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Известия Саратовского университета. Новая серия. Серия: Математика
Механика. Информатика. 2024. Т. 24, вып. 2. С. 162-172
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2024:
vol. 24, iss. 2, pp. 162-172
https://mmi.sgu.ru
https://doi.org/10.18500/1816-9791-2024-24-2-162-172 EDN: WNJQOX
Article
A highly accurate difference method for solving the Dirichlet problem of the Laplace equation on a rectangular parallelepiped with boundary values in
A. A. Dosiyev
Western Caspian University, 31 Istiglaliyyat St., Baku AZ1001, Azerbaijan Adiguzel A. Dosiyev, [email protected], https:/orcid.org/0000-0001-9154-8116
Abstract. A three-stage difference method for solving the Dirichlet problem of Laplace's equation on a rectangular parallelepiped is proposed and justified. In the first stage, approximate values of the sum of the pure fourth derivatives of the solution are defined on a cubic grid by the 14-point difference operator. In the second stage, approximate values of the sum of the pure sixth derivatives of the solution are defined on a cubic grid by the simplest 6-point difference operator. In the third stage: the system of difference equations for the sought solution is constructed again by using the 6-point difference operator with the correction by the quantities determined in the first and the second stages. It is proved that the proposed difference solution to the Dirichlet problem converges uniformly with the order O(h6(| lnh\ + 1)), when the boundary functions on the faces are from C7'1 and on the edges their second, fourth, and sixth derivatives satisfy the compatibility conditions, which follows from the Laplace equation. A numerical experiment is illustrated to support the analysis made.
Keywords: finite difference method, 3D Laplace equation, cubic grids on parallelepiped, 14-point averaging operator, error estimations For citation: Dosiyev A. A. A highly accurate difference method for solving the Dirichlet problem of the Laplace equation on a rectangular parallelepiped with boundary values in Ck'1. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2024, vol. 24, iss. 2. pp. 162-172. https://doi.org/10.18500/1816-9791-2024-24-2-162-172 EDN: WNJQOX
This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0)
Научная статья УДК 518.517.944/947
Разностный метод высокой точности при решении задачи Дирихле для уравнения Лапласа на прямоугольном параллелепипеде с граничными значениями в Cк,г
А. А. Досиев
Западно-Каспийский университет, Азербайджан, AZ1001, г. Баку, ул. Истиглалият, д. 31
Досиев Адигезал Ахмед оглу, доктор физико-математических наук, преподаватель кафедры механики и математики, [email protected], https://orcid.org/0000-0001-9154-8116
Аннотация. В работе предлагается и обосновывается трехэтапный разностный метод для решения задачи Дирихле уравнения Лапласа на прямоугольном параллелепипеде. На первом этапе приближенное значение суммы из чистых четвертых производных решения определяется 14-точечным разностным оператором на кубической сетке. На втором этапе приближенное значение суммы из чистых шестых производных решения определяется простейшим 6-точечным разностным оператором. На третьем этапе система разностных уравнений для искомого решения конструируется также с помощью 6-точечного разностного оператора с коррекцией по результатам первого и второго этапов. Доказано, что предложенная разностная схема решения для задачи Дирихле сходится со скоростью O(h6(\ lnh\ + 1)), ^гда граничные функции на гранях из C7,1, а на ребрах их вторые, четвертые и шестые производные удовлетворяют условию согласования, вытекающего из уравнения Лапласа.
Ключевые слова: конечно разностный метод, 3D уравнения Лапласа, кубические сетки в параллелепипеде, 14-точечный оператор усреднения, оценки погрешности
Для цитирования: Dosiyev A. A. A highly accurate difference method for solving the Dirichlet problem of the Laplace equation on a rectangular parallelepiped with boundary values in Ck,x [Досиев А. А. Разностный метод высокой точности при решении задачи Дирихле для уравнения Лапласа на прямоугольном параллелепипеде с граничными значениями C] // Известия Саратовского университета. Новая серия. Серия: Математика. Механика. Информатика. 2024. Т. 24, вып. 2. С. 162-172. https:// doi.org/10.18500/1816-9791-2024-24-2-162-172, EDN: WNJQOX
Статья опубликована на условиях лицензии Creative Commons Attribution 4.0 International (CC-BY 4.0)
Introduction
A highly accurate method is one of the powerful tools for reducing the number of unknowns, which is the main problem in the numerical solution of differential equations to get reasonable results. In the most of approximations to get highly accurate results the difference operators with a high number of patterns 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 the bandwidth of the matrices of these systems of difference equations. As it was shown by R. E. Tarjan [1], in the case of the Gaussian elimination method the bandwidth elimination for n x n matrices with the bandwidth b, the computational cost is of order O(b2n).
One of the effective methods of increased accuracy which uses the simplest finite difference approximation by correcting the right-hand-side term with the application of the high order differences of the numerical solution of differential equation, was proposed by L. Fox [2] without theoretical justification. Some modification of Fox's approach was given by Woods [3]. A theoretical justification of Fox's method was presented by Volkov in [4,5]. From Volkov's results in the case of the Dirichlet problem for Poisson's equation on a rectangular domain n, it follows that the approximate solution obtained by the q-th correction of the right-hand side of the 5-point
scheme, the convergence order in the uniform metric is O(h2q), h is the mesh step, when the exact solution u has (2q + 2)-th derivatives on n satisfying a Holder condition with exponent A e (0,1), i.e., u e C2q+2'A(n).
In Berikelashvili and Midodashvili [6] it is proved that the corrected 5-point difference scheme on the rectangular grid is convergent at the rate O(|h|m), |h|2 = h2 + h|, in the discrete L2-norm, provided that the exact solution belongs to the Sobolev space W2m, m e [2,4].
In Volkov [7] a 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 functions on the faces of a parallelepiped have the sixth derivatives satisfying the Holder condition, and on the edges, besides the continuity they satisfy the compatibility condition for second derivatives, which results from the Laplace equation. It was proved that by using a simple 7-point scheme in two stages the order of uniform error can be improved up to O(h4lnh-1). From the conditions imposed on the boundary functions in [7], it does not follow as it was mistakenly declared in [6] that the exact solution belongs to C6,x (n).
Moreover, as it was shown in [8], the theoretical justification of the difference schemes needs special attention when the boundary values of a solution belong to the Holder classes C21-1,1 and 21 — 2 order derivatives satisfy the conjunction condition followed from the Laplace equation. In this case, some of 21 order derivatives may be unbounded near the boundary of the solution domain, and for the rate of convergence of the 27-point difference solution, when l = 3, O(h6(|| lnh| + 1)) of order is obtained.
In this paper, a three-stage difference method constructed a special combination of 15-point and 7-point schemes for solving the Dirichlet problem of Laplace's equation on a rectangular parallelepiped is proposed and justified. It is proved that the obtained difference method converges uniformly with an order of O(h6(|| lnh| + 1)) when the boundary functions on the faces are from C7,1, and on the edges their second, fourth, and sixth derivatives satisfy the compatibility conditions which follows from the Laplace equation.
A numerical experiment is illustrated to support the analysis made.
1. The Dirichlet problem on rectangular parallelepiped
Let R = {(x2, x2,x3) : 0 < x < a, i = 1,2,3} be an open rectangular parallelepiped, Tj (j = 1,2,..., 6) be its faces including the boundaries such that r^ for j = 1,2,3 (for j = 4,5,6) belongs to the plane xj = 0 (to the plane xj-3 = aj-3). Let r = u6=1 r be the boundary of the parallelepiped, let y be the union of the edges of R, and let rj = Tj\y and yuv = n Tv. We say that f e Ck,x(D) if f has continuous k—th derivatives on D satisfying a Holder condition with exponent A e (0,1], which is a Lipschitz condition when A = 1.
We consider the boundary value problem
Au = 0 on R, u = on rj, j = 1, 2,..., 6, (1)
where A = d2/dx2 + d2/dx2 + d2/dx2, pj are given functions.
Assume that
Pj e C7'1 (rj), j = 1, 2,..., 6, (2)
Pu = Pv on y^v , (3)
t+t+se-- - '4)
д4 ^ + д4 ^ _ д4 + д4 ^ dt4 + dt2, dt2 _ dt4 + dt2 dt2
д ^U _ д ^V , д ^U /ГЧ
+ Л+2 iU2 _ + Л+2iU2 0n YUV,
v?
дЧ + д6^ + д6^ _ д6^у + + д6^ n dt6 + дИ„ + д*4. дt2 дй at4 + д*6, + at4, дИ, 1uv ' (D)
where 1 < ^ < v < 6, v — ^ = 3, t^v is an element in y^v, t^ and tv is an element of the normal to y^v on the face and rv, respectively.
Let ) = 3{j/3} + 1, where {a} is the fractional part of a.
Lemma 1. In the open parallelepiped R it holds that
d4u(xi,x2,X3) = d4u(xi,x2,X3) + d4u(xi,X2,X3) +2d4u(xi, x2,X3) (7)
dX4 " d<(j) d<(j+i) dxJ(j )dxJ(j+i)'
d6u(x1?x2, X3) d6u(xi, x2, X3) d6u(xi?x2, X3)
j 6(j ) 6(j+1)
_3d6u(xi,x2,X3) _ 3d6u(xi,X2,X3) (8)
dx6(j ) dx6(j+1) dx6(j) dx6(j+i)
where u ¿s the solution to the Dirichlet problem (1).
Proof. The proof directly follows from the Laplace equation. □
On R, we define the functions
vk = vk(xi ,X2,X3) = -dx2h . k = 2,3 (9)
j = i j
where u is the solution to the Dirichlet problem (1).
Lemma 2. The functions (9) coincide with the unique continuous solution on R of the boundary value problems
Avk = 0 on R, vk = Vj on r, j = 1, 2,..., 6, k = 2, 3, (10)
where
,2 V2 (x x ) dVj (x6(j) ,x6(j+i)) + d4Pj (x6(j) ,x6(j+i)) +
= (x6(j) ,x6(j+i)) =-dx4-+-dx-+
, d4Pj (x6(j) ,x6(j+i)) (11)
+ dx2 dx2 ' (11) °X6(3)°X6(3+i)
3 , 3 (x x ) d 6 Pj (x6(j ),x6(j+i)) d 6 Pj (x6(j ) ,x6(j+i)) ( )
= (x6(j) ,x6(j+i)) =--dx4 dx2---dx2 dx4-• (12)
dx6(j ) dx6(j+i) dx6(j) dx6(j+i)
Proof. On the basis of (2)-(6), Theorem 2.1 in [9] it follows that a solution u of problem (1) belongs to the class C7,x(R), 0 < A < 1. Since any order derivatives of a harmonic function are also harmonic, the functions vk, k = 2,3 satisfy Laplace's equation. The boundary conditions in (10) with (11) and (12) follow from (1), Lemma 1 and (9). Then by Theorem 3.1 in [9] each of the functions vk, k = 2, 3 is the unique continuous solution on R of problem (10). □
Lemma 3. Even order derivatives in the form
'u
dx2Pdx2q dx8-2P-2q ' of the solution u of problem (1) are bounded on R.
0 < p < 4, 0 < q < 4 - p, (13)
Proof. Let ш = дгб .We have
Дш = 0 on R, ш = Ф7 on Г,, j = 1, 2,..., 6,
where
Ф,
д У,
j = 2, 3, 5,6,
д6 ^
Ф дхб
_ з - г, ^
д6^7 д6 ^
дх|дх2 дх^дх| дх® '
j = 1,4.
(14)
(15)
From (1)-(6) follows that the boundary functions , j = 1,2,..., 6 defined by (14) and (15) satisfy the conditions
e Ci,i(rJ), ^ = $v on y^v.
Then, on the basis of Theorem 4.1 in [9] the pure second-order derivatives of the function u are bounded in R. Then
sup
(Ж1,Ж2 ,Ж3)gR
д 8u
sup
(Ж1,Ж2 ,Ж3)gR
sup
(Ж1,Ж2 ,Ж3)GR
дх?
д 8u
sup
(Ж1 ,Ж2 ,Ж3)gR
д 2 ш
дх?дх2 д 8u
дх? д 2 ш
< œ,
дх?дх2
= sup
(Ж1 ,Ж2 ,Ж3)gR
= sup
(Ж1 ,Ж2 ,Ж3)GR
дх2 д 2 ш
дх2
<,
<.
Similarly, by taking ш = д;!, and ш derivatives in (13) are proved.
дЖ| the boundedness of the remainder even order
3 □
Lemma 4. Let u be the solution of problem (1), p(xi, x2, x3) be the distance from the current point of R to its boundary and let d/dl = aid/dxi + «2d/dx2 + «3d/dx3, ai + a2 + a3 = 1 be the l — directional differentiation operator. Then
д10 u^i, х2, хз )
д/10
< CoP 2(х?, х2, хз), (х?, х2, хз) G R,
(16)
where c0 is a constant independent of the direction l of the operator d/dl.
Proof. Since any tenth-order derivative of u can be obtained by two times differentiating some of the derivatives of the form (13), on the basis of Lemma 3 from [10, Chapter 4, Sec. 3] and Lemma 3, we have
max max 0^00 o^v <10—^
д10u^i, х2, хз)
дх^дх^дх®
10—^—v
< C1 р2(х1 ,х2,хз), (х1 ,х2,хз) G R.
(17)
□
From (17) follows the inequality (16).
2. O(h6|| ln h\) order accurate approximate solution
Consider a cubic mesh with the mesh size h > 0 formed by the planes x^ = 0, h, 2h, ...(i = 1,2,3). Assume that a^/h > 4 (i = 1,2,3) are integers. Let Dh be the set of mesh nodes, Rh = R n Dh, j = rj n Dh, rh = r n Dh, rjh = rj n Dh, and ^ = r^ U ... U ^. We put R = Rh U ^,
Rh = RhU Vh. Let Rh C Rh be the set of nodes of Rh lying at a distance of kh away from the boundary r of R. It is clear that k = 1, 2, (h), where N(h) = [min{ai, a2, a3}/(2h)].
For the grid functions on Rh, we consider the 6-point difference operator A as
16
Au(x1 ,x2,x3) = 6 up,
p
=i(i)
and the 14-point difference operator S as
1 ( 6 14 I
Su(xi, X2, X3) = — ( 8 UP + XI Uq I ,
\ p=l(i) q=7(3) J
where the sum ^(k) is taken over the grid nodes that are at a distance of \Jkh from the point (xi,x2,x3), up and uq are the values of u at the corresponding grid points. Consider two systems of grid equations
vh = Avh + gh, on Rh, vh = 0 on r'h, (18)
vh = AVh + gh, on Rh, Vh = 0 on rh, (19)
where gh and gh are given functions and |gh| < gh on Rh.
Lemma 5. The solutions vh and Vh to systems (18) and (19) satisfy the inequality
|vh| < vh on Rh.
Proof. The proof of Lemma 5 follows from the comparison theorem (see [11, Chapter 4]). □
2.1. The first stage
Let vh be a solution of the following finite difference problem
vh = Svh on Rh, vh = fy on rjh, j = 1, 2,..., 6, (20)
where fy, j = 1,2,..., 6 are functions defined in (11).
Let c,c1 ,c2,... denote positive constants independent of the nearby multiplier, of which some possibly have identical values.
Lemma 6. The following estimation holds
max _ |vh - v21 < c1 h4(| lnh| + 1),
(X1,X2 ,x3 )eRh
where v2 is the function (9) when k = 2 and vh is the solution of the system of grid equations (20).
Proof. By Lemma 2,
Av2 = 0 on R, v = fy2 on r, j = 1, 2,..., 6,
where functions fy2 defined by (11). For the error function
el = vh - v2, (21)
we have
e2 = Seh + (Sv2 - v2) onRh, e2h = 0 on rh.
Let eh be represented as
eh = eh'1 + eh-2 + ... + (h>, (22)
2 k
where <sh' , 1 < k < N(h) is the solution of system
2,k o 2,k , >fc T-, 2,k n -n
eh' = Seh' + Ck on Rh, eh =0 on ^,
with
k = i Sv2 — v2 on Rk, \0 on Rh\Rh.
By virtue of Lemma 4 in [12], we have
max |4'k| < 5k max |Sv2 — v2|, 1 < k < N(h). (23)
(xi ,X2'X3 )eRh (xi 'X2 ,X3
To estimate Sv2 — v2 on Rh, for k = 1, 2, ...,N(h), first we note that, from (9) and Lemma 4 follows
д6V2(xi ,X2 ,X3)
д/6
< C2
д10 u(xi ,X2 ,X3 )
д/10
< C3P 2 (xi, X2, X3), (xi, X2, X3) G R. (24)
Let x0 = (x10 ,x20 ,x30) be a node of the grid Rh0 C Rh, where k0 be an arbitrary integer number 2 < k0 < N(h) and let r6(x1,x2,x3; x0) be the Lagrange remainder corresponding to this point in Taylor formula
v2(X1, X2, X3) = P5(x1, X2, X3; X0) + r6(x1, X2, X3; X0), (25)
where
Sp5(X10, X20, X30; X0) = v2(X10, X20, X30). (26)
Then on the basis of (24), we have
h6 = ^ (k0h)2 C4 ^
From (25)-(27), we obtain
Sr6 (xio, X20, X30; xo) < c^ _ C4—2. (27)
h4
max |Sv2 — v21 < C4, 2 < k < N(h). (28)
(xi ,X2 ,X3)GRh k2
Let x0 = (x10,x20,x30) be a node of the grid Rh C Rh, and the nodes of operator S lie at the distance h or \p3h from this point. We estimate r6 at the nodes of the operator S. To do this we take a node (x10 — h, x20 — h, x30 + h) and consider the continuous function
J2(s) _ v2 ^xi0 - ' X20 - ' X30 + , < s <
(29)
of one variable s. By estimation (24), we have
6v (s)
ds6
< c5 (л/э^ - s) , 0 < s^h. (30)
The function
sss
¿6 _ Г6 ( Xi0 - , X20 - , X30 + X0
is the remainder term of the representation of the function (29) around the point s = 0 by Taylor's formula with the fifth order polynomial.
By using integral form of the remainder term and (30), we obtain (see [8])
(s s s \
xio - ,X20 - ,X30 + x0j < cah4. (31)
For the remaining nodes of the operator S the estimation (31) can be obtained analogically. Since the maximum norm of the operator S is equal to one, we have
|Sr6(xi0,x20,x30;xo)| < C7h4. (32)
By (25), (26) and (32), we obtain
max |Sv2 - v21 < c8 h4. (33)
(xi ,X2 ,X3)gRh
On the basis of (21)-(23), (28) and (33), we have
N (h)
max |vh - v21 < eg h4 V 1 < cih4 (|| ln h| + 1)
(xi ,X2 ,X3)gRh f—i k
□
2.2. The second stage
Let V3 be a solution to the following finite difference problem
v3 = Av3 on Rh, = on rjh, j = 1, 2,..., 6, (34)
where j j = 1,2,..., 6 are functions defined by (12). Lemma 7. On Rh, it holds that,
max ^ |v3 - v31 < c2h2(| lnh| + 1), (35)
(X1,X2 ,X3 )ER/fl
where v3 is the function (9) for k = 3, v3 is a solution to system (34). Proof. By Lemma 2, we have
Av3 = 0 on R, v3 = on r, j = 1, 2,..., 6, (36)
where functions define boundary values in (12), and from (2)-(6) it follows that
G C1,1 (r), 0 < A < 1, j = 1,2,..., 6, (37)
^ = ^ on y^v , (38)
on the basis of (36)-(38) that satisfy the conditions of Theorem 5.1 in [9] which follows estimation (35). □
2.3. The third stage
Let v2 and v^ be the solution of the difference problems (20)) and (34) respectively. We approximate the solution of the given Dirichlet problem (1) on the grid Rh as a solution u h of the following difference problem
h4 2 h6
—v2--
36 h 720
u h = ^ on rjh, j = 1, 2,..., 6. (40)
uh = Auh - — v2 - —-on Rh, (39)
Theorem 1. Under the conditions (2)-(6), the estimation
max ^ |uh — u| < c3h6(|| lnh| + 1), (41)
(xi ,X2'X3)GRh
is valid, where u is the solution of the Dirichlet problem (1) uh is the solution of system (39), (40).
Proof. Under the smoothness properties of the boundary values specified in (2)-(6), the solution u of the Dirichlet problem (1) has eighth-order partial derivatives that are continuous on R, and by using Taylor's formula with the remainder term in the Lagrange form for each
(x1 ,x2 ,x3) e Rh, we obtain
h4 h6
u(x1, X2, X3) = Au(x1, X2, X3) — 36v2 — 720v3 — r(x1, X2, X3), (42)
where vk, k = 2,3 are the functions defined by (9)
max |r(x1,x2, X3)| < c4h8. (43)
(xi ,x2 ,x3)eR
We put
£h = Uh - u on Rh,
where uh is the solution of the finite difference problem (39), (40).
From (39) and (42), and taking into account that uh = u = pj on Tjh, we obtain the following
system of difference equations for the error eh:
h4 h6
eh = Aeh + 36 (v2 — vl) + 720 (v3 — vh) + r on Rh, (44)
eh = 0 on rh. (45)
On the basis of Lemma 6 , Lemma 7, and the estimation (43)), we obtain
h4 o o. h6
36(V2 - V2) + 72Ô(v3 - V3)+ Г
< C5h8(|| ln h|| +1),
where c5 = max {c1 /36, c2/720, c4}.
Furthermore, from Lemma 5 it follows that for the solution <sh of problem (44), (45) the following estimation is true
kh| < (46)
where ëh is a solution of the problem
ëh = Aëh + C5h8(| lnh| + 1) on Rh, ëh > 0 on rh. (47)
It is easy to check that the function ëh = c5h6(|| lnh| + 1)(12 - r2), where l = y/al + a2 + a3, and r = у/x2 + x2 + x2 is a solution of problem (47). Then from (46), follows (41). □
3. Numerical results
Let R = {(xi, X2, X3) : 0 < хг < 1, i = 1, 2, 3}, and let Г, j = 1, 2,..., 6 be its faces.
Au = 0 on R, u = ^(x1,x2,x3) on Г, (48)
where
p(xb x2, x3) = e3xi cosh(4x2) cos(5x3) + (x8 - 28x1x2 + 70x1x| - 28x2x6 + x8) tan-1 ^^ + 170 Научный отдел
+ (8x[x2 - 56x5x3 + 56x1 x2 - 8xixf) ln yjxf + xf,
is the exact solution of problem (48) and ^ G C7,1 (r). We use the following notations:
lU - U
max |Uh — U
Em
|Uh - U2-m y^h
||Uh - U2-(m+1) '
where U is the trace of the exact solution of the continuous problem on , and Uh is its approximate value obtained by the proposed method.
The numerical results given in Table show that the maximum error of the approximate solution obtained by the proposed method absolute values convergent of order O(h6 lnh), since
26 >En > 26n/(n + 1).
Table. Numerical results for Prob em (48)
h = 2-n maxnh |u2-n — u| En 26 n/(n + 1)
2- -3 1.537D - 07 48.394 48.000
2- -4 3.176D - 09 60.231 51.200
2" -5 5.273D - 11 62.625 53.333
2- -6 8.420D - 13 63.071 54.857
2" -7 1.335D - 14 - -
Conclusion
A new three-stage difference method with an accuracy of order O(h6(| lnh| + 1)), where h is mesh size, is proposed and justified by using one fourth-order and two second-order schemes for the approximate solution of the 3D Laplace's equation. It is assumed that the boundary functions on the faces are from C7,1, and on the edges, their second, fourth, and sixth derivatives satisfy the compatibility conditions, which follows from the Laplace equation.
The idea of this method can be used to design a new scheme with an order of convergence O(h8 (| ln h| + 1)), when ^ G C9'1 (r-), j = 1,..., 6.
Moreover, from the estimation (41) the multiplier | ln h| can be removed by replacing in (2) the condition ^ G C7'1 (r) with the condition ^ G C8'A(r), 0 < A < 1.
The proposed method can be applied when parallelepiped is used as one of the covering figures in some version of domain decomposition methods [13], in the composite grids method for problems in polyhedra and a prism with polygonal base (see [14, 15]). Furthermore, this method can be used to highly approximate the derivatives of the unknown solution of Laplace's equation (see [16-19]).
References
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Поступила в редакцию / Received 23.03.2023 Принята к публикации / Accepted 29.08.2023 Опубликована / Published 31.05.2024
