- Email: [email protected]
On a method for solving elliptic difference equations
B. S. Pol’skii
26
REFERENCES 1. FRANK-KAMENETSKII, D. A., Diffision and heat transfer in chemical kinetics (Diffuziya i teploperedacha v khimicheskoi kinetike), Izd-vo Akad. Nauk SSSR, Moscow, 194’). 2. FRIEDMAN, A., Partil differential equations of ptrrabolictype, Prentice-Hall, 1964. 3. AKRAMOV, T. A., On stabilization of solutions of the system of partial differential equations describing the kinetics of reversible chemical reactions, in: Dynamics of a confinuous medium (Dinamika sploshnoi sredy), SO Akad. Nauk SSSR, No. 26, Novosibirsk, 1976. 4. GO~KOVSKAYA, V. T., OZERKOVSKAYA, N. I., and FILLIPE~O, V. P., On the numerical solution of systems of different~ equations of chemical kinetics the taking d~fusion into account, in: Mathematicalprobfems of chemistry (Matem. probi. khimii), VTs SO Akad. Nauk SSSR, Part I, Novosrbirsk, 1975. 5. IL’IN, A. M., KALASHNIKOV, A. S., and OLEINIK, 0. A., Linear second-order equations of parabolic type, Usp. mat. nauk, 17, No. 3, 3-146, 1952. 6. KRUZHKOV, S. N., A priori estimates and some properties of solutions of elliptic and parabolic equations, Marem. sb., 65 (107), No. 4,522-569,1964. 7. KRUZHKOV, S. N., Non-linear parttil different&z1equations (lectures), Part I,
Izd-voMGU,Moscow,1970.
8. LADYZHENSKAYA, 0. A., SOLONNIKOV, V. A., and URALTSEVA, N. N., Linewand quasi-linear eqr8arionsofparabo~jc type (Lineinye i kvazilineinye uravneniya parabolicheskogo Moscow, 1967.
tipa), Nauka,
9. KRUZHKOV, S. N., eon-~jnearpartialdifferentialequations(Neline~ye uravneniya s chastnymi proizvodnymi), Part 2, Izd-vo MGU, Moscow, 1970.
U.S.S.R. Comput. Maths.Math.Phys. Vol. 21, No. 1, pp. 26-32, Printed in Great flritain
1981.
0041~5553/81/010026-07307.50/O 01982. Pergamon Press Ltd.
ON A METHOD FOR SOLVING ELLIPTIC DIFFERENCE EQUATIONS* B. S. POL’SKR Riga (Received 15 December 1978; revised 27 December 1979)
A REDCAP analogue of the Schwarz alternating procedure is described for solving elliptic difference equations in which the coefficient k (x, v) of the leading derivative varies strongIy in narrow domains. It is shown with the aid of model problems that the rate of convergence of the method is independent of k (x, y), and an asymptotic estimate is obtained for the number of arithmetic operations required to solve the problem with accuracy e. 1. In the rectangle D- { (5, y) : OdsGzcl,, problem
*Zh. vj&hisZ.Mar. mar. Fiz., 21, I, 29-34,
1981
N . 2} we consider the boundary value O
27
Solving elliptic difference equations
Mp=-
y”)
M,u =
au kZ2) dy
+oi(x)u(x,o)=~i(x),
23
+a*wUb,Z2)=~*(5),
x=(0,11).
(0, 41,
We assume that cr,>O, a,>O, k (x, y) >k,>O. We introduce into the set D a uniform mesh with steps h,=ZJN,, construct the difference scheme (LhU)
ij=.fij,
zEOj=(vi)
j,
O
u.ho=
(T2)j7
(Jfihu)x= (~1)i,
h,=Z,lN,.
In this mesh we
O
O
(5)
(6) (7)
where Lh, Mihr MZh are the usual rive-point and two-point difference approximations of the operators L, Ml, and M2. If the function k(x, JJ) varies strongly in the domain D, the efficiency of the well-known iterative methods for solving the system of linear equations (5)-(8) (see [ 1,2]), with the exception of the method described in [3], is seriously reduced. The reasons for this are as follows: the conditionality number of system (5)-(8) is increased; and it is difficult to choose the iterative parameters such that the efficiency of the method is optimized. Problems quite often arise in applications (in the theory of heat conduction and the theory of semiconductor devices), in which k (x, JJ) varies strongly in a narrow part of the domain D. In this case, to solve system (5)-(8), a numerical analogue of the Schwarz alternating procedure [4] has proved very successful. The possibilities of this procedure for the numerical solution of elliptic boundary value problems in domains with complex geometry were demonstrated earlier in [ 5,6].
FIG. I
B. S. Pol’skii
28
2. We introduce the notation (see Fig. 1)
Da={ (5, y) : ya-=G,
O
DF{(x, y) : y,
o
We assume that if
(5, VI =Di,
kzb,y),
if
(2, Y>=Dzr
k,=const,
if
(5, Y>=Ds,
1
k(x,y)=
T,,,, m=l,
k,=const,
2, 3, 4, hits the mesh nodes with numbers i,,, , and (rl-yj)
(9)
<‘I,.
We define sequences ~1” and v2n as follows: O
(LhUi*)ij=fil,
&
O
(10)
(13) (14)
(MlhUSn)
;=
($i) i,
(“?.hC?_n) i=
($2)
i,
i,CiCi,,
(16)
n=l,2,3 ,..., and ~20 is a given initial approximation. Thus, “1” is found as the solution of the difference boundary value problem in the domain D1 U 03 with boundary conditions (1 I)-( 13), and ~2”) as the solution of the difference boundary value problem in the domain Dq with boundary conditions (15), (16). The iterative process (IO)-(16) is the numerical analogue of the Schwarz method [4]. Theorem 1 Let k(x, y) be given by (9), and let u be the solution of problem (5)-(8). estimates
IlVi”-UIlcfg”-lIluzO-~llcr
-Ul/c~gnllvZo-l.zIIC, IIvzn
n-l,
We then have the
2,. . . ,
(17)
Solving elliptic difference equations
29
where (18)
i.e. q < 1 and is independent of k (x, y), h,, and hY. Roof. We define the functions WI” and ~2” as: WZn=p2n-u.
WI n=vln-U,
Obviously, ~1” is the solution of the problem O
(AhWin)ij=Oy
(win)
iS
O
o>=O, (Win)h.f_+=O, OGjG Nz,
(w*“) irF (W2n-‘)id, (a”) ‘J- G-7 CJ, . . O
OGjGV*, &
where L& is the five-point difference approximation of the Laplace operator [2], while ~2” is the solution of the problem
Gw2”) i1=0, (WZ.“) it>= (wi”)
OcjcN,,
i,
(MlhWZn)i=Oy
i-0,
(M2hWZn)
OqGN,,
(Win)id9
( u721) id=
itI,
i,
We put & n=max[
max
maxl
I (W,“)iJIy
osz:jGNa &
n=max[
max
n=O,
(W2n)ijl],
1,.
I (W,“)ijl,
max
Ocj4S2
I(Win)
IiJ],
n=l,
OCj
We define function WC-J n as follows: for 0 Q j
It is easily seen that won is the solution of the problem
( AhWon) ij=0, ( WOn)Oj=O,
OCi
i,
hr,j=O,
i,j=Rz’,
(~U~")irj=(Wo~)
where(Misdo”) 20,
OcjtN,,
OGjGN,,
Odj
("ihwOn)i=(Ot)i(WOn)iO, OCiCi,,
. . ,
Ocj<.h2
(I"2hU)On)L=(62)i(Won)t~*,
i,
(MthWO”) 130.
By the maximum principle [2],
l twS”>iJl<ij,
n-l,
2,. . . (
OCKiz,
2,.
, .
.
B. S. Pol ‘skii
30 Hence it follows that
R,“GqR;-‘,
m=1,2,.
..,
(19)
where 4 is given by (18). We also obtain from the maximum principle the inequalities
IIUAnil&RF-‘, /IW2nllc~Rjn,
(20)
n-l,
2, . . *
(21)
n-l,
2,. . .
(22)
It follows in particular from (21) that
Rz”BRIW, But then, we find from (19) and (22) that R~n~qR~-i
Ri"fq”Rz’,
n=l,2,...
.
(23)
n--l, 2,. , , .
(24)
Using (20) and (2 1) in conjunction with (23), we obtain Il&‘IIe~qQn--&“, Since
Il~~“li~~~~&~,
Rzof IIuz”- uilc, we obtain from (24) the inequalities (17). B t
‘i
i 0
Y,QY*
t=
FIG. 2 3. Let us now discuss ways of actually solving problems (lo)-( 13) and (14) -( 16), i.e. finding functions vl n and ~2”. Since k (x, u) = const in domainsD1 and D2, we can find VIn effectively either by an iterative method [l ,2], or by a fast non-iterative method [7]. To find uzn, matrix pivotal condensation [2] is a very suitable method, since the number of nodes with respect to the x coordinate is small. Notice that, when finding ~2” with n > 2, the number of operations in the matrix pivotal condensation method is substantially reduced if we store as a preliminary the part of the pivotal condensation coefficients that were evaluated for n = 1 (see [2 I). If an iterative method is used to find ~1n, it can be shown (in the same way as in [8]) that the 4 in estimates (17) has to be replaced by 41 = (I t E 4 t E, where e is the accuracy to which VIn is found.
Solvingelliptic difference equations
31
Expression (18) shows that, as Tj ‘0, y&+1, , the quantity q decreases, i.e. the rate of convergence of the Schwarz method increases. However, the number of operations for finding 19” (when matrix pivotal condensation is used) then greatly increases. The question arises of choosing 71 and 74 in such a way as to minimize the number of arithmetic operations needed to solve the problem ta accuracy E. To examine this, we take an elementary example, namely, the problem (see Fig. 2) with l,=l,=l,
y:=y3=0..3,
~,=0.%-6,
k,. if
k(x.y)=
x
‘3.
-t
if
~,=0.51+6;
(5, y) ED,, (x,
y)=Ds.
(25) (26)
Here, the domain D2 degenerates into the line x = 0.51, on which the usual matching conditions are specified. We also assume that (31i==const,
o-=s
u~2=COrlSt,
0.51cac1,
o2j=const,
OtsC0.51,
1 o~:=COllst,
0.51
1
G(5) (3:(x) =
(27)
T%eorem 2 Let conditions (25)-(27) hold? let K be the number of nodes with respect to each coordinate. and let u be the solution of problem (5)--(g). Assume that VI”, defined by Eqs. (lo)-(13), is found by the method of cyclical reduction [7], while ~2“, defined by Eqs. (14)-(16). is found b} matrix pivotal condensation. Then. if 6=0..?/,\‘-‘;, to obtain the estimates
it is sufficient to make nxN”> In E -I iterations. II being independent of kl/k3. The number of arithmetic operations here is W (N, E) xv2 log, K In E-‘. hoof. From conditions (25) and expression (18). we obtain 0.51-b = I-N-‘“, Q= _ 0.52 but it then follows from estimates (17) that, to satisfy inequalities (28) it is sufficient to perform nxh”” In E-’ iterations. To fmd the function vl n by cyclical reduction we require XN2 log, N arithmetic operations. The number of nodes in the interval [yl ,741 is hr/, so that, to find 19 1 by matrix pivotal condensation, we require X (N’“) 3N arithmetic operations. To find ~2”. n > 2, the number of operations in matrix pivotal condensation is reduced to X (IV) *N, since part of the pivotal condensation coefficients has been evaluated for n = 1. Hence the total number of arithmetic operations is W(h’,
E) X (hr*
log: N) nSh7”2-tN2nX~‘2
log, N In E-‘.
32
B. S. Pol’skii
4. Let us give an example of the solution of a concrete problem. We took the square zcXn, yz=9n/20, ys=l ln!20. In relation (9), k,=5. IO”, ks==lW’, ks(s, y) -c exp [ -4 (~-0.5~) ‘1. The boundary conditions were ~I=rps==$l=~l=ol==u~=O. Obviously, the function u = 0 is a solution of this problem. As the initial approximaTion we chose the function vzo(z, y) =sin 5 sin y+ sin 392 sin 39y+sin 20s sin 20y+sin 2 sin 20y+sin 39t sin y. The number of nodes with respect to each direction isN = 41. To solve the problem, we took yI and y2 equal to 6n/20 and 14n/20 respectively, i.e. domain Dq contained 17X 41 nodes. The function IQ” was only roughly determined, using 10 iterations of over-relaxation [I] with iterative parameter o=:!(l+ain h)-‘, which is not optimal for domains D1 and D3 ; it is in fact extremely difficult to optimize this parameter when solving actual problems. The function v2n was found by matrix pivotal condensation; the pivotal condensation coefficients c~i(see (2 J) were only evaluated for n = 1; these were stored and then used for subsequent computations. The computations stopped when max ( I/u,~/~c, llvz”ll~)
1.988 27.2 1.996 86 For comparison, the same problem was solved by over-relaxation, the value of the optimal parameter being found experimentally to accuracy 0.001. The results are given in Table 1, where nI is the number of iterations of the Schwarz method, tI is the computing time in minutes for the Schwarz method, n2 is the number of iterations in overrelaxation, w is the optimal iterative parameter, and t2 is the computing time in minutes for over-relaxation. The medium-speed GE-415 computer was used; it can be seen that the Schwarz method is very efficient for large values of c. i?ansluted by
D. E. Brown
REFERENCES 1.
FEDORENKO,R. P. Iterative methods for solving elliptic difference equations, Vsp. Mut. Nauk, 28, No. 2, 120-182,1973.
2. SAMARSKII, A. A., Theory of difference schemes (Teoriya raznostnykh skhem), Nauka, Moscow, 1977. 3. KUCHEROV, A. B., and NIKOLAE_V, E. S., Alternately-triangular
iterative method for solving mesh elliptic equations in a rectangle, Zk. vyckisl. Muf. mut. Fiz., 16, No. 5, 1164-1174,1976.
4. KANTOROVICH, L. V., and KRYLOV, V. I., Approximate methods of hi&heranalysis(Priilizhennye metody vysshego analiza), Fizmatgiz, Moscow-Leningrad,
1962.
5. D'YAKONOV, E. G., On a method of solving Poisson’s equation, Dokl. Akad. Nauk SSSR, 143, No. 1, 21-24,1962.
6. MILLER,K., Numerical analogs to the Schwarz alternating procedure, Numer. Math., 7,91-103,196s. 7. DORR, F. W., The direct solution of the discrete Poisson equation on a rectangle, SL4MRev. 12,No.2, 248-263,197O.
8. D’YAKONOV, E. G., An iterative method for solving systems of finitedifference NaukSSSR, 138, No. 3,522-525,196l.
equations, Dokl. Akud.
26
REFERENCES 1. FRANK-KAMENETSKII, D. A., Diffision and heat transfer in chemical kinetics (Diffuziya i teploperedacha v khimicheskoi kinetike), Izd-vo Akad. Nauk SSSR, Moscow, 194’). 2. FRIEDMAN, A., Partil differential equations of ptrrabolictype, Prentice-Hall, 1964. 3. AKRAMOV, T. A., On stabilization of solutions of the system of partial differential equations describing the kinetics of reversible chemical reactions, in: Dynamics of a confinuous medium (Dinamika sploshnoi sredy), SO Akad. Nauk SSSR, No. 26, Novosibirsk, 1976. 4. GO~KOVSKAYA, V. T., OZERKOVSKAYA, N. I., and FILLIPE~O, V. P., On the numerical solution of systems of different~ equations of chemical kinetics the taking d~fusion into account, in: Mathematicalprobfems of chemistry (Matem. probi. khimii), VTs SO Akad. Nauk SSSR, Part I, Novosrbirsk, 1975. 5. IL’IN, A. M., KALASHNIKOV, A. S., and OLEINIK, 0. A., Linear second-order equations of parabolic type, Usp. mat. nauk, 17, No. 3, 3-146, 1952. 6. KRUZHKOV, S. N., A priori estimates and some properties of solutions of elliptic and parabolic equations, Marem. sb., 65 (107), No. 4,522-569,1964. 7. KRUZHKOV, S. N., Non-linear parttil different&z1equations (lectures), Part I,
Izd-voMGU,Moscow,1970.
8. LADYZHENSKAYA, 0. A., SOLONNIKOV, V. A., and URALTSEVA, N. N., Linewand quasi-linear eqr8arionsofparabo~jc type (Lineinye i kvazilineinye uravneniya parabolicheskogo Moscow, 1967.
tipa), Nauka,
9. KRUZHKOV, S. N., eon-~jnearpartialdifferentialequations(Neline~ye uravneniya s chastnymi proizvodnymi), Part 2, Izd-vo MGU, Moscow, 1970.
U.S.S.R. Comput. Maths.Math.Phys. Vol. 21, No. 1, pp. 26-32, Printed in Great flritain
1981.
0041~5553/81/010026-07307.50/O 01982. Pergamon Press Ltd.
ON A METHOD FOR SOLVING ELLIPTIC DIFFERENCE EQUATIONS* B. S. POL’SKR Riga (Received 15 December 1978; revised 27 December 1979)
A REDCAP analogue of the Schwarz alternating procedure is described for solving elliptic difference equations in which the coefficient k (x, v) of the leading derivative varies strongIy in narrow domains. It is shown with the aid of model problems that the rate of convergence of the method is independent of k (x, y), and an asymptotic estimate is obtained for the number of arithmetic operations required to solve the problem with accuracy e. 1. In the rectangle D- { (5, y) : OdsGzcl,, problem
*Zh. vj&hisZ.Mar. mar. Fiz., 21, I, 29-34,
1981
N . 2} we consider the boundary value O
27
Solving elliptic difference equations
Mp=-
y”)
M,u =
au kZ2) dy
+oi(x)u(x,o)=~i(x),
23
+a*wUb,Z2)=~*(5),
x=(0,11).
(0, 41,
We assume that cr,>O, a,>O, k (x, y) >k,>O. We introduce into the set D a uniform mesh with steps h,=ZJN,, construct the difference scheme (LhU)
ij=.fij,
zEOj=(vi)
j,
O
u.ho=
(T2)j7
(Jfihu)x= (~1)i,
h,=Z,lN,.
In this mesh we
O
O
(5)
(6) (7)
where Lh, Mihr MZh are the usual rive-point and two-point difference approximations of the operators L, Ml, and M2. If the function k(x, JJ) varies strongly in the domain D, the efficiency of the well-known iterative methods for solving the system of linear equations (5)-(8) (see [ 1,2]), with the exception of the method described in [3], is seriously reduced. The reasons for this are as follows: the conditionality number of system (5)-(8) is increased; and it is difficult to choose the iterative parameters such that the efficiency of the method is optimized. Problems quite often arise in applications (in the theory of heat conduction and the theory of semiconductor devices), in which k (x, JJ) varies strongly in a narrow part of the domain D. In this case, to solve system (5)-(8), a numerical analogue of the Schwarz alternating procedure [4] has proved very successful. The possibilities of this procedure for the numerical solution of elliptic boundary value problems in domains with complex geometry were demonstrated earlier in [ 5,6].
FIG. I
B. S. Pol’skii
28
2. We introduce the notation (see Fig. 1)
Da={ (5, y) : ya-=G,
O
DF{(x, y) : y,
o
We assume that if
(5, VI =Di,
kzb,y),
if
(2, Y>=Dzr
k,=const,
if
(5, Y>=Ds,
1
k(x,y)=
T,,,, m=l,
k,=const,
2, 3, 4, hits the mesh nodes with numbers i,,, , and (rl-yj)
(9)
<‘I,.
We define sequences ~1” and v2n as follows: O
(LhUi*)ij=fil,
&
O
(10)
(13) (14)
(MlhUSn)
;=
($i) i,
(“?.hC?_n) i=
($2)
i,
i,CiCi,,
(16)
n=l,2,3 ,..., and ~20 is a given initial approximation. Thus, “1” is found as the solution of the difference boundary value problem in the domain D1 U 03 with boundary conditions (1 I)-( 13), and ~2”) as the solution of the difference boundary value problem in the domain Dq with boundary conditions (15), (16). The iterative process (IO)-(16) is the numerical analogue of the Schwarz method [4]. Theorem 1 Let k(x, y) be given by (9), and let u be the solution of problem (5)-(8). estimates
IlVi”-UIlcfg”-lIluzO-~llcr
-Ul/c~gnllvZo-l.zIIC, IIvzn
n-l,
We then have the
2,. . . ,
(17)
Solving elliptic difference equations
29
where (18)
i.e. q < 1 and is independent of k (x, y), h,, and hY. Roof. We define the functions WI” and ~2” as: WZn=p2n-u.
WI n=vln-U,
Obviously, ~1” is the solution of the problem O
(AhWin)ij=Oy
(win)
iS
O
o>=O, (Win)h.f_+=O, OGjG Nz,
(w*“) irF (W2n-‘)id, (a”) ‘J- G-7 CJ, . . O
OGjGV*, &
where L& is the five-point difference approximation of the Laplace operator [2], while ~2” is the solution of the problem
Gw2”) i1=0, (WZ.“) it>= (wi”)
OcjcN,,
i,
(MlhWZn)i=Oy
i-0,
(M2hWZn)
OqGN,,
(Win)id9
( u721) id=
itI,
i,
We put & n=max[
max
maxl
I (W,“)iJIy
osz:jGNa &
n=max[
max
n=O,
(W2n)ijl],
1,.
I (W,“)ijl,
max
Ocj4S2
I(Win)
IiJ],
n=l,
OCj
We define function WC-J n as follows: for 0 Q j
It is easily seen that won is the solution of the problem
( AhWon) ij=0, ( WOn)Oj=O,
OCi
i,
hr,j=O,
i,j=Rz’,
(~U~")irj=(Wo~)
where(Misdo”) 20,
OcjtN,,
OGjGN,,
Odj
("ihwOn)i=(Ot)i(WOn)iO, OCiCi,,
. . ,
Ocj<.h2
(I"2hU)On)L=(62)i(Won)t~*,
i,
(MthWO”) 130.
By the maximum principle [2],
l twS”>iJl<
n-l,
2,. . . (
OCKiz,
2,.
, .
.
B. S. Pol ‘skii
30 Hence it follows that
R,“GqR;-‘,
m=1,2,.
..,
(19)
where 4 is given by (18). We also obtain from the maximum principle the inequalities
IIUAnil&RF-‘, /IW2nllc~Rjn,
(20)
n-l,
2, . . *
(21)
n-l,
2,. . .
(22)
It follows in particular from (21) that
Rz”BRIW, But then, we find from (19) and (22) that R~n~qR~-i
Ri"fq”Rz’,
n=l,2,...
.
(23)
n--l, 2,. , , .
(24)
Using (20) and (2 1) in conjunction with (23), we obtain Il&‘IIe~qQn--&“, Since
Il~~“li~~~~&~,
Rzof IIuz”- uilc, we obtain from (24) the inequalities (17). B t
‘i
i 0
Y,QY*
t=
FIG. 2 3. Let us now discuss ways of actually solving problems (lo)-( 13) and (14) -( 16), i.e. finding functions vl n and ~2”. Since k (x, u) = const in domainsD1 and D2, we can find VIn effectively either by an iterative method [l ,2], or by a fast non-iterative method [7]. To find uzn, matrix pivotal condensation [2] is a very suitable method, since the number of nodes with respect to the x coordinate is small. Notice that, when finding ~2” with n > 2, the number of operations in the matrix pivotal condensation method is substantially reduced if we store as a preliminary the part of the pivotal condensation coefficients that were evaluated for n = 1 (see [2 I). If an iterative method is used to find ~1n, it can be shown (in the same way as in [8]) that the 4 in estimates (17) has to be replaced by 41 = (I t E 4 t E, where e is the accuracy to which VIn is found.
Solvingelliptic difference equations
31
Expression (18) shows that, as Tj ‘0, y&+1, , the quantity q decreases, i.e. the rate of convergence of the Schwarz method increases. However, the number of operations for finding 19” (when matrix pivotal condensation is used) then greatly increases. The question arises of choosing 71 and 74 in such a way as to minimize the number of arithmetic operations needed to solve the problem ta accuracy E. To examine this, we take an elementary example, namely, the problem (see Fig. 2) with l,=l,=l,
y:=y3=0..3,
~,=0.%-6,
k,. if
k(x.y)=
x
‘3.
-t
if
~,=0.51+6;
(5, y) ED,, (x,
y)=Ds.
(25) (26)
Here, the domain D2 degenerates into the line x = 0.51, on which the usual matching conditions are specified. We also assume that (31i==const,
o-=s
u~2=COrlSt,
0.51cac1,
o2j=const,
OtsC0.51,
1 o~:=COllst,
0.51
1
G(5) (3:(x) =
(27)
T%eorem 2 Let conditions (25)-(27) hold? let K be the number of nodes with respect to each coordinate. and let u be the solution of problem (5)--(g). Assume that VI”, defined by Eqs. (lo)-(13), is found by the method of cyclical reduction [7], while ~2“, defined by Eqs. (14)-(16). is found b} matrix pivotal condensation. Then. if 6=0..?/,\‘-‘;, to obtain the estimates
it is sufficient to make nxN”> In E -I iterations. II being independent of kl/k3. The number of arithmetic operations here is W (N, E) xv2 log, K In E-‘. hoof. From conditions (25) and expression (18). we obtain 0.51-b = I-N-‘“, Q= _ 0.52 but it then follows from estimates (17) that, to satisfy inequalities (28) it is sufficient to perform nxh”” In E-’ iterations. To fmd the function vl n by cyclical reduction we require XN2 log, N arithmetic operations. The number of nodes in the interval [yl ,741 is hr/, so that, to find 19 1 by matrix pivotal condensation, we require X (N’“) 3N arithmetic operations. To find ~2”. n > 2, the number of operations in matrix pivotal condensation is reduced to X (IV) *N, since part of the pivotal condensation coefficients has been evaluated for n = 1. Hence the total number of arithmetic operations is W(h’,
E) X (hr*
log: N) nSh7”2-tN2nX~‘2
log, N In E-‘.
32
B. S. Pol’skii
4. Let us give an example of the solution of a concrete problem. We took the square zcXn, yz=9n/20, ys=l ln!20. In relation (9), k,=5. IO”, ks==lW’, ks(s, y) -c exp [ -4 (~-0.5~) ‘1. The boundary conditions were ~I=rps==$l=~l=ol==u~=O. Obviously, the function u = 0 is a solution of this problem. As the initial approximaTion we chose the function vzo(z, y) =sin 5 sin y+ sin 392 sin 39y+sin 20s sin 20y+sin 2 sin 20y+sin 39t sin y. The number of nodes with respect to each direction isN = 41. To solve the problem, we took yI and y2 equal to 6n/20 and 14n/20 respectively, i.e. domain Dq contained 17X 41 nodes. The function IQ” was only roughly determined, using 10 iterations of over-relaxation [I] with iterative parameter o=:!(l+ain h)-‘, which is not optimal for domains D1 and D3 ; it is in fact extremely difficult to optimize this parameter when solving actual problems. The function v2n was found by matrix pivotal condensation; the pivotal condensation coefficients c~i(see (2 J) were only evaluated for n = 1; these were stored and then used for subsequent computations. The computations stopped when max ( I/u,~/~c, llvz”ll~)
1.988 27.2 1.996 86 For comparison, the same problem was solved by over-relaxation, the value of the optimal parameter being found experimentally to accuracy 0.001. The results are given in Table 1, where nI is the number of iterations of the Schwarz method, tI is the computing time in minutes for the Schwarz method, n2 is the number of iterations in overrelaxation, w is the optimal iterative parameter, and t2 is the computing time in minutes for over-relaxation. The medium-speed GE-415 computer was used; it can be seen that the Schwarz method is very efficient for large values of c. i?ansluted by
D. E. Brown
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FEDORENKO,R. P. Iterative methods for solving elliptic difference equations, Vsp. Mut. Nauk, 28, No. 2, 120-182,1973.
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iterative method for solving mesh elliptic equations in a rectangle, Zk. vyckisl. Muf. mut. Fiz., 16, No. 5, 1164-1174,1976.
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