎ0 β‘ πΎ4 , πΎ5 β‘ πΎ1). Also let the boundary of Ξ
be defined by πΎ = β πΎπ4π=1 .
The arclength along πΎ is denoted by π , and π π is the value of π at the beginning of πΎπ. We denote
by π β πΆπ,π(π·) if π has π β π‘β deravatives on π· satisfying HΓΆlder condition, where exponent
π β (0,1).
-
7
We consider the following boundary value problem
βπ’ = 0 on Ξ , π’ = ππ(π ) on πΎπ, π = 1,2,3,4 (2.1)
where ββ‘ π2 ππ₯2 + π2 ππ¦2ββ , ππ are given functions of π .
Assume that
ππ β πΆ7,π(πΎπ), 0 < π < 1, π = 1,2,3,4, (2.2)
ππ(2π)(π π) = (β1)
πππβ1(2π)(π π), π = 0,1,2,3. (2.3)
Lemma 2.1 The solution π’ of problem (2.1) is from πΆ7,π(π±).
The proof of Lemma 2.1 follows from Theorem 3.1 by (Volkov, 1969).
Lemma 2.2 Let π(π₯, π¦) be the distance from the current point of open rectangle π± to its
boundary and let π ππβ β‘ πΌ π ππ₯β + π½ π ππ¦β , πΌ2 + π½2 = 1. Then the next inequality holds
|π8π’(π₯, π¦)
ππ8| β€ πππβ1(π₯, π¦), (π₯, π¦) β π±, (2.4)
where π is a constant independent of the direction of differentiation π ππβ , and π’ is a solution of
problem (2.1).
Proof. We choose an arbitrary point (π₯0, π¦0) β Ξ . Let π0 = π(π₯0, π¦0), and π0 β Ξ Μ be the closed
circle of radius π0 centered at (π₯0, π¦0). Consider the harmonic function on Ξ
π£(π₯, π¦) =π7π’(π₯, π¦)
ππ7β
π7π’(π₯0, π¦0)
ππ7. (2.5)
-
8
By Lemma 2.1, π’ β πΆ7,π(Ξ Μ ), for 0 < π < 1. Then for the function (2.5) we have
max(π₯,π¦)βοΏ½Μ οΏ½0
|π£(π₯, π¦)| β€ π0π0π, (2.6)
where π0 is a constant independent of the point (π₯0, π¦0) β Ξ or the direction of π ππβ . Since π’ is
harmonic in Ξ , by using estimation (2.6) and applying Lemma 3 from (Mikhailov, 1978) we
have
|π
ππ(π7π’(π₯, π¦)
ππ7β
π7π’(π₯0, π¦0)
ππ7)| β€ π1
π0π
π0.
or
|π8π’(π₯, π¦)
ππ8| β€ π1π0
πβ1(π₯0, π¦0),
where π1is a constant independent of the point (π₯0, π¦0) β Ξ or the direction of π ππβ . Since the
point (π₯0, π¦0) β Ξ is arbitrary, inequality (2.4) holds true. β
2.2 Difference Equations for the Dirichlet Problem and a Pointwise Estimation
Let β > 0, and πππ{π ββ , π ββ } β₯ 6 where π ββ and π ββ are integers. A square net on Ξ is
assigned by Ξ β, with step β, obtained by the lines π₯, π¦ = 0, β, 2β,β¦ .
The set of nodes on πΎπis denoted by πΎπβ, and let
πΎβ = β πΎπβ4
π=1 , Ξ Μ β = Ξ β βͺ πΎβ.
Let the averaging operator π΅ be defined as following
-
9
π΅π’(π₯, π¦) =1
20[4(π’(π₯ + β, π¦) + π’(π₯ β β, π¦) + π’(π₯, π¦ + β)
+π’(π₯, π¦ β β)) + π’(π₯ + β, π¦ + β) + π’(π₯ + β, π¦ β β)
+π’(π₯ β β, π¦ + β) + π’(π₯ β β, π¦ β β)]. (2.7)
Let π, π0, π1, β¦ be constants which are independent of β 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 = π΅π’β on Ξ β, π’β = ππ on πΎπ
β, π = 1,2,3,4. (2.8)
By the maximum principle system (2.8) has a unique solution (Samarskii, 2001).
Let Ξ πβ be the set of nodes of grid Ξ β whose distance from πΎ is πβ. It is obvious that 1 β€ π β€
π(β), where
π(β) = [1
2βπππ{π, π}],
(2.9)
[π] is the integer part of π.
We define for 1 β€ π β€ π(β) the function
πβπ = {
1, π(π₯, π¦) = πβ,
0, π(π₯, π¦) β πβ.
(2.10)
Consider the following systems
πβ = π΅πβ + πβ on Ξ β, πβ = 0 on πΎ
β, (2.11)
οΏ½Μ οΏ½β = π΅οΏ½Μ οΏ½β + οΏ½Μ οΏ½β on Ξ β, οΏ½Μ οΏ½β = 0 on πΎ
β, (2.12)
where πβ and οΏ½Μ οΏ½β are given function, and |πβ| β€ οΏ½Μ οΏ½β on Ξ β.
-
10
Lemma 2.3 The solution πβ and οΏ½Μ οΏ½β of systems (2.11) and (2.12) satisfy the inequality
|πβ| β€ οΏ½Μ οΏ½β on Ξ Μ β .
The proof of Lemma 2.3 follows from comparison Theorem see Chapter 4 (Samarskii, 2001).
Lemma 2.4 The solution of the system
π£βπ = π΅π£β
π + πβπ on Ξ β, π£β
π = 0 on πΎβ (2.13)
satisfies the inequality
π£βπ(π₯, π¦) β€ πβ
π, 1 β€ π β€ π(β), (2.14)
where πβπ is defined as follows
πβπ = πβ
π(π₯, π¦) = {
6π
β, 0 β€ π(π₯, π¦) β€ πβ ,
6π, π(π₯, π¦) > πβ . (2.15)
Proof. By virtue of (2.7) and (2.15) and in consider of Fig. (2.1), we have for 0 β€ π = πβ,
π΅πβπ =
1
20[4(6π + 6(π β 1) + 6(π β 1) + 6π)
+6(π β 1) + 6π + 6(π β 1) + 6(π β 1)]
= 6π β66
20,
which leads to
πβπ β π΅πβ
π =66
20> 1 = πβ
π.
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11
Figure 2.1: The selected region in Ξ is π = πβ
In consider of Fig. (2.2) for π > πβ, then
π΅πβπ =
1
20[4(6π + 6π + 6π + 6π) + 6π
+6π + 6π + 6π = 6π,
which leads to
π΅πβπ = πβ
π .
-
12
Figure 2.2: The selected region in Ξ is π > πβ
In consider of Fig. (2.3) for π < πβ, then
π΅πβπ =
1
20[4(6π + 6(π β 2) + 6(π β 1) + 6(π β 1))
+6(π β 1) + 6π + 6(π β 2) + 6(π β 2)]
= 6π β126
20,
which leads to
πβπ β π΅πβ
π =126
20> 1 = πβ
π.
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13
Figure 2.3: The selected region in Ξ is π < πβ
In consider of Fig. (2.1) for π < πβ, then
π΅πβπ =
1
20= [4(6(π β 1) + 6(π β 1) + 6(π β 2) + 6π)
+6π + 6π + 6(π β 2) + 6(π β 2)]
= 6π β 6 = 6(π β 1),
which leads to
πβπ = π΅πβ
π.
-
14
Figure 2.4: The selected region in Ξ is π < πβ
From the above calculations we have
πβπ = π΅πβ
π + πβπ on Ξ β, πβ
π = 0 on πΎβ, π = 1,β¦ ,π(β), (2.16)
where |πβπ| β₯ 1. On the basis of (2.10), (2.13), (2.16) and the Comparison Theorem from
(Samarskii, 2001), we obtain
|π£βπ| β€ πβ
π for all π, 1 β€ π β€ π(β). β
Lemma 2.5 The following equality is true
π΅π7(π₯0, π¦0) = π’(π₯0, π¦0)
where π7 is the seventh order Taylorβs polynomial at (π₯0, π¦0), π’ is a harmonic function.
-
15
Proof. The seventh order Taylorβs polynomial at (π₯0, π¦0) has the form
π7(π₯, π¦) = π’(π₯, π¦) + β (ππ’
ππ₯+
ππ’
ππ¦) +
β2
2!(π2π’
ππ₯2+ 2
π2π’
ππ₯ππ¦+
π2π’
ππ¦2) +
β3
3!(π3π’
ππ₯3
+3π3π’
ππ₯2ππ¦+ 3
π3π’
ππ₯ππ¦2+
π3π’
ππ¦3) +
β4
4!(π4π’
ππ₯4+ 4
π4π’
ππ₯3ππ¦+ 6
π4π’
ππ₯2ππ¦2
+4π4π’
ππ₯ππ¦3+
π4π’
ππ¦4) +
β5
5!(π5π’
ππ₯5+ 5
π5π’
ππ₯4ππ¦+ 20
π5π’
ππ₯3ππ¦2+ 20
π5π’
ππ₯2ππ¦3
+5π5π’
ππ₯ππ¦4+
π5π’
ππ¦5) +
β6
6!(π6π’
ππ₯6+ 6
π6π’
ππ₯5ππ¦+ 15
π6π’
ππ₯4ππ¦2+ 20
π6π’
ππ₯3ππ¦3
+15π6π’
ππ₯2ππ¦4+ 6
π6π’
ππ₯ππ¦5+
π6π’
ππ¦6) +
β7
7!(π7π’
ππ₯7+ 7
π7π’
ππ₯6ππ¦+ 21
π7π’
ππ₯5ππ¦2
+35π7π’
ππ₯4ππ¦3+ 35
π7π’
ππ₯3ππ¦4+ 21
π7π’
ππ₯2ππ¦5+ 7
π7π’
ππ₯ππ¦6+
π7π’
ππ¦7). (2.17)
Then according to (2.7) and (2.17) we have
π΅π7(π₯0, π¦0) =1
20[4(π7(π₯0 + β, π¦0) + π7(π₯0 β β, π¦0) + π7(π₯0, π¦0 + β)
+π7(π₯0, π¦0 β β)) + π7(π₯0 + β, π¦0 + β) + π7(π₯0 + β, π¦0 β β)
+π7(π₯0 β β, π¦0 + β) + π7(π₯0 β β, π¦0 β β)]
= π’(π₯0, π¦0) +β4
40
π2
ππ₯2(π2π’(π₯0, π¦0)
ππ₯2+
π2π’(π₯0, π¦0)
ππ¦2)
+β4
40
π2
ππ¦2(π2π’(π₯0, π¦0)
ππ₯2+
π2π’(π₯0, π¦0)
ππ¦2) +
3β6
5 Γ 6!
π4
ππ₯4(π2π’(π₯0, π¦0)
ππ₯2+
π2π’(π₯0, π¦0)
ππ¦2)
+3β6
5 Γ 6!
π4
ππ¦4(π2π’(π₯0, π¦0)
ππ₯2+
π2π’(π₯0, π¦0)
ππ¦2) +
2β6
5 Γ 5!
π4
ππ₯2ππ¦2(π2π’(π₯0, π¦0)
ππ₯2+
π2π’(π₯0, π¦0)
ππ¦2).
Since π’ is harmonic, we obtain
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16
π΅π7(π₯0, π¦0) = π’(π₯0, π¦0) β
Lemma 2.6 The inequality holds
max(π₯,π¦)βΞ πβ
|π΅π’ β π’| β€ πβ7+π
π1βπ, π = 1,2, β¦ ,π(β),
(2.18)
where π’ is a solution of problem (2.1).
Proof. Let (π₯0, π¦0) be a point of Ξ 1β, and let
Ξ 0 = {(π₯, π¦): |π₯ β π₯0| < β, |π¦ β π¦0| < β }, (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 π΅π’(π₯0, π¦0). We represent a solution of problem (2.1) in some neighborhood of
(π₯0, π¦0) β Ξ 1β by Taylorβs formula
π’(π₯, π¦) = π7(π₯, π¦) + π8(π₯, π¦), (2.20)
where π7(π₯, π¦) is the seventh order Taylorβs polynomial, π8(π₯, π¦) is the remainder term, by
Lemma 2.5 we have
π΅π7(π₯0, π¦0) = π’(π₯0, π¦0). (2.21)
Now, we estimate π8 at the nodes of the operator π΅. We take node (π₯0 + β, π¦0 + β) which is
one of the eight nodes of π΅, and consider the function
-
17
οΏ½ΜοΏ½(π ) = π’ (π₯0 +π
β2, π¦0 +
π
β2) , β β2β β€ π β€ β2β (2.22)
of one variable π . By virtue of Lemma 2.2, we have
|π8οΏ½ΜοΏ½(π )
ππ 8| β€ π2(β2β β π )
πβ1, 0 β€ π β€ β2β . (2.23)
We represent function (2.22) around the point π = 0 by Taylorβs formula
οΏ½ΜοΏ½(π ) = π7(π ) + οΏ½ΜοΏ½8(π ), (2.24)
where
π7(π ) = π7 (π₯0 +π
β2, π¦0 +
π
β2) (2.25)
is the seventh order Taylorβs polynomial of the variable π , and
οΏ½ΜοΏ½8(π ) = π8 (π₯0 +π
β2, π¦0 +
π
β2) , 0 β€ |π | β€ β2β (2.26)
is the remainder term. On the basis of continuity of οΏ½ΜοΏ½8(π ) on the interval [ββ2β, β2β], it follows
from (2.26) that
π8 (π₯0 +π
β2, π¦0 +
π
β2) = lim
πβ+0οΏ½ΜοΏ½8(β2β β π). (2.27)
Applying an integral representation for οΏ½ΜοΏ½8 we have
-
18
οΏ½ΜοΏ½8(β2β β π) =1
7!β« (β2β β π β π‘)
7οΏ½ΜοΏ½8(π‘)ππ‘
β2ββπ
0
, 0 < π β€β
β2 .
Using estimation (2.23), we have
|οΏ½ΜοΏ½8(β2β β π)| β€ π3 β« (β2β β π β π‘)7(β2β β π‘)
πβ1ππ‘
β2ββπ
0
β€ π41
7!β« (β2β β π‘)
6+πππ‘
β2ββπ
0
β€ πβπ+7, 0 < π β€β
β2 . (2.28)
From (2.26)-(2.28) yields
|π8(π₯0 + β, π¦0 + β| β€ π1βπ+7, (2.29)
where π1 is a constant independent of the taken point (π₯0, π¦0) on Ξ 1β. Proceeding in a similar
manner, we can find the same estimates of π8 at the other vertices of square (2.19) and at the
centers of its sides. Since the norm of π΅ in the uniform metric is equal to unity, we have
|π΅π8(π₯0, π¦0)| β€ π5βπ+7. (2.30)
where π5 is a constant independent of the taken point (π₯0, π¦0) on Ξ 1β . From (2.20), (2.21),
(2.30) and linearity of the operator π΅, we obtain
|π΅π’(π₯0, π¦0) β π’(π₯0, π¦0)| β€ πβπ+7, (2.31)
for any (π₯0, π¦0) β Ξ 1β.
-
19
Now let (π₯0, π¦0) β Ξ πβ, 2 β€ π β€ π(β) and π8(π₯, π¦) be the Lagrange remainder corresponding
to this point in Taylorβs formula (2.20). Then π΅π8(π₯0, π¦0) can be expressed linearly in terms of
a fixed number of eighth derivatives of π’ at some point of the open square Ξ 0 , which is a
distance of πβ 2β away from the boundary of Ξ . The sum of the coefficients multiplying the
eighth derivatives does not exceed πβ8, which is independent of π (2 β€ π β€ π(β)). By using
Lemma 2.2, we have
|π΅π8(π₯0, π¦0)| β€ πβ8
(πβ)1βπ= π
βπ+7
π1βπ, (2.32)
where π is a constant independent of π (2 β€ π β€ π(β)). On the basis of (2.20), (2.21), (2.31),
and (2.32) follows estimation (2.18) at any point (π₯0, π¦0) β Ξ πβ, 1 β€ π β€ π(β). β
Theorem 2.1 The next estimation holds
|π’β β π’| β€ ππβ6,
where π is a constant independent of π and β, π’ is the exact solution of problem (2.1), π’β is the
solution of the finite difference problem (2.8) and π = π(π₯, π¦) is the distance from the current
point (π₯, π¦) β π±β to the boundary of rectangle π±.
Proof. Let
πβ(π₯, π¦) = π’β(π₯, π¦) β π’(π₯, π¦), (π₯, π¦) β Ξ Μ β. (2.33)
Putting π’β = πβ + π’ into (2.8), we have
πβ = π΅πβ + (π΅π’ β π’) on Ξ β, πβ = 0 on πΎ
β. (2.34)
We represent a solution of system (2.34) as follows
-
20
πβ = β πβπ
π(β)
π=1
, π(β) = [1
2βmin{π, π}], (2.35)
where πβπ is a solution of the system
πβπ = π΅πβ
π + πβπ on Ξ β, πβ = 0 on πΎ
β, π = 1,2, β¦ , π(β); (2.36)
πβπ = {
π΅π’ β π’ on Ξ βπ ,
0 on Ξ β Ξ βπβ .
(2.37)
By virtue of (2.36), (2.37) and Lemma 2.4, for each π , 1 β€ π β€ π(β), follows the
inequality
|πβπ(π₯, π¦)| β€ πβ
π(π₯, π¦) max(π₯,π¦)βΞ πβ
|(π΅π’ β π’)| on Ξ Μ β . (2.38)
On the basis of (2.33), (2.35) and (2.38) we have
|πβ| β€ β|πβπ|
π(β)
π=1
β€ β πβπ(π₯, π¦) max
(π₯,π¦)βΞ πβ|(π΅π’ β π’)|
π(β)
π=1
= β πβπ(π₯, π¦) max
(π₯,π¦)βΞ πβ|(π΅π’ β π’)|
π
ββ1
π=1
+ β πβπ(π₯, π¦) max
(π₯,π¦)βΞ πβ|(π΅π’ β π’)|,
π(β)
π=π
β
(π₯, π¦) β Ξ πβ .
By definition (2.15) of the function πβπ, we have
-
21
β πβπ(π₯, π¦) max
(π₯,π¦)βΞ πβ|(π΅π’ β π’)| β€ 6πβ8 β
π
(πβ)1βπ
π
ββ1
π=1
π
ββ1
π=1
, (2.39)
β πβπ(π₯, π¦) max
(π₯,π¦)βΞ πβ|(π΅π’ β π’)|6πβ8 β
π
(πβ)1βπ
π(β)
π=π
β
.
π(β)
π=π
β
(2.40)
Then from (2.39)-(2.40) we have
|πβ(π₯, π¦)| β€ 6πβ7+π β ππ
π
ββ1
π=1
+ 6πβ6+ππ β1
π1βπ
π(β)
π=π
β
β€ 6πβ7+π
(
1 + β« π₯π
π
ββ1
1
ππ₯
)
+ 6πβ6+ππ ((π
β)
πβ1
+ β« π₯πβ1
π(β)
π
β
ππ₯)
β€ πβ7+π + πβ7+π ((π
ββ 1)
π+1
π + 1β
1
π + 1) + πβ7ππ + πβ6+ππ (
(π
β)π
πβ
π
β
π)
β€ πβ7+π +π
π + 1β7+π (
π
ββ 1)
π+1
βπ
π + 1β7+π + πβ7ππ
+π
πβ6+ππ
(π
β)
π
πβ
π
π(π
β)
π
β6+ππ
β€ πβ7+π + π1β7+π (
π
ββ 1)
π+1
β π1β7+π + πβ7ππ + π2β
6πππ β π2β6ππ+1
= πβ6π(π₯, π¦), (π₯, π¦) β Ξ Μ β.
Theorem 2.1 is proved. β
-
22
2.3 Approximation of the First Derivatives
Let π’ be a solution of the boundary value problem (2.1). We put π =ππ’
ππ₯. It is obvious that the
function π is a solution of boundary value problem
βπ = 0 on Ξ , π = ππ on πΎπ, π = 1,2,3,4, (2.41)
where ππ =ππ’
ππ₯ on πΎπ, π = 1,2,3,4.
Let π’β be a solution of finite difference problem (2.8). We define the following operators ππ£β,
π£ = 1,2,3,4,
π1β(π’β) =1
60β[β147π1(π¦) + 360π’β(β, π¦) β 450π’β(2β, π¦)
+400π’β(3β, π¦) β 225π’β(4β, π¦) + 72π’β(5β, π¦)
β10π’β(6β, π¦)] on πΎ1β, (2.42)
π3β(π’β) =1
60β[147π3(π¦) β 360π’β(π β β, π¦)
+450π’β(π β 2β, π¦) β 400π’β(π β 3β, π¦) + 225π’β(π β 4β, π¦)
β72π’β(π β 5β, π¦) + 10π’β(π β 6β, π¦)] on πΎ3β, (2.43)
ππβ(π’β) =πππ
ππ₯ on πΎπ
β, π = 2,4. (2.44)
Lemma 2.7 The next inequality holds
|ππβ(π’β) β ππβ(π’)| β€ πβ6, π = 1,3,
where π’β is the solution of problem (2.8), and π’ is the solution of problem (2.1).
-
23
Proof. It is obvious that ππβ(π’β) β ππβ(π’) = 0 for π = 2,4.
For π = 1, by (2.42) and Theorem 2.1 we have
|π1β(π’β) β π1β(π’)| = |1
60β[(β147π1(π¦) + 360π’β(β, π¦)
β450π’β(2β, π¦) + 400π’β(3β, π¦) β 225π’β(4β, π¦) + 72π’β(5β, π¦)
β10π’β(6β, π¦)) β (β147π1(π¦) + 360π’(β, π¦) β 450π’(2β, π¦)
+400π’(3β, π¦) β 225π’(4β, π¦) + 72π’(5β, π¦)10π’(6β, π¦))]|
β€1
60β[|π’β(β, π¦) β π’(β, π¦)| + 450|π’β(2β, π¦) β π’(2β, π¦)|
+400|π’β(3β, π¦) β π’(3β, π¦)| + 225|π’β(4β, π¦) β π’(4β, π¦)|
+72|π’β(5β, π¦) β π’(5β, π¦)| + 10|π’β(6β, π¦) β π’(6β, π¦)|]
β€1
60β[360(πβ)β6 + 450(2πβ)β6 + 400(3πβ)β6 + 225(4πβ)β6
+72(5πβ)β6 + 10(6πβ)β6].
β€ π7β6.
The same inequality is true for π = 3. β
Lemma 2.8 The inequality holds
max(π₯,π¦)βπΎπ
β|ππβ(π’β) β ππ| β€ π8β
6 , π = 1,3,
where ππβ, π = 1,3 are the functions defined by (2.42) and (2.43), ππ =ππ’
ππ₯ on πΎπ,
π = 1,3.
Proof. From Lemma 2.1 follows that π’ β πΆ7,0(Ξ Μ ). Then at the end points (0, πβ) β πΎ1β and
(π, πβ) β πΎ3β of each line segment {(π₯, π¦): 0 β€ π₯ β€ π , 0 < π¦ = πβ < π} expressions (2.42) and
(2.43) give the sixth order approximation of ππ’
ππ₯ respectively.
From the truncation error formulae follows that
-
24
max(π₯,π¦)βπΎπ
β|ππβ(π’) β ππ| β€ max
(π₯,π¦)βΞ Μ
1
7!|π7π’
ππ₯7| β(2β)(3β)(4β)(5β)(6β)
β€ π9β6, k = 1,3. (2.45)
On the basis of Lemma 2.7 and estimation (2.45)
max(π₯,π¦)βπΎπ
β|ππβ(π’β) β ππ| β€ max
(π₯,π¦)βπΎπβ|ππβ(π’β) β ππβ(π’)| + max
(π₯,π¦)βπΎπβ|ππβ(π’) β ππ|
β€ π7β6 + π9β
6 = π8β6, π = 1,3.β
Consider the finite difference problem
π£β = π΅π£β on Ξ β, π£ = ππβ on πΎπ
β, π = 1,2,3,4, (2.46)
where ππβ , π = 1,2,3,4 are defined by the formulas (2.42)-(2.44).
Since the boundary values ππβ for π = 1,3 are defined by the solution of finite-difference
problem (2.46) which assumed to be known and ππβ =πππ
ππ₯, π = 2,4 are calculated by the
boundary functions ππ , π = 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
βπ = 0 on Ξ , π = Ξ¦π on πΎπ, π = 1,2,3,4,
where Ξ¦π in the given function, which satisfy the following conditions
Ξ¦π β πΆ6,π(πΎπ), 0 < π < 1, (2.47)
-
25
Ξ¦π2π(π π) = (β1)
πΞ¦πβ12π (π π), π = 0,1,2. (2.48)
Let πβ be a solution of the finiteβdifference problem
πβ = π΅πβ on Ξ β, πβ = Ξ¦πβ on πΎπ
β , π = 1,2,3,4, (2.49)
where ππβ is the value of ππ on πΎπβ, π = 1,2,3,4. It is clear that the error function πβ = πβ β π
is a solution of the boundary value problem
πβ = π΅πβ + (π΅πβ β π) on Ξ β, πβ = 0 on πΎπ
β , π = 1,2,3,4. (2.50)
As follows from Theorem 12 in (Dosiyev, 2003) the following estimation
max(π₯,π¦)βΞ Μ β
|πβ β π| β€ πβ6, (2.51)
is true.
Theorem 2.2 The following estimation holds
max(π₯,π¦)βΞ Μ
|π£β βππ’
ππ₯| β€ π0β
6,
where π’ is the solution of problem (2.1), and π£β is the solution of finite difference problem
(2.46).
Proof. Let
πβ = π£β β π£ on Ξ Μ β, (2.52)
-
26
where π£ =ππ’
ππ₯. From (2.46) and (2.52), we have
πβ = π΅πβ + (π΅π£ β π£) on Ξ β, πβ = ππβ(π’β) β π£ on πΎπ
β,
π = 1,3, πβ = 0 on πΎπβ, π = 2,4. (2.53)
We represent
πβ = πβ1 + πβ
2 (2.54)
where
πβ1 = π΅πβ
1 on Ξ β, πβ1 = ππβ(π’β) β π£ on πΎπ
β, π = 1,3,
πβ1 = 0 on πΎπ
β, π = 2,4; (2.55)
πβ2 = π΅πβ
2 + (π΅π£ β π£) on Ξ β, πβ2 = 0 on πΎπ
β, π = 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₯,π¦)βΞ β
|πβ1| β€ max
π=1,3max
(π₯,π¦)βπΎπβ|ππβ(π’β) β π£| β€ π1β
6. (2.57)
From (2.50) follows that πβ2 in (2.56) is a solution of problem (2.50) with the boundary
value of π£ satisfy the equations (2.47), (2.48). By the estimation (2.51), we obtain
max(π₯,π¦)βΞ Μ β
|πβ2| β€ π2β
6 (2.58)
By virtue of (2.54), (2.57) and (2.58) the proof is completed. β
-
27
2.4 Approximation of the Pure Second Derivatives
We denote π =π2π’
ππ₯2. The function π is harmonic on Ξ , on the basis of Lemma 2.1 π is
continuous on Ξ Μ , and it is a solution of the following Dirichlet problem
βπ = 0 on Ξ , π = ππ on πΎπ , π = 1,2,3,4 (2.59)
where
ππ =π2ππππ₯2
, π = 2,4, (2.60)
ππ = βπ2Οπππ¦2
, π = 1,3. (2.61)
From the continuity of the function π on Ξ Μ , and from (2.2), (2.3) and (2.60), (2.61) it follows
that
ππ β πΆ5,π(πΎπ), 0 < π < 1, π = 1,2,3,4. (2.62)
ππ(2π)
(π π) = (β1)πππβ1
2π (π π), π = 0,1,2 , π = 1,2,3,4. (2.63)
Let πβ be a solution of the finite difference problem
πβ = π΅πβ on Ξ β, πβ = ππ on πΎπ
β, π = 1,2,3,4, (2.64)
where ππ are functions determined by (2.60) and (2.61).
-
28
Lemma 2.9 The next inequality holds true
|π8π(π₯, π¦)
ππ8| β€ π1π
πβ3(π₯, π¦), (π₯, π¦) β Ξ (2.65)
where π1is a constant independent of the direction of differentiation π ππβ .
Proof. We choose an arbitrary point (π₯0, π¦0) β Ξ . Let π0 = π(π₯0, π¦0) and π β Ξ Μ be the closed
circle of radius π0 centered at (π₯0, π¦0), consider the harmonic function on
π£(π₯, π¦) =π5π(π₯, π¦)
ππ5β
π5π(π₯0, π¦0)
ππ5. (2.66)
By (2.62), π =π2π’
ππ₯12 β πΆ
5,π(Ξ Μ ), for, 0 < π < 1. Then for the function (2.66) we have
max(π₯,π¦)βοΏ½Μ οΏ½0
|π£(π₯, π¦)| β€ π0π0π, (2.67)
where π0 is a constant independent of the point (π₯0, π¦0) β Ξ or the direction of π ππβ . Since π’
is harmonic in Ξ , by using estimation (2.67) and applying Lemma 3 from (Mikhailov, 1978)
we have
|π3
ππ3(π5π(π₯, π¦)
ππ5β
π5π(π₯0, π¦0)
ππ5)| β€ π
π0π
π03 ,
or
|π8π(π₯, π¦)
ππ8| β€ ππ0
πβ3(π₯0, π¦0),
where π is a constant independent of the point (π₯0, π¦0) β Ξ or the direction of π ππβ . Since the
point (π₯0, π¦0) β Ξ is arbitrary, inequality (2.65) holds true. β
-
29
Lemma 2.10 The inequality holds
max(π₯,π¦)βΞ πβ
|π΅π β π| β€ πβ5+π
π3βπ, π = 1,2, β¦ ,π(β), (2.68)
where π’ is a solution of problem (2.1).
Proof. Let (π₯0, π¦0) be a point of Ξ 1β β Ξ β, and let
Ξ 0 = {(π₯, π¦): |π₯ β π₯0| < β , |π¦ β π¦0| < β }, (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(π₯0, π¦0). We represent a solution of problem (2.59) in some neighborhood
of (π₯0, π¦0) β Ξ 1β by Taylorβs formula
π(π₯, π¦) = π7(π₯, π¦) + π8(π₯, π¦), (2.70)
where π7(π₯, π¦) is the seventh order Taylorβs polynomial, π8(π₯, π¦) is the remainder term, and
π΅π7(π₯0, π¦0) = π(π₯0, π¦0). (2.71)
Now, we estimate π8 at the nodes of the operator π΅. We take node (π₯0 + β, π¦0 + β) which is
one of the eight nodes of π΅, and consider the function
αΏΆ(π ) = π (π₯0 +π
β2, π¦0 +
π
β2) , β β2β β€ π β€ β2β (2.72)
of one variable π . By virtue of Lemma 2.9, we have
-
30
|π8αΏΆ(π )
ππ 8| β€ π2(β2β β π )
πβ3, 0 β€ π β€ β2β (2.73)
we represent function (2.72) around the point π = 0 by Taylorβs formula as
αΏΆ(π ) = π7(π ) + οΏ½ΜοΏ½8(π ), (2.74)
where
π7(π ) = π7 (π₯0 +π
β2, π¦0 +
π
β2) (2.75)
is the seventh order Taylorβs polynomial of the variable π , and
οΏ½ΜοΏ½8(π ) = π8 (π₯0 +π
β2, π¦0 +
π
β2) , 0 β€ |π | β€ β2β, (2.76)
is the remainder term. On the basis of continuity of οΏ½ΜοΏ½8(π ) on the interval [ββ2β, β2β], it follows
from (2.76) that
π8 (π₯0 +π
β2, π¦0 +
π
β2) = lim
πβ+0οΏ½ΜοΏ½8(β2β β π). (2.77)
Applying an integral representation for οΏ½ΜοΏ½8 we have
οΏ½ΜοΏ½8(β2β β π) =1
7!β« (β2β β π β π‘)
7οΏ½ΜοΏ½8(π‘)ππ‘
β2ββπ
0
, 0 < π β€β
β2. (2.78)
Using estimation (2.73), we have
-
31
|οΏ½ΜοΏ½8(β2β β π)| β€ π3 β« (β2β β π β π‘)7(β2β β π‘)
πβ3ππ‘
β2ββπ
0
β€ π41
7!β« (β2β β π‘)
4+πππ‘
β2ββπ
0
β€ πβπ+5, 0 < π β€β
β2. (2.79)
From (2.76)-(2.79) yields
|π8(π₯0 + β, π¦0 + β| β€ π1βπ+5,
where π1 is a constant independent of the taken point (π₯0, π¦0) on Ξ 1β. Proceeding in a similar
manner, we can find the same estimates of π8 at the other vertices of square (2.69) and at the
centers of its sides. Since the norm of π΅ in the uniform metric is equal to unity, we have
|π΅π8(π₯0, π¦0)| β€ π5βπ+5, (2.80)
where π5 is a constant independent of the taken point (π₯0, π¦0) on Ξ 1β . From (2.70), (2.71),
(2.80) and linearity of the operator π΅, we obtain
|π΅π(π₯0, π¦0) β π(π₯0, π¦0)| β€ πβπ+5, (2.81)
for any (π₯0, π¦0) β Ξ 1β.
Now let (π₯0, π¦0) β Ξ πβ, 2 β€ π β€ π(β) and π8(π₯, π¦) be the Lagrange remainder corresponding
to this point in Taylorβs formula (2.70). Then π΅π8(π₯0, π¦0) can be expressed linearly in terms of
a fixed number of eighth derivatives ofπ’ at some point of the open square Ξ 0, which is a distance
of πβ 2β away from the boundary of Ξ . The sum of the coefficients multiplying the eighth
-
32
derivatives does not exceed πβ8, which is independent of π, (2 β€ π β€ π(β)). By Lemma 2.9,
we have
|π΅π8(π₯0, π¦0)| β€ πβ8
(πβ)3βπ= π
βπ+5
π3βπ, (2.82)
where π is a constant independent of π, (2 β€ π β€ π(β)). On the basis of (2.70), (2.71), (2.81)
and (2.82) follows estimation (2.68) at any point (π₯0, π¦0) β Ξ πβ, 1 β€ π β€ π(β). β
Theorem 2.3 The estimation holds
max(π₯0,π¦0)βΞ Μ β
|πβ β π| β€ π12 βπ+5, (2.83)
where π =π2π’
ππ₯2, π’ is the solution of problem (2.1), and πβ is the solution of the finite
difference problem (2.64).
Proof. Let
πβ = πβ β π, (2.84)
where πβ, and π is a solution of problem (2.64) and (2.59) respectively. Then for πβ, we have
πβ = π΅πβ + (π΅π β π) on Ξ β, πβ = 0 on πΎ
β. (2.85)
We represent a solution of system (2.85) as follows
πβ = β πβ, π π(β) = [
1
2βmin{π, π}] ,
π(β)
π=1
(2.86)
-
33
where πβπ is a solution of the system
πβπ = π΅πβ
π + πβπ on Ξ β, πβ
π = 0 on Ξ³β, π = 1,2, β¦ ,π(β) (2.87)
πβπ = {
π΅π β π on Ξ βπ ,
0 on Ξ β Ξ βπβ .
(2.88)
By virtue of (2.87), (2.88) and Lemma 2.4 for each π , 1 β€ π β€ π(β), follows the inequality
|πβπ(π₯, π¦)| β€ πβ
π(π₯, π¦) max(π₯,π¦)βΞ β
|(π΅π β π)|, on Ξ Μ β. (2.89)
On the basis of (2.84), (2.86) and (2.89) we have
max(π₯,π¦)βΞ β
|πβ| β€ β 6π max(π₯,π¦)βΞ β
|(π΅π β π)|
π(β)
π=1
β€ β 6ππ13β5+π
π3βπ
π(β)
π=1
β€ 6πβ5+π [1 + β« π₯πβ2
π(β)
1
ππ₯]
β€ 6π14β5+π [1 +
π₯πβ1
π β 1|1
π
β
]
= 6π14β5+π [1 + (
π
β)
πβ1 1
π β 1β
1
π β 1]
= 6π14β5+π + 6π1β
6ππβ1 β 6π1β5+π
β€ π4β5+π.
Theorem 2.3 is proved. β
-
34
CHAPTER 3
ON THE HIGH ORDER CONVERGENCE OF THE DIFFERENCE SOLUTION OF
LAPLACEβS 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 π(πβ6) with the weight function π, where π = π(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
π(β6|ln β|) . The approximation of the pure second derivatives are obtained with uniform
accuracy π(β5+π), 0 < π < 1.
3.1 The Dirichlet Problem in a Rectangular Parallelepiped
Let π = {(π₯1, π₯2, π₯3): 0 < π₯π < ππ , π = 1,2,3} be an open rectangular parallelepiped; Ξπ (π =
1,2, β¦ ,6) be its faces including the edges; Ξπ for π = 1,2,3 (for π = 4,5,6) belongs to the plane
π₯π = 0 (to the plane π₯πβ3 = ππβ3), let Ξ = β Ξj6j=1 be the boundary of the parallelepiped, let πΎ
be the union of the edges of π , and let Ξππ = Ξπ βͺ Ξπ . We say that π β πΆπ,π(π·) , if π has
continuous π β π‘β deravatives on π· satisfying HΓΆlder condition with exponent π β (0,1).
We consider the following boundary value problem
βπ’ = 0 on π , π’ = ππ on Ξπ, π = 1,2, β¦ ,6, (3.1)
-
35
where ββ‘π2
ππ₯12 +
π2
ππ₯22 +
π2
ππ₯32, ππ are given functions.
Assume that
ππ β πΆ7,π(Ξπ), 0 < π < 1, π = 1,2, β¦ ,6, (3.2)
ππ = ππ on Ξ³ππ , (3.3)
π2ππ
ππ‘π2+
π2ππππ‘π2
+π2ππ
ππ‘ππ2= 0 on Ξ³ππ , (3.4)
π4ππ
ππ‘π4+
π4ππ
ππ‘π2ππ‘ππ2=
π4ππππ‘π4
+π4ππ
ππ‘π2ππ‘ππ2 on Ξ³ππ , (3.5)
π6ππ
ππ‘π6
+π6ππ
ππ‘π4ππ‘ππ2+
π6ππ
ππ‘π4ππ‘π2=
π6ππ
ππ‘π2ππ‘π4+
π6ππ
ππ‘π6 +
π6ππ
ππ‘π4ππ‘ππ2 on Ξ³ππ . (3.6)
Where 1 β€ π < π β€ 6, π β π β 3, tππ is an element in Ξ³ππ , tπ and tπ is an element of the
normal to Ξ³ππ on the face Ξπ and Ξπ, 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 . Then, it is
differentiated twice with respect to tπ . We have
π2ππ
ππ‘π2=
π2ππππ‘π2
π2
ππ‘π2(π2ππ
ππ‘π2) =
π2
ππ‘π2(π2ππππ‘π2
)
-
36
π4ππ
ππ‘π2ππ‘π2=
π4ππππ‘π2ππ‘π2
. (3.7)
(3.4) is differentiated twice with respect to tπ
π2
ππ‘π2(π2ππ
ππ‘π2+
π2ππππ‘π2
+π2ππ
ππ‘ππ2) = 0.
We have
π4ππ
ππ‘π4+
π4ππππ‘π2ππ‘π2
+π4ππ
ππ‘ππ2 ππ‘π2= 0. (3.8)
(3.4) is differentiated twice with respect to tπ
π2
ππ‘π2(π2ππ
ππ‘π2+
π2ππππ‘π2
+π2ππ
ππ‘ππ2) = 0.
We have
π4ππ
ππ‘π2ππ‘π2+
π4ππππ‘π4
+π4ππ
ππ‘ππ2 ππ‘π2= 0. (3.9)
From (3.7), (3.8) and (3.9) follows
π4ππ
ππ‘π4+
π4ππ
ππ‘π2ππ‘ππ2=
π4ππππ‘π4
+π4ππ
ππ‘π2ππ‘ππ2.
(3.7) is differentiated twice with respect to tππ
-
37
π2
ππ‘ππ2(
π4ππ
ππ‘π2ππ‘ππ2) =
π2
ππ‘ππ2(
π4ππππ‘π2ππ‘π2
)
π6ππ
ππ‘π2ππ‘π2ππ‘ππ2=
π6ππππ‘π2ππ‘π2ππ‘ππ2
. (3.10)
(3.5) is differentiated twice with respect to tπ follows
π2
ππ‘π2(π4ππ
ππ‘π4+
π4ππ
ππ‘π2ππ‘ππ2) =
π2
ππ‘π2(π4ππππ‘π4
+π4ππ
ππ‘π2ππ‘ππ2),
π6ππ
ππ‘π6
+π6ππ
ππ‘π4ππ‘ππ2=
π6ππππ‘π2ππ‘π4
+π6ππ
ππ‘π2ππ‘π2ππ‘ππ2. (3.11)
(3.5) is differentiated twice with respect to tπ follows
π2
ππ‘π2(π4ππ
ππ‘π4+
π4ππ
ππ‘π2ππ‘ππ2) =
π2
ππ‘π2(π4ππππ‘π4
+π4ππ
ππ‘π2ππ‘ππ2)
π6ππ
ππ‘π4ππ‘π2+
π6ππ
ππ‘π2ππ‘π2ππ‘ππ2=
π6ππ
ππ‘π6 +
π6ππ
ππ‘π4ππ‘ππ2. (3.12)
From (3.10), (3.11) and (3.12) follows
π6ππ
ππ‘π6
+π6ππ
ππ‘π4ππ‘ππ2+
π6ππ
ππ‘π4ππ‘π2=
π6ππ
ππ‘π2ππ‘π4+
π6ππ
ππ‘π6 +
π6ππ
ππ‘π4ππ‘ππ2.
Lemma 3.1 The solution π’ of the problem (3.1) is from πΆ7,π(οΏ½Μ οΏ½),
The proof of Lemma 3.1 follows from Theorem 2.1 (Volkov, 1969).
-
38
Lemma 3.2 Let π = (π₯1, π₯2, π₯3) be the distance from the current point of the open
parallelepiped π to its boundary and let π ππβ β‘ πΌ1π
ππ₯1+ πΌ2
π
ππ₯2+ πΌ3
π
ππ₯3, πΌ1
2 + πΌ22 + πΌ3
2 = 1.
Then the next inequality holds
|π8π’(π₯1, π₯2, π₯3)
ππ8| β€ πππβ1(π₯1, π₯2, π₯3), (π₯1, π₯2, π₯3) β π , (3.13)
where π is a constant independent of the direction of differentiation π ππβ , π’ is a solution of
problem (3.1).
Proof. We choose an arbitrary point (π₯10, π₯20, π₯30) β R. Let π0 = π(π₯10, π₯20, π₯30), and π0 β RΜ
be the closed ball of radius π0 centred at (π₯10, π₯20, π₯30). Consider the harmonic function on π
π£(π₯1, π₯2, π₯3) =π7π’(π₯1, π₯2, π₯3)
ππ7β
π7π’(π₯10, π₯20, π₯30)
ππ7. (3.14)
As it follows from Theorem 2.1 in (Volkov, 1969) the solution π’ of problem (3.1) which
satisfies the conditions (3.3)-(3.6) belongs to the class πΆ7,π(RΜ ), for 0 < π < 1. Then for the
function (3.14) we have
max(π₯1,π₯2,π₯3)βοΏ½Μ οΏ½0
|π£(π₯1, π₯2, π₯3)| β€ π0π0π, (3.15)
where π0 is a constant independent of the point (π₯10, π₯20, π₯30) β π or the direction of π ππβ .
By using estimation (3.15) and applying Lemma 3 from (Mikeladze, 1978) we have
|π
ππ(π7π’(π₯1, π₯2, π₯3)
ππ7β
π7π’(π₯10, π₯20, π₯30)
ππ7)| β€ π
π0π
π0,
or
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39
|π8π’(π₯1, π₯2, π₯3)
ππ8| β€ ππ0
πβ1(π₯1, π₯2, π₯3),
where π is a constant independent of the point (π₯10, π₯20, π₯30) β π or the direction of π ππβ . Since
the point (π₯10, π₯20, π₯30) β π is arbitrary, inequality (3.13) holds true. β
Let π£ be a solution of the problem
βπ£ = 0 on π , π£ = Ξ¨π on Ξπ, π = 1,2, β¦ ,6, (3.16)
where Ξ¨π, π = 1,2, β¦ ,6 are given functions and
Ξ¨π β πΆ5,π(Ξπ), 0 < π < 1, π = 1,2, β¦ ,6, Ξ¨π = Ξ¨π on Ξ³ππ, (3.17)
π2Ξ¨π
ππ‘π2+
π2Ξ¨πππ‘π2
+π2Ξ¨π
ππ‘ππ2= 0 on Ξ³ππ, (3.18)
π4Ξ¨π
ππ‘π4+
π4Ξ¨π
ππ‘π2ππ‘ππ2=
π4Ξ¨πππ‘π4
+π4Ξ¨π
ππ‘π2ππ‘π£π2 on Ξ³ππ . (3.19)
Lemma 3.3 The next inequality is true
|π8π£(π₯1, π₯2, π₯3)
ππ8| β€ π1π
πβ3(π₯1, π₯2, π₯3), (π₯1, π₯2, π₯3) β π , (3.20)
where π1 is a constant independent of the direction of differentiation π ππβ .
Proof. We choose an arbitrary point (π₯10, π₯20, π₯30) β R. Let π0 = π(π₯10, π₯20, π₯30) and π β RΜ
be the closed ball of radius π0 centred at (π₯10, π₯20, π₯30). Consider the harmonic function on π
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40
π(π₯1, π₯2, π₯3) =π5π£(π₯1, π₯2, π₯3)
ππ5β
π5π£(π₯10, π₯20, π₯30)
ππ5. (3.21)
As it follows from Theorem 2.1 in (Volkov, 1969) the solution π’ of problem (3.1), which
satisfies the conditions (3.17)-(3.19) belongs to the class πΆ5,π(RΜ ), 0 < π < 1. Then for
the function (3.21) we have
max(π₯1,π₯2,π₯3)βοΏ½Μ οΏ½0
|π(π₯1, π₯2, π₯3)| β€ π0π0π, (3.22)
where π0 is a constant independent of the point (π₯10, π₯20, π₯30) β R or the direction of π ππβ .
Since π’ is harmonic in π , by using estimation (3.22) and applying Lemma 3 in (Mikeladze,
1978) we have
|π3
ππ3(π5π£(π₯1, π₯2, π₯3)
ππ5β
π5π£(π₯10, π₯20, π₯30)
ππ5)| β€ π
π0π
π03 ,
or
|π8π£(π₯1, π₯2, π₯3)
ππ8| β€ ππ0
πβ3(π₯1, π₯2, π₯3),
where π is a constant independent of the point (π₯10, π₯20, π₯30) β R or the direction of π ππβ .
Since the point (π₯10, π₯20, π₯30) β R is arbitrary, inequality (3.20) holds true. β
3.2 27-Point Finite Difference Method for the Dirichlet Problem
Let β > 0, and ππ ββ β₯ 6 where π = 1,2,3, β¦ ,6 integers. We assign π β a cubic grid on π , with
step β, obtained by the planes π₯π = 0, β, 2β,β¦ , π = 1,2,3. Let π·β be a set of nodes of this grid,
Rβ = π β© π·β, Ξπβ = Ξπ β© π·
β, and Ξβ = Ξ1β βͺ Ξ2
β βͺ β¦βͺ Ξ6β.
Let the operator β be defined as follows by (Mikeladze, 1938).
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41
βπ’(π₯1, π₯2, π₯3) =1
128(14 β π’π + 3 β π’π
18
π=7(2)
6
π=1(1)
+ β π’π
26
π=19(3)
),
(π₯1, π₯2, π₯3) β π , (3.23)
where the sum β(π) is taken over the grid nodes that are at a distance of βπβ from the point
(π₯1, π₯2, π₯3), and π’π, π’π and π’π are the values of π’ at the corresponding grid points.
The red points have distance β from the center point, the white points have
distance β2β from the center point and the blue points have distance β3β
from center point.
Let π, π0, π1, β¦ be constants which are independent of β 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 = βπ’β on Rβ, π’β = ππ on Ξπ
β, π = 1,2, β¦ ,6. (3.24)
Figure 3.1: 26 points around center point using
operator β
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42
By the maximum principle system (3.24) has a unique solution.
Let π πβ be the set of the grid nodes Rβ whose distance from Ξ is πβ. It is obvious that 1 β€ π β€
π(β), where
π(β) = [1
2βπππ{π1, π2, π3}], (3.25)
[π] is the integer part of π.
We define for 1 β€ π β€ π(β) the function
πβπ = {
1, π(π₯1, π₯2, π₯3) = πβ,
0, π(π₯1, π₯2, π₯3) β πβ. (3.26)
Consider two systems of grid equations
π£β = π΄π£β + πβ, on π β, π£β = 0 on Ξβ, (3.27)
οΏ½Μ οΏ½β = π΄οΏ½Μ οΏ½β + οΏ½Μ οΏ½β, on π β, οΏ½Μ οΏ½β = 0 on Ξβ, (3.28)
where πβ and οΏ½Μ οΏ½β are given functions and |πβ| β€ οΏ½Μ οΏ½β on π β.
Lemma 3.4 The solution π£β and οΏ½Μ οΏ½β to systems (3.23) and (3.31) satisfy the inequality
|π£β| β€ οΏ½Μ οΏ½β on π β.
Proof. The proof of Lemma 3.4 follows from the comparison theorem (Samarskii, 2001).
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43
Lemma 3.5 The solution of the system
π£βπ = βπ£β
π + πβπ on Rβ, π£β
π = 0 on Ξβ (3.29)
satisfies the inequality
π£βπ(π₯1, π₯2, π₯3) β€ πβ
π, 1 β€ π β€ π(β), (3.30)
where πβπ is defined as follows
πβπ = πβ
π(π₯1, π₯2, π₯3) = {6π
β, 0 β€ π(π₯1, π₯2, π₯3) β€ πβ ,
6π, π(π₯1, π₯2, π₯3) > πβ . (3.31)
Proof. By virtue of (3.23) and (3.31), and in consider of Fig 3.2, we have for 0 β€ π = πβ
βπβπ =
1
128[14(2 Γ 6(π β 1) + 4 Γ 6π) + 3(7 Γ 6(π β 1) + 5 Γ 6π)
+6 Γ 6(π β 1) + 2 Γ 6π] =1
128[504π β 168
+216π β 126 + 48π β 36] =1
128[768π β 330]
= 6π β330
128,
which leads to
πβπ β βπβ
π =330
128> 1 = πβ
π.
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44
In consider of Fig 3.3 for π > πβ, then
βπβπ =
1
128[14(6 Γ 6π) + 3(12 Γ 6π) + 8 Γ 6π] =
768
128π = 6π,
which leads to
πβπ β βπβ
π = 0.
Figure 3.2: The selected region in π is π = πβ
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45
In consider of Fig 3.4 for π < πβ, then
βπβπ =
1
128[14(6(π β 2) + 4 Γ 6(π β 1) + 6π) + 3(4 Γ 6(π β 2)
+4 Γ 6(π β 1) + 4 Γ 6π) + 4 Γ 6(π β 2) + 4 Γ 6π]
= 504π β 504 + 216π β 216 + 48π β 48
=768
128π β
768
128= 6π β 6 = 6(π β 1),
which leads to
πβπ β βπβ
π = 0.
Figure 3.3: The selected region in π is π > πβ
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46
From the above calculations we have
πβπ = βπβ
π + πβπ on Rβ, πβ
π = 0 on Ξβ, π = 1,β¦ ,π(β), (3.32)
where |πβπ| β₯ 1.
On the basis of (3.26), (3.31), (3.32) and Lemma 3.4, we obtain
π£βπ β€ πβ
π for all π, 1 β€ π β€ π(β).
β
Introducing the notation π₯0 = (π₯10, π₯20, π₯30) , we used Taylorβs formulate to represent the
solution of Dirichlet problem around some point π₯0 β π β
π’(π₯1, π₯2, π₯3) = π7(π₯1, π₯2, π₯3; π₯0) + π8(π₯1, π₯2, π₯3; π₯0), (3.33)
where π7 is the seventh order Taylorβs polynomial and π8(π₯1, π₯2, π₯3; π₯0) is the remainder term.
Figure 3.4: The selected region in π is π < πβ
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47
Here,
π7(π₯10, π₯20, π₯30; π₯0) = π’(π₯10, π₯20, π₯30; π₯0) + π8(π₯10, π₯20, π₯30; π₯0) = 0.
Lemma 3.6 It is true that
βπ’(π₯10, π₯20, π₯30) = π’(π₯10, π₯20, π₯30) + βπ8(π₯10, π₯20, π₯30; π₯0), (π₯10, π₯20, π₯30) β
π β,
Proof. Let π7(π₯10, π₯20, π₯30; π₯0) be a Taylorβs polynomial.
By Direct calculations we have
βπ7(π₯10, π₯20, π₯30) = π’(π₯10, π₯20, π₯30) +1
128[π2
ππ₯12 (
π2π’(π₯10, π₯20, π₯30)
ππ₯12
+π2π’(π₯10, π₯20, π₯30)
ππ₯22 +
π2π’(π₯10, π₯20, π₯30)
ππ₯32 ) +
π2
ππ₯22 (
π2π’(π₯10, π₯20, π₯30)
ππ₯12
+π2π’(π₯10, π₯20, π₯30)
ππ₯22 +
π2π’(π₯10, π₯20, π₯30)
ππ₯32 ) +
π2
ππ₯32 (
π2π’(π₯10, π₯20, π₯30)
ππ₯12
+π2π’(π₯10, π₯20, π₯30)
ππ₯22 +
π2π’(π₯10, π₯20, π₯30)
ππ₯32 )] +
1
1536[π4
ππ₯14 (
π2π’(π₯10, π₯20, π₯30)
ππ₯12
+π2π’(π₯10, π₯20, π₯30)
ππ₯22 +
π2π’(π₯10, π₯20, π₯30)
ππ₯32 ) +
π4
ππ₯24 (
π2π’(π₯10, π₯20, π₯30)
ππ₯12
+π2π’(π₯10, π₯20, π₯30)
ππ₯22 +
π2π’(π₯10, π₯20, π₯30)
ππ₯32 ) +
π4
ππ₯34 (
π2π’(π₯10, π₯20, π₯30)
ππ₯12
+π2π’(π₯10, π₯20, π₯30)
ππ₯22 +
π2π’(π₯10, π₯20, π₯30)
ππ₯32 ) + 4
π4
ππ₯12ππ₯2
2 (π2π’(π₯10, π₯20, π₯30)
ππ₯12
+π2π’(π₯10, π₯20, π₯30)
ππ₯22 +
π2π’(π₯10, π₯20, π₯30)
ππ₯32 ) + 4
π4
ππ₯12ππ₯3
2 (π2π’(π₯10, π₯20, π₯30)
ππ₯12
+π2π’(π₯10, π₯20, π₯30)
ππ₯22 +
π2π’(π₯10, π₯20, π₯30)
ππ₯32 ) + 4
π4
ππ₯22ππ₯3
2 (π2π’(π₯10, π₯20, π₯30)
ππ₯12
+π2π’(π₯10, π₯20, π₯30)
ππ₯22 +
π2π’(π₯10, π₯20, π₯30)
ππ₯32 )]
= π’(π₯10, π₯20, π₯30).
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48
Since π’ is harmonic function, all terms on the right-hand side of this equality vanished except
the first term. Thus
βπ7(π₯10, π₯20, π₯30) = π’(π₯10, π₯20, π₯30).
Combining this with (3.33) and recalling the linearity of β, Lemma 3.6. is proved. β
Lemma 3.7 Let π’ be a solution of problem (3.1) The inequality holds
max(π₯1,π₯2,π₯3)βRπβ
|βπ’ β π’| β€ πβ7+π
π1βπ, π = 1,β¦ ,π(β). (3.34)
Proof. Let (π₯01, π₯02, π₯03) be a point of π 1β, and let
π 0 = {(π₯1, π₯2, π₯3): |π₯π β π₯π0| < β, π = 1,2,3 } (3.35)
be an elementary cube, some faces of which lie on the boundary of the rectangular parallelepiped
π .
On the vertices of π 0, and on the center of its faces and edges lie the nodes of which the function
values are used to evaluate β(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) β R1β by Taylorβs formula
π’(π₯1, π₯2, π₯3) = π7(π₯1, π₯2, π₯3; π₯0) + π8(π₯1, π₯2, π₯3; π₯0), (3.36)
where π7(π₯1, π₯2, π₯3; π₯0) is the seventh order Taylorβs polynomial, π8(π₯1, π₯2, π₯3; π₯0) is the
remainder term. Taking into account the function π’ is harmonic, hence by Lemma 3.6 we have
βπ7(π₯10, π₯20, π₯30; π₯0) = π’(π₯10, π₯20, π₯30). (3.37)
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49
Now, we estimate π8 at the nodes of the operator β. We take node (π₯10 + β, π₯20, π₯3