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The iterative improvement of eigenvalues
Improvement of eigenvalues
123
w ,,”= -q2 sin nz ,,, conditions (22) were infringed. The initial values were u,“=sin nz,, A’=-30qc, where q=2h-’ sin (xh / 2) ; we took u l equal to 3h. To avoid obtaining the max trivial solution, we divided the function U, by a constant, in such a way that ,,=,,Z.,,,,Y_~~~= 11 u ,,< 1. The number of steps was 288. The variation when, after the next step, ,,l__in,itx,_,_, of Am is indicated in Table 4. Translated by D. E. Brown REFERENCES
of ordinary differential equations, McGraw, 1955.
1.
CODDINGTON, E. A., and LEVINSON, N., Theory
2.
PALTSEV, B. V., Series expansion of the solution of the Dirichlet problem and the mixed problem for the biharmonic equation, in solutions of splitting problems, Zh. vjkhisl. Mar. mat. Fiz., 6, No. 1, 43-51.1966.
THE ITERATIVE IMPROVEMENT OF EIGENVALUES* G. P. ASTRAKHANTSEV Leningrad (Received 22 January 1974; revised 21 October 1974)
A MODIFICATION of the Newton method of finding the Ritz approximation ph to the eigenvalue p of a completely continuous operator is described. It is shown that if/~ is many-valued the iterative process ensures convergence to neither $ nor to the Ritz eigenvector, but in the limit the Rayleigh ratio for the vectors obtained approximates to /J with the same accuracy as @.
Introduction Let T be a linear, completely continuous, self-conjugate, non-negative (for simplicity) operator in the Hilbert space H, and let the eigenvalue p of the equation
(1) have multiplicity 1: p,ZQ
. . . >p=pk=p*,-I=
. . . =p*;:--l>pkl
>...
Here I is the unit operator. We consider a family of spaces H,,=H, dense in the limit, that is, lim I% =zb h4
where ph is the orthoprojector onto Hh. It is known that the eigenvalues equation
are approximations to p [ 1] . *Zh. vjkhirl. Mat. mat. Fiz., 16,1,131-139,
1976.
cLkh, - . . $+~_iof the
The eigenvectors 8.ihsatisfy the equation
where p (u”) denotes the Rayleigh ratio calculated on the vector nh y
[ 13 is the scalar product in H. In this paper we construct B ~~d~cati~ of Newton’s method for Eq f2), which guarantees &Itbc case l=l ~~~r~~~ to the ~~~nvector and to the ~~nv~~~ $A& In the case of a rn~~~~e x1-‘) differs &gawahe &=- I$ we obtain a sequence ZF,,,EH~ such that ufnm) for m=O@ from p by a quantity of order sl (h) 1 where a, (Jb) is the order of the approximation # of the nurnber p. The main element in the realization of the integration step Is the solution of an equation of the form
The result obtained is used to find the eigenvahres for the variational-difference approximation (see [2j) of two-dinrensional elliptic problems on a net of step k>O, If &. (3) is solved with an papacy of up to ez & is a faMy sm& number) after a mrmber N(al, hj + of arjt~eti~ jr +th o~rat~ons, theE after O( fN’+&+] la Es-‘) o~~~o~s we construct F (u”] f a~ro~at~~
We show that for sufficiently small ho the vector U, differs little from uo.
Improvement of eigenvalues
125
Lemma 1 For small h, the following inequality holds:
Roof: From Eq. (1 .l), written in the form (1) llm+l--Ulll
(Uo+U~‘)+(T~-T~)u~+(~o-~C1(U,))u,l=O,
(*)+s[(P-po)
we find the vector ugll: (i)
(Th-T~)u,+(~o-~(u,))u~)].
%?I+1-4
(1.4)
The operator S, acting in Hh , is bounded, and its norm is given by
IISIl=max{ (j~~--pl\h_i)-*,(cLkAI+rp~)-~), and for small h,, the norm of S depends on the distance between /.Q and Irr n Pk--1: !.&+I. Using the inequality ([l] , p. 589)
llP-Pll~2”ql
(Z-P)
T”ll 7
for the first term from (1.4) we have ]JS(Th--T4) (u,+u:‘)
Il<]]Sll[]l (z-Ph~)Thu,)l+2”‘1)u~‘(III (I-P”o)Thll].
We estimate p (u,) -PO. Using the fact that u. is a vector of unit length and
uol=[ wyho)Uo,
1 w-po)uo,
uol=O,
we obtain for p (u,) 2p0 the equation
P(Ud-Po= =
uho)
[u,,u
%I,
] m
u?nl
2b7??‘,(~h-~o)uol+bd”, 1+ [ IZE)) uy
This equation and the inequality
imply
[
(Th-CLo>U7Fl
(1.5)
126
G. P. Artrakhantrev
Therefore, for u#jl
we obtain
Using the completeness in the limit of the spaces Hh, it is easy to prove by induction that for small ho (1.7) implies that
We show that the sequence u, has a limit.
Lemma 2 The following inequality holds for small ho:
Roof From Eq. (1.4) we obtain (U::, -r&Z’) = -S{ (P-P)
(u:‘-z&) 0.9)
1(aP)-u~~‘l,)+p~)flL(un-*)-~(n~> I}.
+rPo-~(%-~) For P(G)-p(u,_,)
we have
Since the norm of u,==u~+u$),
given by
is greater than 1, then
lj&&-p(~m-i)
1~(1+llk?Jl) Ilu~‘-U~~,llll~*-~(~m-*~ II.
Then Eq. (1.9) and the inequalities (1 S), (1.6) imply that
+llw?II (l+II&n-*ll) IP-_Cl(%-dll}. Since Lemma 1 and the completeness in the limit of Hh imply that the expression in curly brackets tends to zero as h,,+O the statement of the lemma holds f& small ho. Hence, the limit of urn exists
Improvement of eigenvalues
127
and IIUm+i-~hII~Qm(l--Q)-lIIU1-uoll. We show that the Rayleigh ratio calculated on u, is an approximation to the eigenvalue pk. 7Reorem 1 We consider u,,,EH,,, defined by Eqs. (1.1) and (1.2), where as the initial approximation ~0 we take the Ritz approximation uOh~~Hho to the eigenvector corresponding to the I-fold eigenvalue & of &. (1.1). The inequality
I CL(~~)-~CLkI~C!{qm+Clk-iII(z-Ph)Tu,IIII
(I-Ph)Tcpll},
(1.10)
holds, where cp is some eigenvector for the number & (see (1.12)), and C, is a constant. (Here and below the letter C with or without a subscript will denote constants, and one letter may denote different constants.) hoof. We have the identity
U’--~(um))unv~“1 ~-~~l-~i[um,
[ Turn-~IJ.G,,, qd=[
[urn, cphl+P~m,
-(pr-p(um))
(P-(P~],
(1.11)
where cp and ‘ph are arbitrary vectors from H and Hh. We denote by @ the orthoprojector onto the subspace of eigenvectors corresponding to the number &. We take as cp a vector from Q"H,such that @p=uO, that is, (1.12)
(P=Q%.
We estimate the norm of the operator inverse to Q which establishes a one-to-one correspondence between Q"H and QH. Let z be an arbitrary normed vector from Q”H, I-1
Z=
Il
2 at2=1,
aiw,
11z112 =
i=O
i-0
where Wiis an orthonormed basis in Q’H. Then I-l
aiwi-ai (wrQw0,
Qz=E f-0
lail Ilwt--QwiII.
IlQzll~~ -E i-0
G. P. Astrakhantsev
128
For small h, the following estimate holds ([I] , p. 595)
Il~~~~~~ll~[l+~i(~~)lll(~~~hO)~ill~&(h,)‘0, ho+O. Therefore, for Qz we obtain I-t IV&II=
-z
~l+s,(~,)lII(z-~~)will~~,, i=o
which implies the inequality IIQ-‘II
Then Qcph=Qqr=zzo.
(1.13)
We can now show that the first term in (1 .l 1) tends to zero. Indeed, using (1 .13), we obtain
1[ix,-p(uw,)~m,
cp”] ~4rp”llII (Z-Q) (~*-~(~~m)~~mlI
+I [(T-_CL(zLJ)&,
hII I-
(1.14)
From (1 .l) and (1.8) it follows that
II (Z-Q) (T’*-~(u,))u,lf~l~~h”-~allQn~li~~-~~lI fmax(
pi-lo,
~0) qmllt.h--UOll,
(1.15)
and for the second term in (1.14)
I[ (T---CL (%I)) Gl,%- u,+t&nI I- I1(T-J.&) 1%n,&? 1I dll(Z--Q)
(1.16)
(J’“-p(~,))zdIIld?II.
Using (1.3), we have for (1.14)
I [ (h4~J)~mr cp”l~~ll (Z-Q) V*-pbm>)~mll x (lb,(‘) Il+llcpll)~max(pi-po, p0)qm x IISII II (Z-p”“) ThUoII[z,- ‘+allsllll (Z-P) Thuoll]=C,q”.
[Kmcp-cp”l =o,
[u,,q*]=l+[u,?, (Z-Q)q++-llu:’
II(1+6(~,))Il(Z-~ho)cpll~~zr
we obtain from (1.11),(1.17) I/.&&--l+r?-l
[CZqfn+P!i-ill (z-p*)
TutnIl II (z--P*) TqllI,
which completes the proof of the theorem. Remarks. 1. It is obvious from the inequalities (1.15), (1.16) that in the case of a single eigenvalue (Z= 1)
(1.17)
Improvement of eigenvaluer
lim //(P-p(u,))n,!i= m-.m
and consequently, I
129
0
converges to pkh.
2. If in the identity (1 .I 1) we define the vector cp by the equations
Q%J=-Q-‘~Qu-Q”hl,
[z-Q”l~=[T-~lkl-‘(I-Q~)lEm,
then, using the technique of [ 1 ] (the proof of Theorem 2), it can be shown that in the limit u, gives an approximation of the eigenvector from QVZ. Namely, for m= O(ln[.el!(1-P) TQp~~ll I-‘) the following estimate holds: ll~~-Q~~ll~ll (I-Ph)TQ"uhll (pk-‘+e+b(ho)).
(1.18)
From the inequality (1.18) it is easy to obtain that Ip(h)-pkIG
c
Ipk+i-pkl,
max i.-O,i,...,l--i
that is, that p(u(u,) approximates /ok just as well as ph, obtained by the Ritz method.
2. Application of the method to a variational-difference scheme 1. In a two-dimensional bounded domain a with boundary S let the eigenvalue problem
+
J
J
mucpds=A
ucpdS1
vcpdvw,* (Q) .
(2-l)
P
S
be considered. We assume that the inequality
YBO, Vzv=W,’(Q) ,
B(u, u> Wlull&,,,
issatisfied and that the data of problem (2.1) are sufficiently smooth. If in the space H of functions from WZ’ (52) we introduce the scalar product by the formula [u, rp] =B (u, cp), problem (2.1) can be written in the equivalent form: pZZL=TLf,
p=3L-‘,
where the operator T is defined by the equation
[Tu, cpl=
Ju’p a
Vu, ‘p=H.
P
We will describe the well-known process of constructing a variational-difference scheme for problem (2.1). We cover the plane of the variables (x1, x2) by a square net of step h and subdivide each square into two triangles diagonally, making an angle a/4 with the x1 -axis. We denote by Hh the set of continuous functions linear on the triangles. We assume that ho=@, where p is an integer. Then a function of & is also a function of Hh.
130
G. l? Astrakhantsev
We compare with problem (2.1) the equation of the Ritz method:
Vqhd?,.
B (uh, rp”) =hh j uhqhd&-J
(2.2)
P
We consider an iterative process for improving an eigenvalue & of multiplicity 1 of Eq. (2.1). As the initial approximation we take hk’b=3L0 and t.&+Hho, corresponding to the problem
B (uo’@, @) =&h” j uohq+&I
Vcp+&.
P
(2.3)
Let up=Hb, j=O,1, . . . . Z- 1, be eigen-functions, orthonormed by the scalar product [ , ] , corresponding to the eigenvalues k 2+j of Eq. (2.3). 2. We indicate the sequence of operations as a result of which we obtain the function u, + 1 from I& =Hh From Eq. (2.4) we find zhO&Yho:
Z-l +
Z(s
u,,,uj’bdQ- pmB(urn,4’9
j-0
Ifit is required that
P
B(zho, ujho) =O,j=O,1,
1B (UP,$9 =F (urn,cp’b)
. . . , Z-l,
(2.4)
then this function zb is unique.
We solve approximately the equation
-poB(z,cp)=F(um, cp)
"(PEHh,
(2.5)
that is, we find a function zh such that -poB(~h,
c,j)=F(u,,, q)-F(um,b)
where A is a linear operator on @!I,,
v'?EH~l,
(2.6)
small in norm:
We define zlho from the equation
B @iho, q+“>=B (zh,cp”O) v"cp"=H,. ‘Ihen ugii
will be found from the following equation:
(2.7)
Improvement of eigenvalues
131
1-i (‘1 um+l=um-
z j=o
uy B(u,,
u” ) +z~+z”-zlho,
(2 -8)
and for u, + 1 we have (I’
U*+1’wn+1
(2.9)
+u,.
3. Using the results of the preceding paragraph, it is easy to obtain an estimate of the rate of convergence. We have the following theorem. l7teorem
2
Let Xk be an eigenvalue of multiplicity 1 of Eq. (2.1). We consider urn defined by formulas (2.4) and (2.6)--(2.9). Then the inequality 1p,,,-Ak-’
/
m>C
In h-‘.
(2.10)
holds for sufficiently small h, and e2. Roof. With the scalar product [ , ] introduced by us the iterative process considered is included within the scope of that discussed in the preceding paragraph, except for the fact that Eq. (2.5) is not solved exactly. This leads to Eq. (1 .l) appearing as follows: (1) Um+l--&l(i) + ( P@-pa) --I (Z-Q)
+/A;’ (I-Pho)A’[
[ThUm-&&I
(Th-~m)um]=O,
where the operator A* is determined from the equation
[A*u, q+HAcp, u)
Qu, cp~H!,.
Assuming e,=liA*ll to be a fairly small number, it is easy to obtain that the statements of Lemmas 1 and 2 are true (the proof does not involve anything fundamentally new), and consequently, so also is Theorem 1. In order to use the estimate (1 .lO) it is sufficient to know the smallness of the norm 11( I-Ph) Z’ll, that is, II(I-Ph)
Tll=
sup
{[u,
ul-‘%+Acpl-‘”[U(cp-q+).dO},
where ~‘EH,, is defined by the equation B(cph, q”) ==B(q,
$“j
Q qh= H,,.
Any function of Wzz (Q) is approximated by functions from Hh of order h, that is, the inequality (see [2], Theorem 2.3.1)
min ]]r?uhll
uhai h
holds, and then (see [3], Theorem 3.1)
W$(Q)
~cltl141wv,~(o,,
A. V. Babakou and L. I. Severinov
132
Therefore, we have
which implies the statement of the theorem. Remark. Suppose we are given the multiplicity 1 of the number & and Eq. (2.2). Suppose we solve Eq. (2.6) after a number N(e2, h) of arithmetic operations. Considering that to find u,+~ from the remain~g equations we need 0(k2) operations, we obtain that for (2.10) to be correct it is necessary to perform C In h-1[h-2+N(e 2r h) ] arithmetic operations. If for the approximate solution of (2.5) the method of R. P. Fedorenko [4] is used, then N( e2, h) =O (h-2) (see [S] ), and consequently, to obtain xk with accuracy h2 it iS necessary to perform O(h-’ In h-‘) operations.
REFERENCES 1.
VAINIKKO, G. M., Evaluation of the error of the Bubnov-GaIerkin method in an eigenvalue problem. Zh. vj&hisl.Mat. mat. Fiz., 5,4,X%1--607,1965.
2.
OGANESYAN, L. A., RI~IND, V. Ya. and RUKHOVETS, L. A., Va~tionald~ference methods of solving elliptic equations. (Part 1). In: Differential equations and their application (Diferents. ur-niya i ikh prlmenenie). No. 5. In-t fiz. i matem. Akad. Nauk Lit. SSR, Vil’nyus, 1973.
3.
AUBIN, J. P., Behaviour of the error of the approximate solutions of boundary value problems for linear elliptic operators by Galerkin’s and finite difference methods. Ann. Scuola norm. Super. P&a. Sci fir mat., 21,599-637,1967.
4.
FEDORENKO, R. P., A relaxation method for solving elliptic difference equations. Zh. vj%zisl.Mat. mat. Fiz., 1,5,922-927,196l.
5.
ASTRAKHANTSEV, G. P., An iterative method of solving elliptic net problems. Zh. ujchisl. Mat. mat. Fiz., l&2,439-448,197l.
THE STATIONARY VARIANT OF THE FLOW METHOD FOR SOLVING PROBLEMS OF THE MECHANICS OF A CONTINUOUS MEDIUM* A. V. BABAKOV and L. I. SEVERINOV Moscow (Received 11 December
1974)
THE METHOD of successive approximations for solving a system of non-linear algebraic equations arising from the use of the flow method to calculate the flows of a viscous heatconducting gas is described. The proposed method is compared with the build-up method previously used. The new method is more efficient, by more than an order of magnitude of the number of iterations, than the build-up method. In the example given about 1200 iterations were required to attain the desired accuracy of solution of a system of 20X30X4 essentially non-linear equations. In [l ] a numerical method for solving problems of the mechanics of a cont~uous medium was proposed, which was based on a difference approximation of the conservation laws written down for each cell of the difference net in terms of surface integrals and flow density vectors. The method was illustrated by calculating the flow of a viscous heat-conducting gas past a body of finite dimensions. The stationary problem is solved by the build-up method in extremely simple form. To attain the desired accuracy a large number of time steps are required (of the order of tens and thousands) and consequently, considerable computer time. It is probably more useful to *Zh. vychisl. Mat. mat. Fiz.. 16,1,140-151,1976.
123
w ,,”= -q2 sin nz ,,, conditions (22) were infringed. The initial values were u,“=sin nz,, A’=-30qc, where q=2h-’ sin (xh / 2) ; we took u l equal to 3h. To avoid obtaining the max trivial solution, we divided the function U, by a constant, in such a way that ,,=,,Z.,,,,Y_~~~= 11 u ,,< 1. The number of steps was 288. The variation when, after the next step, ,,l__in,itx,_,_, of Am is indicated in Table 4. Translated by D. E. Brown REFERENCES
of ordinary differential equations, McGraw, 1955.
1.
CODDINGTON, E. A., and LEVINSON, N., Theory
2.
PALTSEV, B. V., Series expansion of the solution of the Dirichlet problem and the mixed problem for the biharmonic equation, in solutions of splitting problems, Zh. vjkhisl. Mar. mat. Fiz., 6, No. 1, 43-51.1966.
THE ITERATIVE IMPROVEMENT OF EIGENVALUES* G. P. ASTRAKHANTSEV Leningrad (Received 22 January 1974; revised 21 October 1974)
A MODIFICATION of the Newton method of finding the Ritz approximation ph to the eigenvalue p of a completely continuous operator is described. It is shown that if/~ is many-valued the iterative process ensures convergence to neither $ nor to the Ritz eigenvector, but in the limit the Rayleigh ratio for the vectors obtained approximates to /J with the same accuracy as @.
Introduction Let T be a linear, completely continuous, self-conjugate, non-negative (for simplicity) operator in the Hilbert space H, and let the eigenvalue p of the equation
(1) have multiplicity 1: p,ZQ
. . . >p=pk=p*,-I=
. . . =p*;:--l>pkl
>...
Here I is the unit operator. We consider a family of spaces H,,=H, dense in the limit, that is, lim I% =zb h4
where ph is the orthoprojector onto Hh. It is known that the eigenvalues equation
are approximations to p [ 1] . *Zh. vjkhirl. Mat. mat. Fiz., 16,1,131-139,
1976.
cLkh, - . . $+~_iof the
The eigenvectors 8.ihsatisfy the equation
where p (u”) denotes the Rayleigh ratio calculated on the vector nh y
[ 13 is the scalar product in H. In this paper we construct B ~~d~cati~ of Newton’s method for Eq f2), which guarantees &Itbc case l=l ~~~r~~~ to the ~~~nvector and to the ~~nv~~~ $A& In the case of a rn~~~~e x1-‘) differs &gawahe &=- I$ we obtain a sequence ZF,,,EH~ such that ufnm) for m=O@ from p by a quantity of order sl (h) 1 where a, (Jb) is the order of the approximation # of the nurnber p. The main element in the realization of the integration step Is the solution of an equation of the form
The result obtained is used to find the eigenvahres for the variational-difference approximation (see [2j) of two-dinrensional elliptic problems on a net of step k>O, If &. (3) is solved with an papacy of up to ez & is a faMy sm& number) after a mrmber N(al, hj + of arjt~eti~ jr +th o~rat~ons, theE after O( fN’+&+] la Es-‘) o~~~o~s we construct F (u”] f a~ro~at~~
We show that for sufficiently small ho the vector U, differs little from uo.
Improvement of eigenvalues
125
Lemma 1 For small h, the following inequality holds:
Roof: From Eq. (1 .l), written in the form (1) llm+l--Ulll
(Uo+U~‘)+(T~-T~)u~+(~o-~C1(U,))u,l=O,
(*)+s[(P-po)
we find the vector ugll: (i)
(Th-T~)u,+(~o-~(u,))u~)].
%?I+1-4
(1.4)
The operator S, acting in Hh , is bounded, and its norm is given by
IISIl=max{ (j~~--pl\h_i)-*,(cLkAI+rp~)-~), and for small h,, the norm of S depends on the distance between /.Q and Irr n Pk--1: !.&+I. Using the inequality ([l] , p. 589)
llP-Pll~2”ql
(Z-P)
T”ll 7
for the first term from (1.4) we have ]JS(Th--T4) (u,+u:‘)
Il<]]Sll[]l (z-Ph~)Thu,)l+2”‘1)u~‘(III (I-P”o)Thll].
We estimate p (u,) -PO. Using the fact that u. is a vector of unit length and
uol=[ wyho)Uo,
1 w-po)uo,
uol=O,
we obtain for p (u,) 2p0 the equation
P(Ud-Po= =
uho)
[u,,u
%I,
] m
u?nl
2b7??‘,(~h-~o)uol+bd”, 1+ [ IZE)) uy
This equation and the inequality
imply
[
(Th-CLo>U7Fl
(1.5)
126
G. P. Artrakhantrev
Therefore, for u#jl
we obtain
Using the completeness in the limit of the spaces Hh, it is easy to prove by induction that for small ho (1.7) implies that
We show that the sequence u, has a limit.
Lemma 2 The following inequality holds for small ho:
Roof From Eq. (1.4) we obtain (U::, -r&Z’) = -S{ (P-P)
(u:‘-z&) 0.9)
1(aP)-u~~‘l,)+p~)flL(un-*)-~(n~> I}.
+rPo-~(%-~) For P(G)-p(u,_,)
we have
Since the norm of u,==u~+u$),
given by
is greater than 1, then
lj&&-p(~m-i)
1~(1+llk?Jl) Ilu~‘-U~~,llll~*-~(~m-*~ II.
Then Eq. (1.9) and the inequalities (1 S), (1.6) imply that
+llw?II (l+II&n-*ll) IP-_Cl(%-dll}. Since Lemma 1 and the completeness in the limit of Hh imply that the expression in curly brackets tends to zero as h,,+O the statement of the lemma holds f& small ho. Hence, the limit of urn exists
Improvement of eigenvalues
127
and IIUm+i-~hII~Qm(l--Q)-lIIU1-uoll. We show that the Rayleigh ratio calculated on u, is an approximation to the eigenvalue pk. 7Reorem 1 We consider u,,,EH,,, defined by Eqs. (1.1) and (1.2), where as the initial approximation ~0 we take the Ritz approximation uOh~~Hho to the eigenvector corresponding to the I-fold eigenvalue & of &. (1.1). The inequality
I CL(~~)-~CLkI~C!{qm+Clk-iII(z-Ph)Tu,IIII
(I-Ph)Tcpll},
(1.10)
holds, where cp is some eigenvector for the number & (see (1.12)), and C, is a constant. (Here and below the letter C with or without a subscript will denote constants, and one letter may denote different constants.) hoof. We have the identity
U’--~(um))unv~“1 ~-~~l-~i[um,
[ Turn-~IJ.G,,, qd=[
[urn, cphl+P~m,
-(pr-p(um))
(P-(P~],
(1.11)
where cp and ‘ph are arbitrary vectors from H and Hh. We denote by @ the orthoprojector onto the subspace of eigenvectors corresponding to the number &. We take as cp a vector from Q"H,such that @p=uO, that is, (1.12)
(P=Q%.
We estimate the norm of the operator inverse to Q which establishes a one-to-one correspondence between Q"H and QH. Let z be an arbitrary normed vector from Q”H, I-1
Z=
Il
2 at2=1,
aiw,
11z112 =
i=O
i-0
where Wiis an orthonormed basis in Q’H. Then I-l
aiwi-ai (wrQw0,
Qz=E f-0
lail Ilwt--QwiII.
IlQzll~~ -E i-0
G. P. Astrakhantsev
128
For small h, the following estimate holds ([I] , p. 595)
Il~~~~~~ll~[l+~i(~~)lll(~~~hO)~ill~&(h,)‘0, ho+O. Therefore, for Qz we obtain I-t IV&II=
-z
~l+s,(~,)lII(z-~~)will~~,, i=o
which implies the inequality IIQ-‘II
Then Qcph=Qqr=zzo.
(1.13)
We can now show that the first term in (1 .l 1) tends to zero. Indeed, using (1 .13), we obtain
1[ix,-p(uw,)~m,
cp”] ~4rp”llII (Z-Q) (~*-~(~~m)~~mlI
+I [(T-_CL(zLJ)&,
hII I-
(1.14)
From (1 .l) and (1.8) it follows that
II (Z-Q) (T’*-~(u,))u,lf~l~~h”-~allQn~li~~-~~lI fmax(
pi-lo,
~0) qmllt.h--UOll,
(1.15)
and for the second term in (1.14)
I[ (T---CL (%I)) Gl,%- u,+t&nI I- I1(T-J.&) 1%n,&? 1I dll(Z--Q)
(1.16)
(J’“-p(~,))zdIIld?II.
Using (1.3), we have for (1.14)
I [ (h4~J)~mr cp”l~~ll (Z-Q) V*-pbm>)~mll x (lb,(‘) Il+llcpll)~max(pi-po, p0)qm x IISII II (Z-p”“) ThUoII[z,- ‘+allsllll (Z-P) Thuoll]=C,q”.
[Kmcp-cp”l =o,
[u,,q*]=l+[u,?, (Z-Q)q++-llu:’
II(1+6(~,))Il(Z-~ho)cpll~~zr
we obtain from (1.11),(1.17) I/.&&--l+r?-l
[CZqfn+P!i-ill (z-p*)
TutnIl II (z--P*) TqllI,
which completes the proof of the theorem. Remarks. 1. It is obvious from the inequalities (1.15), (1.16) that in the case of a single eigenvalue (Z= 1)
(1.17)
Improvement of eigenvaluer
lim //(P-p(u,))n,!i= m-.m
and consequently, I
129
0
converges to pkh.
2. If in the identity (1 .I 1) we define the vector cp by the equations
Q%J=-Q-‘~Qu-Q”hl,
[z-Q”l~=[T-~lkl-‘(I-Q~)lEm,
then, using the technique of [ 1 ] (the proof of Theorem 2), it can be shown that in the limit u, gives an approximation of the eigenvector from QVZ. Namely, for m= O(ln[.el!(1-P) TQp~~ll I-‘) the following estimate holds: ll~~-Q~~ll~ll (I-Ph)TQ"uhll (pk-‘+e+b(ho)).
(1.18)
From the inequality (1.18) it is easy to obtain that Ip(h)-pkIG
c
Ipk+i-pkl,
max i.-O,i,...,l--i
that is, that p(u(u,) approximates /ok just as well as ph, obtained by the Ritz method.
2. Application of the method to a variational-difference scheme 1. In a two-dimensional bounded domain a with boundary S let the eigenvalue problem
+
J
J
mucpds=A
ucpdS1
vcpdvw,* (Q) .
(2-l)
P
S
be considered. We assume that the inequality
YBO, Vzv=W,’(Q) ,
B(u, u> Wlull&,,,
issatisfied and that the data of problem (2.1) are sufficiently smooth. If in the space H of functions from WZ’ (52) we introduce the scalar product by the formula [u, rp] =B (u, cp), problem (2.1) can be written in the equivalent form: pZZL=TLf,
p=3L-‘,
where the operator T is defined by the equation
[Tu, cpl=
Ju’p a
Vu, ‘p=H.
P
We will describe the well-known process of constructing a variational-difference scheme for problem (2.1). We cover the plane of the variables (x1, x2) by a square net of step h and subdivide each square into two triangles diagonally, making an angle a/4 with the x1 -axis. We denote by Hh the set of continuous functions linear on the triangles. We assume that ho=@, where p is an integer. Then a function of & is also a function of Hh.
130
G. l? Astrakhantsev
We compare with problem (2.1) the equation of the Ritz method:
Vqhd?,.
B (uh, rp”) =hh j uhqhd&-J
(2.2)
P
We consider an iterative process for improving an eigenvalue & of multiplicity 1 of Eq. (2.1). As the initial approximation we take hk’b=3L0 and t.&+Hho, corresponding to the problem
B (uo’@, @) =&h” j uohq+&I
Vcp+&.
P
(2.3)
Let up=Hb, j=O,1, . . . . Z- 1, be eigen-functions, orthonormed by the scalar product [ , ] , corresponding to the eigenvalues k 2+j of Eq. (2.3). 2. We indicate the sequence of operations as a result of which we obtain the function u, + 1 from I& =Hh From Eq. (2.4) we find zhO&Yho:
Z-l +
Z(s
u,,,uj’bdQ- pmB(urn,4’9
j-0
Ifit is required that
P
B(zho, ujho) =O,j=O,1,
1B (UP,$9 =F (urn,cp’b)
. . . , Z-l,
(2.4)
then this function zb is unique.
We solve approximately the equation
-poB(z,cp)=F(um, cp)
"(PEHh,
(2.5)
that is, we find a function zh such that -poB(~h,
c,j)=F(u,,, q)-F(um,b)
where A is a linear operator on @!I,,
v'?EH~l,
(2.6)
small in norm:
We define zlho from the equation
B @iho, q+“>=B (zh,cp”O) v"cp"=H,. ‘Ihen ugii
will be found from the following equation:
(2.7)
Improvement of eigenvalues
131
1-i (‘1 um+l=um-
z j=o
uy B(u,,
u” ) +z~+z”-zlho,
(2 -8)
and for u, + 1 we have (I’
U*+1’wn+1
(2.9)
+u,.
3. Using the results of the preceding paragraph, it is easy to obtain an estimate of the rate of convergence. We have the following theorem. l7teorem
2
Let Xk be an eigenvalue of multiplicity 1 of Eq. (2.1). We consider urn defined by formulas (2.4) and (2.6)--(2.9). Then the inequality 1p,,,-Ak-’
/
m>C
In h-‘.
(2.10)
holds for sufficiently small h, and e2. Roof. With the scalar product [ , ] introduced by us the iterative process considered is included within the scope of that discussed in the preceding paragraph, except for the fact that Eq. (2.5) is not solved exactly. This leads to Eq. (1 .l) appearing as follows: (1) Um+l--&l(i) + ( P@-pa) --I (Z-Q)
+/A;’ (I-Pho)A’[
[ThUm-&&I
(Th-~m)um]=O,
where the operator A* is determined from the equation
[A*u, q+HAcp, u)
Qu, cp~H!,.
Assuming e,=liA*ll to be a fairly small number, it is easy to obtain that the statements of Lemmas 1 and 2 are true (the proof does not involve anything fundamentally new), and consequently, so also is Theorem 1. In order to use the estimate (1 .lO) it is sufficient to know the smallness of the norm 11( I-Ph) Z’ll, that is, II(I-Ph)
Tll=
sup
{[u,
ul-‘%+Acpl-‘”[U(cp-q+).dO},
where ~‘EH,, is defined by the equation B(cph, q”) ==B(q,
$“j
Q qh= H,,.
Any function of Wzz (Q) is approximated by functions from Hh of order h, that is, the inequality (see [2], Theorem 2.3.1)
min ]]r?uhll
uhai h
holds, and then (see [3], Theorem 3.1)
W$(Q)
~cltl141wv,~(o,,
A. V. Babakou and L. I. Severinov
132
Therefore, we have
which implies the statement of the theorem. Remark. Suppose we are given the multiplicity 1 of the number & and Eq. (2.2). Suppose we solve Eq. (2.6) after a number N(e2, h) of arithmetic operations. Considering that to find u,+~ from the remain~g equations we need 0(k2) operations, we obtain that for (2.10) to be correct it is necessary to perform C In h-1[h-2+N(e 2r h) ] arithmetic operations. If for the approximate solution of (2.5) the method of R. P. Fedorenko [4] is used, then N( e2, h) =O (h-2) (see [S] ), and consequently, to obtain xk with accuracy h2 it iS necessary to perform O(h-’ In h-‘) operations.
REFERENCES 1.
VAINIKKO, G. M., Evaluation of the error of the Bubnov-GaIerkin method in an eigenvalue problem. Zh. vj&hisl.Mat. mat. Fiz., 5,4,X%1--607,1965.
2.
OGANESYAN, L. A., RI~IND, V. Ya. and RUKHOVETS, L. A., Va~tionald~ference methods of solving elliptic equations. (Part 1). In: Differential equations and their application (Diferents. ur-niya i ikh prlmenenie). No. 5. In-t fiz. i matem. Akad. Nauk Lit. SSR, Vil’nyus, 1973.
3.
AUBIN, J. P., Behaviour of the error of the approximate solutions of boundary value problems for linear elliptic operators by Galerkin’s and finite difference methods. Ann. Scuola norm. Super. P&a. Sci fir mat., 21,599-637,1967.
4.
FEDORENKO, R. P., A relaxation method for solving elliptic difference equations. Zh. vj%zisl.Mat. mat. Fiz., 1,5,922-927,196l.
5.
ASTRAKHANTSEV, G. P., An iterative method of solving elliptic net problems. Zh. ujchisl. Mat. mat. Fiz., l&2,439-448,197l.
THE STATIONARY VARIANT OF THE FLOW METHOD FOR SOLVING PROBLEMS OF THE MECHANICS OF A CONTINUOUS MEDIUM* A. V. BABAKOV and L. I. SEVERINOV Moscow (Received 11 December
1974)
THE METHOD of successive approximations for solving a system of non-linear algebraic equations arising from the use of the flow method to calculate the flows of a viscous heatconducting gas is described. The proposed method is compared with the build-up method previously used. The new method is more efficient, by more than an order of magnitude of the number of iterations, than the build-up method. In the example given about 1200 iterations were required to attain the desired accuracy of solution of a system of 20X30X4 essentially non-linear equations. In [l ] a numerical method for solving problems of the mechanics of a cont~uous medium was proposed, which was based on a difference approximation of the conservation laws written down for each cell of the difference net in terms of surface integrals and flow density vectors. The method was illustrated by calculating the flow of a viscous heat-conducting gas past a body of finite dimensions. The stationary problem is solved by the build-up method in extremely simple form. To attain the desired accuracy a large number of time steps are required (of the order of tens and thousands) and consequently, considerable computer time. It is probably more useful to *Zh. vychisl. Mat. mat. Fiz.. 16,1,140-151,1976.