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Filomat 32:3 (2018), 791–800

https://doi.org/10.2298/FIL1803791D

Published by Faculty of Sciences and Mathematics,

University of Niˇ

s, Serbia

Available at: http://www.pmf.ni.ac.rs/filomat

14-Point Diﬀerence Operator for the Approximation of the First

Derivatives of a Solution of Laplace’s Equation in a Rectangular

Parallelepiped

Adiguzel A. Dosiyeva, Hediye Sarikayaa

aNear East University, Department of Mathematics, Nicosia, KKTC, Mersin 10, Turkey

Abstract. A 14-point diﬀerence operator is used to construct ﬁnite diﬀerence problems for the approxi-

mation of the solution, and the ﬁrst order derivatives of the Dirichlet problem for Laplace’s equations in a

rectangular parallelepiped. The boundary functions ϕjon the faces Γj,j=1,2, ..., 6 of the parallelepiped

are supposed to have pth order derivatives satisfying the H¨

older condition, i.e., ϕj∈Cp,λ (Γj), 0 < λ < 1,

where p={4,5}.On the edges, the boundary functions as a whole are continuous, and their second and

fourth order derivatives satisfy the compatibility conditions which result from the Laplace equation. For

the error uh−uof the approximate solution uhat each grid point (x1,x2,x3),|uh−u|≤cρp−4(x1,x2,x3)h4is

obtained, where uis the exact solution, ρ=ρ(x1,x2,x3) is the distance from the current grid point to the

boundary of the parallelepiped, his the grid step, and cis a constant independent of ρand h. It is proved

that when ϕj∈Cp,λ,0<λ<1,the proposed diﬀerence scheme for the approximation of the ﬁrst derivative

converges uniformly with order O(hp−1),p∈ {4,5}.

1. Introduction

It is well known that the use of diﬀerence operators with a low number of pattern and with the highest

order of accuracy for the approximate solution of diﬀerential equations reduces the eﬀective realization of

the obtained system of ﬁnite-diﬀerence equations. Moreover, to enlarge the class of applied problems the

convergence of the diﬀerence solutions are preferred to be investigated under the weakened assumptions on

the smoothness of the boundary conditions. All of these become more valuable in 3D problems, especially

the derivatives of the unknown solution are sought.

The application of derivatives arise in many applied problems such as problems in electrophysics in

which the ﬁrst derivatives of the potential function deﬁne the electrostatic ﬁeld [7], and in the fracture

problems where the ﬁrtst derivatives of the stress function deﬁne the components of the tangential stress

[8].

The investigation of approximate derivatives started in [9],where it was proved that 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 hwith which the diﬀerence

solution converges on the given domain. The uniform convergence of the diﬀerence derivatives over the

2010 Mathematics Subject Classiﬁcation. Primary 65M06; Secondary 65M12, 65M22

Keywords. Finite diﬀerence method, approximation of the ﬁrst derivatives, error estimations, Laplace’s equation on parallelepiped

Received: 8 December 2016; 13 February 2017; Accepted: 19 March 2017

Communicated by Allaberen Ashyralyev

Email addresses: adiguzel.dosiyev@emu.edu.tr (Adiguzel A. Dosiyev), sarikayahediye@windowslive.com (Hediye Sarikaya)

A.A. Dosiyev, H. Sarikaya /Filomat 32:3 (2018), 791–800 792

whole grid domain to the corresponding derivatives of the solution for the 2D Laplace equation with the

order O(h2) was proved in [14].In [5],for the ﬁrst and pure second derivatives of the solution of the 2D

Laplace equation special ﬁnite diﬀerence problems were investigated. It was proved that the solution of

these problems converge to the exact derivatives with the order O(h4).

In [17] for the 3D Laplace equation the convergence of order O(h2) of the diﬀerence derivatives to the

corresponding ﬁrst order derivatives of the exact solution is proved. It was assumed that ton the faces

the boundary functions have third derivatives satisfying the H¨

older condition. Furthermore, they are

continuous on the edges, and their second derivatives satisfy the compatibility condition that is implied

by the Laplace equation. Whereas in [16] when the boundary values on the faces of a parallelepiped are

supposed to have the fourth derivatives satisfying the H¨

older condition, the constructed diﬀerence schemes

converge with order O(h2) to the ﬁrst and pure second derivatives of the exact solution. In [5] it is assumed

that the boundary functions on the faces have sixth order derivatives satisfying the H¨

older condition, and

the second and fourth order derivatives satisfy some compatibility conditions on the edges. Diﬀerent

diﬀerence schemes with the use of the 26-point diﬀerence operator are constructed on a cubic grid with

mesh size h,to approximate the ﬁrst and pure second derivatives of the solution of the Dirichlet problem

with order O(h4).

In this paper O(hp−1),p=4,5 order of approximation for the ﬁrst derivatives of the solution of 3D

Laplace’s equation is obtained under weaker assumptions on the smoothness of the boundary functions on

the faces of the parallelepiped than those used in [6]. Moreover, to construct the ﬁnite diﬀerence problems

the diﬀerence operator with a lower number of pattern is used.

Finally, the obtained theoretical results are supported by the illustration of numerical results.

2. Some Properties of a Solution of the Dirichlet Problem on a Rectangular Parallelepiped

Let R={(x1,x2,x3): 0 <xi<ai,i=1,2,3}be an open rectangular parallelepiped; Γj,j=1,2, ..., 6 be

its faces including the edges; Γjfor j=1,2,3 (j=4,5,6) belongs to the plane xj=0 ( xj−3=aj−3),and let

Γ = 6

∪

j=1Γjbe the boundary of R;γµν = Γµ∩Γνbe the edges of the parallelepiped R.Ck,λ(E) is the class of

functions that have continuous kth derivatives satisfying the H¨

older condition with an exponent λ∈(0,1).

Consider the boundary value problem

∆u=0 on R,u=ϕjon Γj,j=1,2, ..., 6 (1)

where ∆≡∂2

∂x2

1

+∂2

∂x2

2

+∂2

∂x2

3

, ϕjare given functions.

Assume that

ϕj∈Cp,λ Γj,0<λ<1,j=1,2, ..., 6,p∈{4,5}(2)

ϕµ=ϕνon γµν,(3)

∂2ϕµ

∂t2

µ

+∂2ϕν

∂t2

ν

+∂2ϕµ

∂t2

µν

=0 on γµν, (4)

∂4ϕµ

∂t4

µ

+∂4ϕµ

∂t2

µ∂t2

µν

=∂4ϕν

∂t4

ν

+∂4ϕν

∂t2

ν∂t2

νµ

on γµν,(5)

where 1 ≤µ < ν ≤6, ν −µ,3,tµν is an element in γµν ,and tµand tνis an element of the normal to γµν on

the face Γµand Γν,respectively.

The following Lemma follows from Theorem 2.1 in [12].

Lemma 2.1. Under conditions (2)−(5),the solution u of the Dirichlet problem (1)belong to the H¨older class Cp,λ(R),

0<λ<1,p∈{4,5}.

A.A. Dosiyev, H. Sarikaya /Filomat 32:3 (2018), 791–800 793

Lemma 2.2. Let ρ(x1,x2,x3)be the distance from the current point of the open parallelepiped R to its boundary and

let ∂

∂l≡α1∂

∂x1+α2∂

∂x2+α3∂

∂x3, α2

1+α2

2+α2

3=1.

Then the next inequality holds

∂6u(x1,x2,x3)

∂l6

≤cρp+λ−6(x1,x2,x3),(x1,x2,x3)∈R and p ∈{4,5}(6)

where u is the solution of the problem (1),c is a constant independent of the direction of derivative ∂

∂l.

Proof. Since u∈Cp,λ(R),p∈{4,5}(Lemma 2.1) the proof Lemma 2.2 follows with the use of Lemma 3 in [10]

(Chap.4, Sec.3)

3. Finite Diﬀerence Problem

We introduce a cubic grid with a step h>0 deﬁned by the planes xi=0,h,2h, ..., i=1,2,3. It is assumed

that the edge lengths of Rand hare such that ai

h≥4 (i=1,2,3) are integers.

Let Dhbe the set of nodes of the grid constructed, Rh=R∩Dh,Rh=R∩Dh,Rk

h⊂Rhbe the set of nodes

of Rhlying at a distance of kh away from the boundary Γof R,and Γh= Γ ∩Dh.

The 14-point diﬀerence operator Son the grid is deﬁned as (see [19])

Su(x1,x2,x3)=1

56

8

6

X

p=1(1)

up+

14

X

q=7(3)

uq

,(x1,x2,x3)∈Rh,(7)

where P(m)is the sum extending over the nodes lying at a distance of m1/2haway from the point (x1,x2,x3)

and upand uqare the values of uat the corresponding nodes.

On the boundary Γof R, we deﬁne continuous on the entire boundary including the edges of R, the

function ϕas follows

ϕ=

ϕ1on Γ1

ϕjon Γj\ j−1

∪

i=1Γi!,j=2,3, ..., 6.(8)

Obviously,

ϕ=ϕjon Γj,j=1,2, ..., 6.

We consider the ﬁnite diﬀerence problem approximating Dirichlet problem (1):

uh=Suhon Rh,uh=ϕon Γh,(9)

where Sis the diﬀerence operator given by (7)and ϕis the function deﬁned by (8). By maximum principle,

the system (9)has a unique solution (see [11], Chap. 4).

In what follows and for simplicity, we denote by c,c1,c2, ... constants, which are independent of hand

the nearest factors, the identical notation will be used for various constants.

Consider two systems of grid equations

vh=Svh+1h,on Rh,vh=0 on Γh,(10)

vh=Svh+1h,on Rh,vh=0 on Γh,(11)

where 1hand 1hare given functions and 1h≤1hon Rh.

Lemma 3.1. The solutions vhand vhof systems (10)and (11)satisfy the inequality

vh≤vhon Rh.

A.A. Dosiyev, H. Sarikaya /Filomat 32:3 (2018), 791–800 794

The proof of Lemma 3.1 is similar to that of the comparison theorem in [11] (Chap.4, Sec.3).

Deﬁne

N(h)=min {a1,a2,a3}

2h,(12)

where [a]is the integer part of a.

Consider for a ﬁxed k,1≤k≤N(h) the systems of grid equations

vk

h=Svk

h+1k

hon Rk

h,vk

h=0 on Γh,(13)

where

1k

h=(1, ρ (x1,x2,x3)=kh,

0, ρ (x1,x2,x3),kh.

Lemma 3.2. The solution vk

hof the system (13)satisﬁes the inequality

vk

h(x1,x2,x3)≤Tk

h,1≤k≤N(h),(14)

where Tk

his deﬁned as

Tk

h=Tk

h(x1,x2,x3)=(5ρ

h,0≤ρ(x1,x2,x3)≤kh,

5k, ρ (x1,x2,x3)>kh.(15)

Proof. By the direct calculation of the expression STk

h,we obtain

Tk

h−STk

h≥(1, ρ (x1,x2,x3)=kh,

0, ρ (x1,x2,x3),kh,(16)

on Rh.On the basis of (13), inequalities (16) and taking the boundary condition Tk

h=0 on Γhinto account,

by Lemma 3.1, we get (14).

Let x0=(x10,x20 ,x30),be some point in Rh.By Taylor’s formula for the solution uof the problem (1)

around the point x0,we have

u(x1,x2,x3)=p5(x1,x2,x3;x0)+r5(x1,x2,x3;x0),(17)

where p5is ﬁfth-degree Taylor polynomial and r5is remainder.

Since uis a harmonic function and Sis linear, by taking into account that Sp5(x10 ,x20,x30;x0)=

u(x10,x20 ,x30) from (17) follows

Su(x10,x20 ,x30)=u(x10 ,x20,x30)+Sr5(x10 ,x20,x30 ;x0).(18)

Lemma 3.3. The following estimation holds

max

(x1,x2,x3)∈Rk

h

|Su −u|≤c4

hp+λ

k6−p−λ,k=1,2, ..., N(h),p∈{4,5},(19)

where u is the solution of the Dirichlet problem (1),S is the diﬀerence operator deﬁned by (7),and N(h)is given by

(12).

A.A. Dosiyev, H. Sarikaya /Filomat 32:3 (2018), 791–800 795

Proof. Let x0=(x10,x20,x30)be some point in R1

h,and let Pmnq =x10 +mh,x20 +nh,x30 +qh, where m,n,q=

0,±1,m2+n2+q2,0,be any point in the pattern of operator S.Then by using the integral form of the

remainder term of Taylor’s formula for each point Pmnq by virtue of Lemma 2.1 and Lemma 2.2, we obtain

r5x10 +mh,x20 +nh,x30 +qh;x0≤chp+λ,p∈ {4,5}.(20)

From the structure (7) of the operator Sfollows that its norm in the uniform metric is equal to one, then

using by (20), we have

|Sr5(x10,x20 ,x30;x0)|≤chp+λ,p∈ {4,5}.(21)

On the basis of (18) and (21) follows the inequality (19), for k=1.Let x0∈Rk

hbe an arbitrary point for

2≤k≤N(h),and let r5(x1,x2,x3;x0)be remainder term of the Taylor formula (17)in the Lagrange form.

Then Sr6(x10,x20 ,x30;x0)can be expressed linearly in terms of the 14 number of sixth derivatives of uat

some points on the open intervals connecting the points of pattern of the operator Swith the point x0.The

sum of the absolute values of the coeﬃcients multiplying the sixth derivatives does not exceed ch6which is

independent k0(2≤k0≤N(h))or the point x0∈Rk0

h.Using the estimation of the sixth derivatives by Lemma

2.2, for all k,2≤k≤N(h),we obtain

|Sr5(x10,x20 ,x30;x0)|≤c1

h6

(kh)6−p−λ=c1

hp+λ

k6−p−λ.(22)

By virtue of (18)and (22)follows the estimation (19).

Theorem 3.4. Assume that the boundary functions ϕjsatisfy conditions (2)−(5). Then at each point (x1,x2,x3)∈Rh

|uh−u|≤c0h4ρp−4,p∈{4,5},(23)

where uhis the solution of the ﬁnite diﬀerence problem (9), u is the exact solution of problem (1),and ρ=ρ(x1,x2,x3)

is the distance from the current point (x1,x2,x3)∈Rhto the boundary of the rectangular parallelepiped R.

Proof. Let εk

h,1≤k≤N(h),be a solution of the system

εk

h=Sεk

h+µk

hon Rh, εk

h=0 on Γh,(24)

where

µk

h=(Su −uon Rk

h

0 on Rh\Rk

h..(25)

Let

εh=uh−u on Rh.(26)

By (9)and (26)the error function εhsatisﬁes the system of equations

εh=Sεh+(Su −u)on Rh, εh=0 on Γh.(27)

We represent a solution of the system (27)as follows

εh=

N(h)

X

k=1

εk

h,(28)

where N(h) deﬁned by (12), εk

h,1≤k≤N(h),is a solution of the system

εk

h=Sεk

h+σk

hon Rh, εk

h=0 on Γh,(29)

A.A. Dosiyev, H. Sarikaya /Filomat 32:3 (2018), 791–800 796

when

σk

h=(Su −uon Rk

h

0 on Rh\Rk

h.(30)

Then on the basis of (28), (29), (30), Lemma 3.2 and Lemma 3.3,for the solution of (27), we have

|εh|≤

N(h)

X

k=1εk

h≤

N(h)

X

k=1

Tk

hmax

(x1,x2,x3)∈Rk

h

|Su −u|≤c1h6

N(h)

X

k=1

Tk

h

(kh)6−p−λ

≤5c1hp+λ

ρ/h−1

X

k=1

k

(k)6−p−λ+5c1h6

N(h)

X

k=ρ/h

ρ/h

(kh)6−p−λ

≤5c1hp+λ

ρ/h−1

X

k=1

k−5+p+λ+5c1hp−1+λρ

N(h)

X

k=ρ/h

k−6+p+λ

≤c2h4ρ−4+p+λ+c3h4ρ≤c4h4ρp−4,p∈{4,5}.(31)

From (26)and (31),for any point (x1,x2,x3)∈Rh, we obtain

|uh−u|=|εh|≤c0h4ρp−4(x1,x2,x3),p∈ {4,5}.

4. Approximation of the First Derivative

4.1. Boundary Function is from C5,λ

Let the boundary functions ϕj,j=1,2, ..., 6,in problem (1)on the faces Γjbe satisﬁed the conditions

ϕj∈C5,λ Γj,0<λ<1,j=1,2, ..., 6,(32)

i.e., p=5 in (2).Let ube a solution of the problem (1)with the conditions (32)and (3)−(5).

We put v=∂u

∂x1,and Φj=∂u

∂x1on Γj,j=1,2, ..., 6.It is obvious that the function vis a solution of the

following boundary value problem

∆v=0 on R,v= Φjon Γj,j=1,2, ..., 6,(33)

where uis a solution of the problem (1)for p=5.

We deﬁne an approximate solution of problem (33) as a solution of the following ﬁnite diﬀerence problem

νh=Sνhon Rh, νh= Φjh (uh) on Γh

j,j=1,2, ..., 6,(34)

where uhis the solution of the problem (9), Φ1h(Φ4h) is the fourth order forward (backward) numerical

diﬀerentiation operator (see [1], [2]) used in [6] with the 26-point diﬀerence operator. On the nodes Γh

p,the

boundary values are deﬁned as Φph(uh)=∂ϕp

∂x1,p=2,3,5,6.

Theorem 4.1. The estimation is true

max

(x1,x2,x3)∈Rh

νh−∂u

∂x1

≤ch4,(35)

where u is the solution of the problem (1), νhis the solution of the ﬁnite diﬀerence problem (34).

A.A. Dosiyev, H. Sarikaya /Filomat 32:3 (2018), 791–800 797

Proof. Let

h=νh−νon Rk,(36)

where ν=∂u

∂x1.From (34)and (36),we have

h=Sh+(Sν−ν)on Rh,

h= Φkh(uh)−νon Γh

k,k=1,4, h=0 on Γh

p,p=2,3,5,6.

We put

h=1

h+2

h,(37)

where

1

h=S1

hon Rh,(38)

1

h= Φkh(uh)−νon Γh

k,k=1,4, 1

h=0 on Γh

q,q=2,3,5,6; (39)

2

h=S2

h+(Sν−ν)on Rh, 2

h=0 on Γh

j,j=1,2, ..., 6.(40)

First, we estimate the diﬀerence Φkh(uh)−νon Γh

k,k=1,4 using the representation

Φkh(uh)−ν=(Φkh (uh)−Φkh(u))+(Φkh(u)−v)(41)

Since Φkh(u),k=1,4 are the fourth order approximation of ∂u/∂x1on Γkand by Lemma 2.1 the ﬁfth order

partial derivatives of the solution uare bounded in R, the diﬀerence Φkh(u)−vhas estimation (see [1], [2])

max

k=1,4max

(x1,x2,x3)∈Γk

k

|Φkh(u)−ν|≤c1h4.(42)

To estimate Φkh(uh)−Φkh (u),we take the fourth order forward formula (k=1),

Φ1h(uh)=1

12h[−25ϕ1(x2,x3)+48uh(h,x2,x3)−36uh(2h,x2,x3)(43)

+16uh(3h,x2,x3)−3uh(4h,x2,x3)] on Γh

1.

Using the pointwise estimation (24) in Theorem 3.4, when p=5,and taking into account the values of the

distance function ρ(x1,x2,x3) in the formula (43), we have

|Φ1h(uh)−Φ1h(u)|≤c2h4.(44)

The estimation (44) is true for the backward formula (k=4),also. On the basis of (41), (42), (44), by using

the maximum principle, for the solution of system (38),(39),we have

max

(x1,x2,x3)∈Rh1

h≤c3h4.(45)

The solution ε2

hof system (40)is the error function of the ﬁnite diﬀerence solution for problem (33),when

the boundary functions Φj=∂u/∂x1,j=1,2, ..., 6,as follows from (2) −(5) satisfy the conditions

Φj∈C4,λ Γj,0<λ<1,j=1,2, ..., 6,

Φµ= Φνon γµν,

∂2Φµ

∂t2

µ

+∂2Φν

∂t2

ν

+∂2Φµ

∂t2

µν

=0 on γµν

Then, on the basis of Theorem 4 in [19] for the error ε2

h, we have

max

(x1,x2,x3)∈Rh2

h≤c4h4.(46)

By virtue of (37),(45),and (46)follows the inequality (35).

A.A. Dosiyev, H. Sarikaya /Filomat 32:3 (2018), 791–800 798

4.2. Boundary Function From C4,λ

Let the boundary functions ϕj∈C4,λ Γj,0< λ < 1,j=1,2, ..., 6,in (1)−(5),i.e., p=4 in (2),and let

v=∂u

∂x1and let Φj=∂u

∂x1on Γj,j=1,2, ..., 6,and consider the boundary value problem:

∆v=0 on R,v= Φjon Γj,j=1,2, ..., 6,(47)

where uis a solution of the boundary value problem (1).

We deﬁne the following third order numerical diﬀerentiation operators Φνh, ν =1,4

Φ1h(uh)=1

6h[−11ϕ1(x2,x3)+18uh(h,x2,x3)−9uh(2h,x2,x3)

+2uh(3h,x2,x3)] on Γh

1,(48)

Φ4h(uh)=1

6h[11ϕ4(x2,x3)−18uh(a1−h,x2,x3)+9uh(a1−2h,x2,x3)

−2uh(a1−3h,x2,x3)] on Γh

4,(49)

and we put

Φph(uh)=∂ϕp

∂x1

,on Γh

p,p=2,3,5,6 (50)

where uhis the solution of the ﬁnite diﬀerence problem (9).

It is obvious that Φj,j=1,2, ..., 6,satisfy the conditions

Φj∈C3,λ Γj,0<λ<1,j=1,2, ..., 6,(51)

Φµ= Φνon γµν,(52)

∂2Φµ

∂t2

µ

+∂2Φν

∂t2

ν

+∂2Φµ

∂t2

µν

=0 on γµν.(53)

Let νhbe the solution of the following ﬁnite diﬀerence problem

νh=Sνhon Rh, νh= Φjh on Γh

j,j=1,2, ..., 6,(54)

where Φjh,j=1,2, ..., 6,are deﬁned by (48)−(50).

Theorem 4.2. Let the boundary function ϕj∈C4,λ (Γj),j=1, ..., 6.The estimation is true

max

(x1,x2,x3)∈Rh

νh−∂u

∂x1

≤ch3,(55)

where u is the solution of the problem (1), νhis the solution of the ﬁnite diﬀerence problem (54).

Proof. The proof of Theorem 4.2 is similar to that of Theorem 4.1, with the following diﬀerences in estimation

for the errors 1

hand 2

hin (37) : (i) putting p=4 in Lemma 2.1 and Theorem 3.4, and taking into account

that the formulae (48) and (49) are the third order, the estimation

max

(x1,x2,x3)∈Rh1

h≤c5h3.

is proved. (ii) on the basis of (51)-(53) and Theorem 2 in [19], we obtain

max

(x1,x2,x3)∈Rh1

h≤c6h3.

Remark 4.3. We have investigated the method of high order approximations of the ﬁrst derivative ∂u/∂x1.

The same results are obtained for the derivatives ∂u/∂xl,l=2,3 analogously, by using the same order

forward and backward formulae in appropriate faces of the parallelepiped.

A.A. Dosiyev, H. Sarikaya /Filomat 32:3 (2018), 791–800 799

5. Numerical Results

Example 5.1. Let R={(x1,x2,x3): 0 <xi<1,i=1,2,3},and let Γj,j=1, ..., 6 be its faces. We consider the

following problem:

∆u=0 on R,u=ϕ(x1,x2,x3)on Γj,j=1, ..., 6,(56)

where

ϕ(x1,x2,x3)=x3−1

22

−

x2

1+x2

2

2

+x2

1+x2

2(5+1

30 )

2.cos 5+1

30.arctan x2

x1 (57)

is the exact solution of this problem,which is in C5,1/30.

We solve the system (9) and (34) to ﬁnd the approximate solution uhfor uand approximate ﬁrst derivative

vhfor ∂u

∂x1respectively.

In Tables 1 and 2 the maximum errors are given. Table 1 shows that the convergence order more than 4

which is corresponds to the product ρin Theorem 3.4. Table 2 justiﬁed estimation (35) in Theorem 4.1, i.e.,

the fourth order convergence.

hkuh−ukRhEm

u

2−37,5172E−09 32,13

2−42,3396E−10 32,66

2−57,1637E−12 32,74

2−62,1883E−13 32,75

2−76,6827E−15

Table1 Errors for the solution in maximum norm

hkvh−vkRhEm

v

2−34.5436E−03 13,40

2−43.3909E−04 14,76

2−52.2975E−05 15,40

2−61.4922E−06 15,70

2−79.5053E−08

Table 2 Errors for the ﬁrst derivative in maximum norm with the fourth-order formulae

Example 5.2. Let ube a solution of problem (56) when the boundary function ϕis chosen from C4,1

30 as

ϕ(x1,x2,x3)=x3−1

22

−

x2

1+x2

2

2

+x2

1+x2

2(4+1

30 )

2.cos 4+1

30.arctan x2

x1.

Table 3 and 4 give the fourth order convergence when boundary function is from C4,1

30 for both, solution

and ﬁrst derivative which are the numerical justiﬁcation of Theorem 3.4 and 4.2 respectively.

hkuh−ukRhEm

u

2−33,4801E−08 16,20

2−42,1486E−09 16,36

2−51,3135E−10 16,37

2−68,0228E−12 16,37

2−74,8998E−13

A.A. Dosiyev, H. Sarikaya /Filomat 32:3 (2018), 791–800 800

Table 3 Errors for the solution in maximum norm

hkvh−vkRhEm

v

2−38,5126E−03 6,49

2−41,3161E−03 7,29

2−51,8065E−04 7,66

2−62,3598E−05 7,83

2−73,0144E−06

Table 4 Errors for the ﬁrst derivative in maximum norm with the third-order formulae

In Tables 1-4 we have used the following notations:

kUh−UkRh=max

Rh

|Uh−U|and Em

U=kU−U2−mkRh

kU−U2−(m+1)kRh

where Ube the exact solution of the continuous

problem, and Uhbe its approximate values on Rh.

6. Conclusion

Three diﬀerent schemes with the 14-point diﬀerence operator are constructed on a cubic grid with

mesh size h, whose solutions separately approximate the solution of the Dirichlet problem for 3D Laplace’s

equation with the order O(h4ρp−4), p∈ {4,5},where ρ=ρ(x1,x2,x3)is the distance from the current point

(x1,x2,x3)∈Rhto the boundary of the rectangular parallelepiped Rand its ﬁrst derivatives with the orders

O(hp−1).

The obtained results can be used to highly approximate the derivatives of the solution of 3D Laplace’s

boundary value problems on a prism with an arbitrary polygonal base and on polyhedra by developing

the combined or composite grid methods [13, 15]. For the 2D case see [3, 4, 18, 20].

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