- Email: [email protected]
A new truncation procedure in the Bazley-Fox method
L. T. Poznyak
20
solution of operator equations of the first kind, Zh. @hid
Mat. mat.
5.
IVANOV, V. K., On the approximate Fiz., 6,‘1089-1094, 1966.
6.
MOROZOV, V. A., On a new approach to the solution of linear equations of the 1st kind with an approximate operator, Proc. of the First Conference of Young Scientists of the Computational MathematicsDept., Izd-vo Mosk. un-ta, 22-28, 1973.
7.
DAVYDOV, YU. B., Estimation of the stability of the solution of a linear operator equation of the fist kind, obtained by the approximate operator method,Matem, zap. Ural’skiiw-t, No. 4, 19-26. 1970.
8.
TIKHONOV, A. N., On the solution of ill posed problems and a method of regularization, Dokl. Akad. Nauk SSSR, 151, No. 3,501-504,1963.
9.
GONCHARSKII,
A. V., LEONOV, A. S., and YAGOLA, A. G., A generaiized discrepancy principle, Zh. v$hisl. Mat. mat. Fiz., 13, No. 2, 294-302, 1973.
10. GONCHARSKII, A. V., LEONOV, A. S., and YAGOLA, A. G., On regularization of ill posed problems with an approximately specified operator, Zh. vphid Mat. mat. Fiz.. 14, No. 4, 1022-1027, 1974. 11. KY FAN and GLICKSBERG, J., Some geometric properties of spheres ln a normed linear space, Duke Math. J., 25, No. 4,553-568,1958. 12. KLEE, V. L., Convexity of Chebyshev sets, Math. Ann., 142, No. 3.,292-304,
1961.
13. VASIN, V. V., Ill posed problems in B spaces and their approximate solution by a variational method, Diss. kand. fii.-matem. n., Sverdlovsk, In-t Matem. i Mekhan., 1970. 14. LISKOVETS, 0. A., Ill posed problems and the stability of quasi-solutions, Sibirskii matem. zh.. 10, No. 2, 273-385,1969.
A NEW TRUNCATION PROCEDURE IN THE BAZLEY-FOX
METHOD*
L. T. POZNYAK Leningrad (Received 4 May 1975) WHEN the Bazley-Fox method is used in the standard form, complicated transcendental equations have to be solved in order to obtain a lower bound for the eigenvalues of a self-adjoint positive definite operator A with a discrete spectrum. While, to avoid this difficulty, Bazley and Fox supplemented their method with several devices, it turned out that the devices do not provide good convergence in certain important classes of problem. The present paper offers a new means of simplifying the approximate equations, to which the Bazley-Fox method leads. It reduces fading lower bounds for the eigenvalues of the operator A to a problem of linear algebra, and has good velocity characteristics.
1. Introduction Suppose we are given in separable Hilbert space H with scalar product (.,.) and norm ( -1,the self-adjoint positive defmite operator (pdo) A with discrete spectrum; we pose the problem of finding the eigenvalue {hi} of A. As usual, we assume that the eigenvalues are arranged in increasing order, allowing for their multiplicity, and that the corresponding eigenelements {u,} are orthonormalized in the energy space HA of the operator A : (I&, Uj) A= 6
*Zh. vFchisl Mat. mat. Fiz., 17, I, 24-41, 1977.
A new truncation procedure in the Barley-Fox method
21
Assume that there is a self-adjoint pdo An in H with known eigenvalues {3.io} and eigenelements {uio},which is semi-similar to the operator A (i.e., the energy spaces HA and HA o consist of the same elements, see [l] ), and is connected with it by the relation (u, v),=(u,
\u,
v)_&-+ (Su, SP),
r=IZ,.
(1-l)
where S is an operator from Hinto a Hilbert space HI with scalar product (.J1 and norm ( l II . Our problem can then be solved by the Bazley-Fox method (see e.g., [2]), which approximates the required eigenvalues from below. For this, we have to choose in H, a sequence of finite-dimensional subspaces {w,} ,asymptotically dense in HI,and after arbitrarily ftig n, we have to evaluate the eigenvalues {ii”) of the pdo A,, generated in H by the closed bilinear form (u, v)aO+(OJu, Su),, where 0, is the orthogonal projector in H, onto W, . Notice that, to evaluate the {hi”} we do not need a knowledge of the explicit form of the operator A, ; the eigenvalues{k”}are fully defined by the above bilinear form, with the aid of which the eigenvalue problem for the operator A, can be written as the identity
(u, v)A,+(o+su, Sv)I=h(u, v),
O+u=H,,,
(l-2)
~vEH,,.
On varying n, we obtain for each eigenvalue Xi a sequence hi”, n=l, 2. . . . , convergent from below to Xi. The rate of this convergence is estimated in [3] . It was shown there #at, for sufficiently large values of n, (l-3) where I, is the identity operator in HI, PC -I1 is the multiplicity of the eigenvalue Xi, s is the least number for which X, = Xi, and C, is a positive constant, independent of n. If we restrict ourselves to the assumptions made above, it becomes necessary to solve complicated transcendental equations in order to determine the eigenvalues hi”, i=l, 2,. . . , The computations can be simplified by imposing extra conditions on the subspaces W, . The condition was originally pointed out by Bazley and Fox [4], and can be stated as whatever the such that
n=l,
SU,~J-.W*,
2,. . . ) for
there exists a number G-N(~).
N(n), (1.4)
In this case, the eigenvalues of the problem (1.2) can be found by solving an algebraic eigenvalue problem. Experience shows that condition (1.4) is extremely rigid, and not many types of problem have as yet been found in which it is satisfied. Moreover, our studies of convergence for some of these problems (see e.g., [5]) h ave revealed that the rate of convergence of hi” to hi may be extremely slow. In addition to the method considered, involving a special choice of test spaces, Bazley and Fox [4,6] found another means for overcoming the difficulties that arise when solving the intermediate problem (1.2). Their new idea was to solve problem (1.2) itself approximately, while still retaining the main aim of obtaining lower bounds for the eigenvalues hi- While realization of this idea demands certain restrictions, these are much less rigid than condition (1.4). The restrictions are as follows:
22
L T. Poznyak
1) D(P) =LHi, 2) W,dl(S'),iz=l, 2,...,where S* is the operator adjoint to S, and the bar denotes the closure operation. Restriction 1) follows from restriction 2) and the condition made at the start, that the sequence {IV,} be asymptotically dense in H, . The need for condition 2) is clear from the type of problem which Bazley and Fox proposed to solve instead of (1.2):
+ (S’O,Su, u) =
Here, utO=(h?)“Q~, H.
i=l,
a(u,
u) (
OZud,
VUEH.
2, . . . , are the eigenelements of the operator A 0, orthonormalized in
Problem (1 S) is obtained from problem (1.2) by replacing the bilinear form (u. Y)~u by the bilinear form, bounded in H, m
c
,
(ajO-hrE+,) (u, u,O)(Q”,u) fd+1
(0,&a, Sv)i
(S'OJu,u),bounded
the
u .
Ha symmetric bounded pdo Au(“):
A;“)
u
(1.6)
H. The second
To the form (1.6) there corresponds in
.. (hj"-hi+i)
=
(u,
u_i0+h~+*u7
j-t
which is called the m-th order truncation of the operator Au. This name originates from the method of obtaining the operator A0(m): in the spectral decomposition
Aou =
E aI0(4 9”) uj
of the operator A, in the space H, the eigenvalues It:+,, hom+Z,. . . have to be replaced by the same eigenvalue X0,+, . The form (1.6), which can now be written as (Afj”‘)~, V) , is called the m-th order truncation of the form (u, Y)Ao, while the method of replacing problem (I .2) by problem (1.5) is known as the truncation method. The condition 2) enables us to write problem (1 S) in the operator form in the space H:
A~m%z+S'O,Su=~u. It can easily be seen, by analyzing the structure of the operator of its eigenvalues amounts to a problem of linear algebra.
Ad""+S*O,S, that determination
While application of the truncation method rarely presents serious difficulties, it does not
A new truncation procedure in the Bazley-Fox method
23
always give good results. Of course the poor results stem from the slow convergence of the method; but it is hard to say what causes this slow convergence, since we know so little as yet about the rate of convergence of the eigenvalues h,anrnof the problem (1 S) to the exact eigenvalues Xi. In [6], Bazley and Fox obtained the estimate 3Lin-)L{“mGC* (n) (c+,)-ie
(1.7)
They made the assumption that A = Au t B, where B is a symmetric positive operator, and they took relation (1.1) in the form (u, v)a=(u,
C)&+(u,
U)B,
so that H, = HB, while S is equal to the identity operator considered as an operator from H into HB. Bazley and Fox did not investigate the dependence of the constant C,(n) on n. Weinelt attempted to explain it in [7]. Making the additional assumption that IV, is the same as the linear hull of the elements ulo, . . . , u,‘, and that the operator B is positive definite and comparable in force with the operator Aob, OGpG1, i.e.,
Weinelt obtained for C,(n) the estimate C,(n)Gc‘(h,0)2P
(1.8)
with a constant C, which is independent of n. This is all that is known about the convergence of Xinm to Ai” for fured n. An idea of the convergence of Xinm to Xi can be gamed by combining (1.7), (1.8) with (1.3). For instance, if B is bounded @ = 0), it follows from (1.3), (1.7), and (1.8) that the double sequence hinm, n, m=l, 2,. . . , is convergent to Xi, at a rate not less than C,(GI+,)-‘+C,
2 I (E-On)&+jlB2, j-0
where E is the identity operator in H. Unfortunately, this case is not typical in practice. And in the caSe of an unbounded operator B, the expressions in question give no satisfactory estimate of the rate of convergence of xinrn to Xi. If the asymptotic behaviour n”, u > 0, of the eigenvalues {LO}, is knmvn, then (1.3), (1.7), and (1.8) can be used to extract from the double sequence h,““, n, m=l , -I***, 7 the ordinary sequences IL?>“‘(~), n=l, 2,. . . ? and to estimate their rate of convergence to Ai; the estimates thereby obtained have a low order. Recall that everything just said about the convergence of Arm to Xi only holds under the above special assumptions about the operators A, Au and the test subspaces W,. In the general situation considered at the start of the section, nothing is known about the convergence of hnm to x, This present state of affairs in the Bazley-Fox method compels us to look for new ways of realizing the basic idea of the method, concerning approximate solution of the intermediate problem (1.2). A new way of simplifying problem (1.2) is described in the present paper. It differs from the method of truncations mainly in the fact that we replace the bilinear form (u, v) in the identity
24
L. T. Poznyak
(1.2) by a larger bilinear form, whereas Bazley and Fox replaced the form (u, v)~ o in it by a smaller bilinear form. The problem that then arises again has the “intermediate” property: its eigenvalues give lower bounds for the eigenvalues Xi. We shall show that the determination of the eigenvalues of the new problem reduces to a problem of linear algebra. Finally, the feature of principal importance is that, under the general assumptions made below, an estimate can be obtained for the rate of convergence of the new approximate eigenvalues to the exact eigenvalues. The efficiency of the estimate is illustrated by an example of a Neumann problem for a two-dimensional secondorder elliptic equation.
2. A new truncation procedure in the Bazley-Fox method Turning to a detailed treatment of the new approximate method for solving problem (1.2), we fust observe that it can be used under the same general assumptions as the Bazley-Fox method itself. The new method amounts to replacing the bilinear form (u, V)in the identity (1.2) by the symmetric bilinear form
cm[
(Ajo)-‘-(~~+l)-‘l(u, qO)
A0
cu;,v>Ao+(hi+i)-l (u, v)
Ao.
(2.1)
j-1
The approximate problem which we propose to solve instead of problem (1.2) then has the form
(4 ~)A,f(~,Jh Sv)*=h{ (hnf+*)-‘(u, v),, (2.2)
It is easily shown that the form (u, u) is less than the form (2.1):
(2.3) :-m+1
ki
The form (2.1) is obviously positive definite in HA ,,, and corresponds in this space to the symmetric bounded pdo (A,-‘) (m):
c[ m
(/lo-‘)
(m)g =
(hjo)-I-
G+d -'I (l-5,
Uj')
&Uj”+
(hri+i)-‘U*
(2.4)
A new truncation procedure in the Bazley-Fox method
25
We aim to emphasize, by the notation (A,- ’ >(m) that this operator is formally obtained from the operator A0 - 1 by means of the same procedure as the operator Ao(“‘l is obtained from the operator A,. In fact, for Au-’ we write the spectral representation A,-5.1
=
c
(5’) --! (72,Uj')
A&j0
j-1
in the space HA O,and then we replace all the eigenvalues of Au- ’ with numbers m t 1, m t 2, . . . , (m) in this representation by the (m + l>th eigenvalue (AR+r) -‘. Hence it is natural to call (A,-') the m-th order truncation of the operator Au- ‘, and calI the present method the method of inverse operator truncations. Let us return to problem (2.2). We shall show that it has eigenvalues representing lower bounds of the eigenvalues of the problem (1.2), and we shall give the method of calculating them. Considering S as an operator from from
HA Ointo
HI, we introduce the adjoint operator s’, acting
HI into HA*: (Su,W)i=(U,S'W)Ao VUEH,,, vw=H,.
It was shown in [3] that the operator s’ is bounded. Using this operator, we write (2.2) as an operator problem in HA0:
(z+S'O,S)u=3L(Ao-')'"'u, O+U~H&,
(2.5)
where I is the identity operator in HAO.We can write the last equation out more fully as d(n) (U, S’W~)&tlijSrWj=h’ (d+i)-‘U usc i,j-i
(2.6) +
c
1=j
[
(ijo>-‘-(Cn+i)-‘I (UTUj”)AoUjO 1t
where LL’~. . . . , Ed(n) iS the basis in w,, matrix
(aij),
D=( (wi? Wj)i),
i, i=i,
2,. . . , a(rz),is the
i, j=1,2,.
. . , d(n).
inVerSe
matrix to the
(2.7)
Symmetric pdo’s in HAM appear in both sides of Eq. (2.5). It is clear from (2.6) that the subspace V, stretched over ulo, . . . , IL,‘, S’wi, . . . , s’wd( ,,), reduces theSt? operators. On solving problem (2.5) in the subspace, we obtain a finite number of eignvalues of finite multiplicity pi”“< . . . G G& ’ where r(n) denotes the dimensional@ of the subspace I’. In the orthogonal complement to V, Eq. (2.6) transforms into the equation u-h (AZ+*) -?.z, so that its spectrum in this complement consists of the unique eigenvalue A, + 1O, whose corresponding eigenelement is the e:exhaust the spectrum of the entire subspace HA06 V. The eigenvalues problem (2.5). We arrange the eigenvalues of the problem (2.5), not exceeding Xo,+1, in increasing order while allowing for their multiplicities: 3L~“m~h~“m~ . . . . Clearly, starting from some number i, not exceeding r(n) + 1, this sequence becomes “stationary”: 3Ljnm=AR+1, Pi. In the case
L. T. Poznyak
26
wfien A,+1o is the least eigenvalue of problem (2.5), the entire
sequence hjnm, j-1,
2,. . . , is
2, . . . .
stationary: h,“m==h~+~,j-1,
On now applying to problem (1.2), (2.2) the familiar comparison theorems, and recahing (2.3), we obtain the estimates of interest: i=l,
~i”“Ghi”Ghi,
2,. . . .
(2.8)
Since Az+1-+m as m-+00, and the right-hand sides of the inequalities (2.8) are independent of m, the number A,,,+1o cannot be the least eigenvalue of the problem (2.2) for sufficiently large m; for such values of m, there must necessarily be eigenvalues of finite multiplicity of problem (2.2) to the left of A,+1 o , the number of which increases without limit as m increases.
In short, we have established that the inverse operator truncation method reduces the determination of lower bounds for the eigenvalues {I.,} to the solution of problem (2.6) in finite-dimensional space V. In turn, this problem is equivalent to the matrix problem Ax=ABx
(2.9)
with symmetric positive matrices A and B of order r(n). The matrices A, B depend on the choice of basis in V. We can assume without loss of generality that uio, . . . , U,‘, SWi, . . . , S’Wen)-m form a basis in V. With this basis,A and B have the block forms E + CD-X / K’ + C’D-ll? -----------K + F’D-‘C 1 M + F’D-‘F
B=
(2.10)
AoK’
-__--------i--
(2.11)
KLoK’ + h0,+,M where
E=(dij), c=
i, j=l,
&=(
2,. . . , m, i=l,
( (S’wt, u4> &)I
2,. . . ,d(n),
K= ( (S’WI, uj”) ao) t
i-l,2
E= ( (S’wi, S’wj) 4,) 7
i-l,
M= ( (S’wiy S’Wj) A) 7
i, j=1,2,.
Lo= ( [ (Ato) --I- od+i) -‘l&d 9
Expressions (2.7), (2.9)-(2.12) operator truncations.
,...,
i, j=1,
(hf")-'6*j),
j=1,2
r(n)-n2,
2,. . .,d(n),
,...,
2,. . . , m, m,
j=l,2,...,m; j=1,
(2.12)
2,. . . ,r(n)-m,
. . ,r(n)-m,
i,j=l,2
,...,
m.
represent the computational formulae for the method of inverse
A new truncation procedure
in the Bazley-Fox
method
27
3. Convergence and convergence rate estimate We shah start our study of the convergence of the approximate eigenvalues hinm, i= 1, 2,. . . , by seeing how they depend on the index m. From the definition of the truncation (Au- l)(m) it is clear that it decreases monotonically with respect to m: (A,-‘) (m)> (Ao-‘) cm+i)aAo-i, m= 1, 2,... . In view of this, for futed n every eigenvalue hinm is monotonically increasing with respect to m: hi”“
m-l,
,
2,. . . .
Let us show that the sequence hiRm, m=l. 2? . . . , converges to Ain. For this, we reduce each of problems (1.2) and (2.2) to an eigenvalue problem for a symmetric bounded operator in HA ,,. We have already taken a step towards this reduction in the case of the problem (2.2), by replacing the identity (2.2) by the equation (2.5). The same step, for the problem (1.2), leads to the following equation in HA o:
(I+S’OJ)
u=l.A,-‘u.
(3.1)
We introduce the notation F,=I+S’O,S. The properties of the operators F, were examined in detail in [3] and they will be used below without reference. If we make the replacement v=F’,o u, in (2.5) and (3.1), it becomes obvious that (Ai”)-‘, i= 1, 2, . . . , are the eigenvalues of the asymmetric completely continuous operator F,?A,-‘Fi’” in f&o, while (hi”“) -I, i=l, 2, are the eigenvalues of the symmetric bounded operator F,” (Ao-‘) cm)F,‘h in the same . *-, space. By a well-known theorem * (see e.g., [l] , p. 258), hymn)
--!_
(hi”)
IF,li’
-‘<
( (&,-l)
(m)-A,-i)F,‘”
1~0.
(3.2)
It is easily shown that 1 (&-‘)
(m)-Ao-il,,o=
(h:+i) -‘.
(3.3)
On further recalling that 1F,-“’ 1Ao< 1, we obtain from (3.2) and (3.3): (Qy
-‘-
(L+l)-‘3
(hi")-'<
or alternatively,
(h?t+i) -ly
,j,in-hinmG?b*nhinm
which shows that Xinm are convergent
to Xin
(3.4)
as m + 00.
On coarsening the inequality (3.4), we obtain an estimate for the rate of convergence of hinm
t0 Ai”
’ h+hr”m~A,z
*The theorem was proved in [l] only one of which is completely
for completely continuous continuous.
(?hn+,)-i,
operators,
but it remains
(3.5)
true for the present
operators,
L. T. Poznyak
28
in which, as distinct from the case of Bazley and Fox’s estimate (1.7), the constant on the right-hand side is independent of n. We can also obtain from (3.4) effective, practically computable estimates for the error introduced by the operation of truncation of the operator Au - l. In fact, an upper bound is easily obtained for the eigenvalue Xi by Ritz’s method; this bound usually holds before application of the Bazley-Fox method (the latter is in fact used to estimate the error of the Ritz method). Denoting by & an upper bound for hi, computed by the Ritz method, we find from (3.4) that ~~n-.liRm~h**m31~(X:+1) -*)
(3.6)
3Lin-hinmGXfz(~+~)-1. The estimate (3.6) is a posteriori
(3.7)
while the estimate (3.7) is a priori
We now turn to a study of the behaviour of the approximate eigenvalue hinm as a function of the two indices n and m. To be more precise, we shah henceforth regard Xi”“’ as a double sequence and examine its convergence regardless of how n and m tend to 00.We shall give the same interpretation to the convergence of other objects (elements, operators) encountered below, dependent on the pair n, m. The convergence of hinm to Xi is proved in an elementary way on the basis of the inequalities (1.3) and (3.5):
At the same time, (3.8) gives an estimate for the rate of convergence of Xinm to hi. This is not a limiting estimate, however. A better estimate, of higher order with respect to m, can be obtained by comparing problem (2.2) directly with the initial problem OklEIZ&
(u, U)A”f(SU. Su),=h(u, v),
VVEHA,
(3.9)
in accordance with the same scheme as was used in [3] for and estimating the error k-hi”” estimating the error of the Bazley-Fox method without truncation. Let us briefly run over the scheme. We first have to reduce the initial problem (3.9) to the equivalent operator problem in the space H,,: (Z+S’S) ~=hA~-‘zz, then the latter, and the approximate equation (2.5), have to be transformed respectively to u=hF-‘Ao-‘1~. n=hF,-‘(A,>-‘) where F=Z+S’S. It is easily seen that F-lA,,-i=A-‘. notation Tnm for the operator F,-’ (ilo-‘) Irn).
(3.10) (m’zz,
(3.11)
For brevity, we introduce the simpler
When estimating the error h~-h,“‘~ an important role is played by the difference R,,,,=T,,,,,-A-!:
(3.12)
and in particular, by its two obvious representations R,,=F-‘S’(Zi-O,)ST,,+F-‘[
(A,q-‘)‘m)-Ao-‘],
(3.13)
A new truncation procedwe in rhe Baziey-Fox merhod
R,,=F,-‘S’(Z,-O,,)SA-lf-F,-‘i
29
(Ao-l)~*~-_40-i],
(3.14)
and the property I%,
as
].4,-0
(3.15)
rl. m-w,
which is proved in the same way as in [3]. The operator R,, characterizes the proximity of the exact problem (3.10) to the approximate problem (3.11); naturally, the error ;*s-)bi”“’ also depends on it. The nature of the latter dependence is proved in the same way as in [3] . In fact, we initially obtain, with the aid of (3.10) and (3.11): (I-&A-l)
uinm= (hi”“) -l (i,,““:-_3,,) ui”x+_thiR,,u,“m,
(3.16)
where uinm IS . the eigenelement of the problem (3.1) corresponding to the eigenvalue Xinm. Then, multiplying (3.16) scalarly in HA by the element Piuinm, we find O= (3uanrr2) -’ (hznm-l~i) ~~~~~~~ Prui71m) A+2vi(R,,uinm, PiuiRrn)_k_
(3.17)
where Pi is the orthogonal projector in HA onto the subspace stretched over uI, u,+~, . . . , usi,. Finally, noting that ( Uinm, Piurnm) _4= 1PJL,“~ 1Az, and for brevity, putting y= 1P+Uinm1a-‘Uinrn! we obtain from (3.17) the required expression 3.~,-~~,nm=3iii,,nm (Rmy, P,y) -4.
(3.18)
Notice that, when obtaining (3.18), we have tacitly assumed that I Piuinm IA>O. We justify this assumption below, for sufficiently large n and m. The next “block” in the scheme of arguments in [3] is to isolate the “unimportant” terms in the scalar product (R,, y, P,y >A. Replacing R,, in accordance with (3.13) and using relation (1 .I), we can write (R,, y, Piy), as the sum (Rnmy, Piy) ,=(O’“‘STnmP
I a,2+ (O’“‘ST,mP”‘y,
0’“‘SPy)
1-k 1 ( (Ao-‘) (m) (3.19)
0’“‘SPiy) *+ ( ( (AO-‘) (m’
-A,-‘) PCi’y, P,y) Ao, PCil=I-Pi.
We obtain the expression for T,,,, from relations (3.12) and (3.14): T,,,,=A-‘+F,,-‘S’O’n’SA-‘+E’,,-‘(
(A,-‘) ‘“;.-Ao-‘),
and we substitute this expression into the first term on the right-hand side of Eq. (3.19). After obvious transformations, we obtain (Rnmy, Piy) A=&-’ IO’“‘SPiy Ii2+ I ( (AG-‘) (m’-Ao-l) ‘“Piy IA02 +I”,-: IF,‘“S’O’“‘SP,y + (O’“‘ST,,P”‘y,
IA?-+ (SF,-’
( (A,-‘! cm’-Ao-l) Piy, 0’“‘SPiy)
0’“‘SPiy) ,+ ( ( (Ao-‘) ‘“‘-A,-1)
i
(3.20)
P”‘y, P*y),,.
This is in fact the required expansion of the scalar product (R,,, y, Piy) A into essential and inessential terms. Let us show that the last three terms in (3.20) are inessential. We shall show, in fact, that each of them has higher order than 8,,,,= I O(“‘SPiy / 1’+ 1( (Ao-‘) (m’-Ao-‘)“P,y 1Ar,2. For the first term, the proof is easy:
L. T. Poznyak
30
I (SF,-’ ( (A,-‘) (n’)-Ao-l) P,y, O’“‘SP,y) i I
~ISlOil (Ao-l)(m’-A,-‘I~oI( (A,-‘)‘“‘--Ao-l)‘~~PiylAoIO(~‘SPiyl~ GISl,,l
(3.21)
(Ao-l)(m’-Ao-lI~~e,,,
where I -lo 1 denotes the norm of the bounded operator acting from H,Q into HI. For the other
two terms, the required bound cannot be obtained directly, since they contain the expression P($, the connection of which with the quantities ( Q’“‘SPjy 11 and 1 ( (A,-‘) (nd’-AO-‘) ‘“P,y 1Ao is as yet unknown. The connection is established by: Lemma The pair n(i), m(i) exists such that, for n > n(i), m > m(i), we have 1PiUintn
1 *>O, (3.22) 1*<(Js ( IO'"'SPiy
Ip(i’y
1I+ 1( (~40~‘)'mi-AO-')'hPiy
1Ao)t
where the constant C, is independent ofn, m, and i. The proof is the same as the proof of Lemma 8 in [3]. We can now easily obtain the required estimates: I ( ((A-‘)
(m’-A,-‘)
P”‘Y, Pi, y)
acI<2CSI (&lo-‘) (m’-Ao-t I :&I,,,
( (09.w,,P(~)y,o(wP,y)
l
1ecs 1oYw,,
(3.23)
loienm.
(3.24)
Notice that the convergence to zero of the quantity ( Ocn)ST,, I o, follows from (3.12), (3.15), and Lemma 1 of [3]. In short, when estimating the rate of convergence Of hinm to Xi we can neglect the last three terms in (3.20). The third term on the right of (3.20) also has no influence on the order of smallness of the error L-k,“” since
A more exact result, which follows from (3.18), (3.20), (3.21), and (3.23)-(3.25), stated as follows.
can be
Theorem 1 If the sequence of subspaces {IV,} is asymptotically dense in HI, then, given any i=l,2,.. .,apairn’(i)> n(i), m’(i)>m(i), canbefoundsuchthat,forn>rz’(i), m>m’(i) we have the two-sided estimate o.5ki
1O(n)Sp,y
<&_hi”“<2CJ,
1 ,2-/-(). jhi2 1 (
(A,-‘)
(n”-~4-1>“‘P$
1AU’
1O’“‘SP{Y l ,2f2h,2 l ( (A,-L)(rn’-&-‘)‘:?p4 I AZ.
(3.26)
where C,=l+13’1,,‘. If we use the same method as in [3] , and expand Pfl in eigenelements u,, . . . , zzScxr corresponding to the eigenvalue Xi, we easily obtain from (3.26) an estimate connecting the error Ac-&“~~ with the errors of approximation of the elements SuJ, . . . , SU~+~.
A new mtncation procedure in fhe Bazley-Fox method
31
Theorem 2 Let the conditions of Theorem 1 hold. Then, for m>m’( i), n>n’( i) (3.27) ( IO(n)SU*+jli2+ I ( (A,-‘) cm’-Ao-‘) ‘ZZ,+jI*.2)7 ?bi-?Linrn
cm)-A,-l)“:=(
(A,-‘) ‘“‘-A,-‘)‘“(&Q,).
In the light of this equation and the estimate (3.3), we have 1((A,-‘)
(A:+,)-‘“I (I-Qm)%lAo.
(m)-Ao-‘)~ujI~,<
If it is assumed that UjED (A0°.5+B), i=l, we can obtain the better estimate
(3.28)
2, . . . , for some 0 > 0, then instead of (3.28)
I _40~(A:+,) -“-‘I (E-Qm) AOp”+’ UjIa I ( (A,-‘) ‘n”-AOwi) “2~~
(3.29)
For, under the condition indicated, 1(z-Q~)B~I~~=IA~-~(E-Q~)A~~~+~IL~I~IA~-~(~-Q~) I
which, in conjunction with (3.28), gives (3.29). We have thus proved:
If the conditions of Theorem 1 hold, r&n’(i), for some /32 0, then
m>m’( i) and l!jED (Ao”~‘+‘), i=l.
h,-h:““‘GCT (Xi) 2 10(n)SU~+jlI’+0 ( (lb,‘) -‘-2b). I=”
2, . . . ,
(3.30)
Note. In the next section we shah consider an example in which the test subspaces form a generalized and not an ordinary sequence {W,}. The extension of the earlier results to this case is trivial: we simply have to replace the index n in all the expressions by the symbolo.
32
L. T. Poznyak
4. Application
the Neumann
The main practical difficulties that arise when using the new truncation operation are connected with the operators’. We previously had to deal with this operator in [3,5], though there it played an auxiliary role. Now, the operator s’ occurs in the computational expressions (2.12). It is only rarely that s’ can be found in explicit, closed form. It is fortunately not essential to be able to do this. The practical problem concerned with s’ may be stated alternatively as: to select the test subspaces W, (or IVat)in such a way that the operator s’ can be simply evaluated on the elements of W, (War).While this last problem is again difficult, the important example given below shows that a solution is possible. Let fi be a circle or rectangle, and aa the boundary of Q, ~=QUaQ. functions.a(s,, z2), aij(Zt, ZZ), i, j=1, 2, satisfying the conditions: k>l,
fZEP(a),
Uijd”+i(52)T
Uij=CZji,
Given in 3 the
(4.1)
ah, 4 >I.
T=COnSt>l, i,j-1
In the space H= L2(s2) we defme the self-adjoint pdo A by the relations
addrv=en~=o
D(A)={u]u.=W,~(Q),
on
anj,
Au=-div PVufau, where 9” (a+<), i, j-1, 2; Yis the unit vector (written in the column form, as are all vectors) of the inward normal to X2. Hence A is the operator of the Neumann problem for a second-order two-dimensional elliptic equation; and its properties are well known. We specify the operator A0 by the expressions D(A,)={~~uEW~~(Q),
au/&=0
Aeu=-4u+u.
on&?},
We have N_4=W2’ (Q),
(u, II),
= JJ (9Vu
Vv+auu)dB,
P H ib=wz*
(S-J),
(u, v> Ao=
(Vu VVSUV)dSZ, JJ 0
so that (u,v)za=(u,u).ko+
(4.2)
JJ[(sa)vuvu+(a-I)WI~R, 0
where8=(6il),
i,i=l,2. Ifweput H,=Lz(~)XL?(Q)XL2(~), where T denotes transposition, u,,==acz/axi, (u*,, uz*, u)T, ~ =
(9-8)‘h
II
0
0 (a-l)‘”
D(S) =Wz’(Q),
II ’
then Eq. (4.2) can be written in the form (I .l) and all the conditions for application of the Barley-Fox method to the problem Au = )uc are satisfied.
su=sc
A new truncation procedure in the Bazley-Fox method
33
In order to find the test subspaces in which we are able to compute the values of the operator s’, we have to analyze in more detail the structure of the space H, . Theorem 3 The
can be written as the orthogonal sum of the three
space H,=L2(Q)XL2(Q)XL2(S2)
subspaces H,=X@Y@Z, where x={w~U’=(Lz,,,
ux2,U)T, u=IVw,l(fi)},
Y={w~w=(u,2.
-t&O)‘,
Z={r.+=@,,,
us,, Au)~, u~W,~ (S2), du/~?v=O on &2}.
U~IT,~(Q)},
Proof. It is easily shown that X, Y are subspaces. Their orthogonality follows from the well-known theorem on the decomposition of the space Lf (52) XL,(Q) into an orthogonal sum ofsubspacesG={gIg= (u%,,u,)*, u~IJV~~(Q)} and i={gjg=(zz,, -uxJT, ZZEW~~(Q)}. For, given any pair ‘p=X, $EY we have (p=(u.,, (9,
cl
i =
r&c,,U)r, JJ
uf=W*‘(Q),
( wx,-wx,)
$= (v-9 --vx,, 0) 3
VEW,’
(s-2)
)
dQ,
G
and since (u.?, uxl) ‘=G, ( r;,, -vZJT~l,
we have (cp, $) 1~0.
It remains to show that (X@ Y) l=Z. We take an arbitrary element w= ( w1 (z,, zZ) , ~22 (x,,L_), IL’~(z~,z~))~E(X@Y)~. Wehave
ss ss
(u,,w~+u,,w~+uw~) dQ=0
vu= Wz’ (s-2) )
(4.3)
P
(v,,w~-vI,w2)
vvdk2’(8).
cm=0
P
By the theorem just mentioned, on decomposition of the space L, (Q) XL, (Q) ,it follows from (4.4) that w,=z=,, w2=zx,, where ZEWZ’ (i-2). On using this fact in the identity (4.3), we get u,,Z,,+U&,+Uw~) as-J=0 rr.Z=w, (Q) . SS( This last identity implies &at 2(x,, x2) is the generalized solution of the Neumann problem 4z=u,3
in S-2,
dz/&=O
on CP2.
(4.5)
Since 52 is a circle or rectangle, Eqs. (4.5) will hold almost everywhere, in 52 and on 22 respectively (see [8] ). The theorem is proved. The set {U/ UEW,~ (S-2)) c%/&=O on d0) brevity we shall henceforth denote it by %.
is a subspace of the space WZ2(R). For
Let us turn to the construction of the test subspaces. They will now depend on the three integer indices Z,p, 4. The set B of all triples a= (2, p, q) will be assumed to be partially ordered
34
L, T. Poznyak
in the natural way: o.Ccc’, if ZGl’, p
-fm
o>=, fafp,
zg=(z,,, z+,, AZ)‘,
ZdY,.
Let us find s’w for ZUEW,. We have
i = JJ (VuVL+UU)ds2 P + J J (ux,j.rz-ux~,) cm+ J J (vuvz+uAz) dW Vu=D (8) , (Su, w) I= (9-‘Su,
rp+l#+Q
l-4
Q
and since the last two integrals vanish in accordance with Theorem 3, we have (Sa, w) I= (zz, v) + But this implies that 5”~ = v. In short, the difficulty arising in applying the inverse operator truncation method can be overcome in practice in the present example. In order for the approximate to converge to the exact eigenvalues, the generalized sequence (IV%} has to be asymptotically dense in HI. Sufficient conditions for this are given by: Theorem 4
If the sequences {L,) , {Ni}, {Cr,} are asymptotically dense in FV2’(Q) , W,’ (‘2) and R, respectively, then the generalized sequence {w,} is asymptotically dense in HI. Roo.J Let II, and QP be the orthogonal projectors in W, l(n) onto Ll and Mp respectively, and let r4 be the orthogonal projector in W-J~(G!)onto U4. We take an arbitrary element wHYi, multiply it on the left by the matrix 8, and on the basis of Theorem 3, decompose%!w into a sum of three mutually orthogonal terms:
~w=(~w),+(~w).f(~w),,
(4.6)
where(%u),=(v,,,
uxl, ujT, v~FVwz’(Q), (9w) Y=(L _fEl, O)T, f=Wz’(Q) ,(sew),= Using the decomposition (4.6), the orthogonality of the elements (5&u) =, (5%~) y, ( %LJ) z and the minimal property of the orthogonal projector Ua, we fmd that ( z=,, z,, AZ) =, z&l.
IW--O~WI,‘9l5e-~l,~(IIV-~,VIIr~,~(P)+llf-~Pfll~~~~!o, (4.7) -+/jA(s-r,s) Since IIAull~~~~,+lli ~1zl!1:,(Q)~211ull~,z(Q), hypotheses.
Iht(Q)
+(I
1v
b--r,z)
1 lk(Q,).
the theorem now follows from (4.7) and the
Given a concrete choice of the subspaces {L,} , {MC}, {Ui) we can estimate the rate of convergence of the approximate to the exact eigenvalues. We shall confine ourselves to the case when 52 is a circle. It can be assumed without loss of generality that the center of the circle is the origin. As Li we take the set of polynomials of degree not higher than i with respect to each of the variables ~1, ~2, while as Vi we take the linear hull of the fust i eigenfunctions of the operator Ao, and we define Mi by the expression I~i={fIf=(51’+5?‘-P3)U,
OEL,}*
A new truncation procedure in the Bazley-Fox mefhod
35
where p is the radius of the circle 52. To estimate the rate of convergence, we use the corollary to Theorem 2, after first replacing the index n in it by the multi-index (Y= (I, p, 4) (see note on the corollary), and replacing the eigenvahres Am0 by their asy m ptotic form in 112.The inequality (3.30) then takes the form Y hi-?L,a’nGC7(Ai)
c
I (Ii--0,)
su s+jl i*+O
(.VZ-i-ze).
(4.8)
j-0
inequality (4.8) holds for the values of fl for which uj=D (Aoo.5+B), j=i, 2, . . . . In the case of a circle, it follows from the assumptions (4.1) that ~~jEJVz”“(Q)(see e.g., [9]). Noting this, and the results of [lo], we can assert that p=0.25-okwhere E > 0 and is arbitrarily small, so that the second term on the right of (4.8) has order o (m-'.'+'). The
Let us find estimates for the quantities 1(Ii-O,) SUj1,‘, j=l, element %S’Uj as the sum of its projections onto XBY, Z: ~suj=(9SUj)
I.+
X+(%eSUj)
2, . , . . We write the
(4.9)
z.
(%!!sUj)
Let Vj, fjy Zj be elements of W,’ (Q) , W2’ (3)
and R, respectively, realizing the projections in question, i.e.,(%%j)x=(V~,.,, ah, Vj)‘, (S!SUj)~~=(f~, -fir,, 0)‘, (SeSUj) z=(~p,, IZ~,Asj)‘. Proceeding in the same way as when obtaining (4.7), we obtain I (Z~~0~)S~jl~2~219e~il~2(llUj~~lUj~l$~~(0)
(4.10) +llfj~~~fjll~~~~~~+ll~j~~*~~ll~*~~~~~~ Estimation of the quantities II~.,-ILu,ll lrtlca). II.fj-Q)pfjIIu*,s(p, and I/sj-T+;I]~~-~~n) is a familiar problem in approximation theory and has in fact been solved. The answer given by this theory depends on the differential and integral properties of the functions Vi,fi, and Zj. Using the decomposition (4.9), the conditions (4.1) and the relation UjEWyk (‘i-2)) it is easily shown that Uj, fjy ZjEVPk f Q). We then obtain directly from the results of [ 11, 121 the estimates
IIVj-lTVjll
Wz’(Q)
=0(1--k--1)
Ilf,-Q)Pf,llw,~c~,=O(p-R-l)
,’
*
(4.11)
(4.12)
We shall initially estimate the quantity II~;--r& IITV2'(Q) by using the minimal property of the orthogonal projector r, and the equivalence of the norms IIuIIw~vQ)ad llAoull,,cP, (see PI >:
Ii+- r 41 P
2
W~z~Q~~~~zj-f:(~j~~~).~~~w,;,., 1-1
(4.13)
then we apply Theorem 2 of [13] to the term closing the chain of inequalities (4.13). The result thus obtained will depend on the value of the parameter u, for which the relation AaZjED (-40"). holds. We proved above that cj=W yk (a). In accordance with [ lo] , this ensures that A OsjED (A o”.5) for k = 1, and .40s,E D (A;O75-o.‘sjfor k> 1. For these cases, Theorem 2 of [ 131 gives respectively (4.14)
L. T. Poznyak
36
Combining the estimates (4.8), (4.10)-(4.15), ;.,-;*,“” = where a(l) =O, a(k) =0.5
we fmally get
~(~-‘A-?)+~(p--?A--3)+o((I--L--n(k~*e)+o(m-I.5+e),
(4.16)
for k> 1.
It should be mentioned that the estimate (4.16) is better than the estimate obtained for the same problem in [S] , where the Bazley-Fox method is used with a special choice of test spaces. Translated by D. E. Brown REFERENCES 1.
MIKHLIN, S. G., Numerical realizationof variationalmethods (Chislennaya realizatsiya variatsionnykh metodov), Nat&a, Moscow, 1966.
2.
GOULD, S. H., Vmiational methods for eigenvalueproblems:Ai1int&uctin intermediate problems, U of Toronto Press, 1966.
3.
POZNYAK, L. T., On the convergence of the Bazley-Fox method in the problem of the eigenvalues of one bilinear form with respect to another, Zh. v*hisL Mat. mat. Fiz., 13, No. 4, 839-853, 1973.
4.
BAZLEY, N. W., and FOX, D. W., Lower bounds to eigenvalues using operator decompositions B*B, Arch Ration Mech Analysts, 10, No. 4,352-360,1962.
5.
POZNYAK, L. T., Application of the Bazley-Fox method to two-dimensional equations.,?%. vFht&! Mat. mat. Fiz., 16, No. 1,83-101, 1976.
6.
BAZLEY, N. W., and FOX, D. W., Truncations in the method of intermediate to eigenvalues,J. Res. NBS, 65B, No. 2,105-111,196l.
I.
WEINELT, W., Uber apriore Fehlerabschatzungen der Eigenwertnaherungen Beitr. Numer. Math, No. 1,195-236, 1974.
8.
LADYZHENSKAYA, 0. A., and URAL’TSEVA, N. N., Linear and quasi-linearequationsof elliptic type (Lineinye i kvazilineinye uravneniya ellipticheskogo tipa), Nauka, Moscow, 1964.
9.
AGMON, S., Lectures on elliptic boundary value problems, Princeton, Van Nostrand Math. Studies, 1965.
10. GRISVARD, P., Caracterisation No. 1,40-63, 1967.
de quelques espaces d’interpohtion,
to the Weinstein method of
secondorder
of the form
elliptic
problems for lower bounds
beim Bazley-Fox-Verfahren,
Arch Ration. Mech. Analysis, 25,
11. IL’IN, V. P., Some inequalities in functional spaces and their application to investigation of the convergence of variational processes, Tr. Matem. in-taAkad. Nauk SSSR, 53,64-127, 1959. 12. SHAPOSHNIKOVA, T. O., Asymptotic estimates for convergence of Ritz’s method in eigenvalue problems, Izv. vuzov. Matematika, No. 6 (121),86-91, 1972. 13. POZNYAK, L. T., Evaluation of lower bounds for eigenvalues of some ordinary differential equations by the Bazley-Fox method, Zh v.%hisLMat. mat. Fiz., 14, No. 4,873-890,1974.
20
solution of operator equations of the first kind, Zh. @hid
Mat. mat.
5.
IVANOV, V. K., On the approximate Fiz., 6,‘1089-1094, 1966.
6.
MOROZOV, V. A., On a new approach to the solution of linear equations of the 1st kind with an approximate operator, Proc. of the First Conference of Young Scientists of the Computational MathematicsDept., Izd-vo Mosk. un-ta, 22-28, 1973.
7.
DAVYDOV, YU. B., Estimation of the stability of the solution of a linear operator equation of the fist kind, obtained by the approximate operator method,Matem, zap. Ural’skiiw-t, No. 4, 19-26. 1970.
8.
TIKHONOV, A. N., On the solution of ill posed problems and a method of regularization, Dokl. Akad. Nauk SSSR, 151, No. 3,501-504,1963.
9.
GONCHARSKII,
A. V., LEONOV, A. S., and YAGOLA, A. G., A generaiized discrepancy principle, Zh. v$hisl. Mat. mat. Fiz., 13, No. 2, 294-302, 1973.
10. GONCHARSKII, A. V., LEONOV, A. S., and YAGOLA, A. G., On regularization of ill posed problems with an approximately specified operator, Zh. vphid Mat. mat. Fiz.. 14, No. 4, 1022-1027, 1974. 11. KY FAN and GLICKSBERG, J., Some geometric properties of spheres ln a normed linear space, Duke Math. J., 25, No. 4,553-568,1958. 12. KLEE, V. L., Convexity of Chebyshev sets, Math. Ann., 142, No. 3.,292-304,
1961.
13. VASIN, V. V., Ill posed problems in B spaces and their approximate solution by a variational method, Diss. kand. fii.-matem. n., Sverdlovsk, In-t Matem. i Mekhan., 1970. 14. LISKOVETS, 0. A., Ill posed problems and the stability of quasi-solutions, Sibirskii matem. zh.. 10, No. 2, 273-385,1969.
A NEW TRUNCATION PROCEDURE IN THE BAZLEY-FOX
METHOD*
L. T. POZNYAK Leningrad (Received 4 May 1975) WHEN the Bazley-Fox method is used in the standard form, complicated transcendental equations have to be solved in order to obtain a lower bound for the eigenvalues of a self-adjoint positive definite operator A with a discrete spectrum. While, to avoid this difficulty, Bazley and Fox supplemented their method with several devices, it turned out that the devices do not provide good convergence in certain important classes of problem. The present paper offers a new means of simplifying the approximate equations, to which the Bazley-Fox method leads. It reduces fading lower bounds for the eigenvalues of the operator A to a problem of linear algebra, and has good velocity characteristics.
1. Introduction Suppose we are given in separable Hilbert space H with scalar product (.,.) and norm ( -1,the self-adjoint positive defmite operator (pdo) A with discrete spectrum; we pose the problem of finding the eigenvalue {hi} of A. As usual, we assume that the eigenvalues are arranged in increasing order, allowing for their multiplicity, and that the corresponding eigenelements {u,} are orthonormalized in the energy space HA of the operator A : (I&, Uj) A= 6
*Zh. vFchisl Mat. mat. Fiz., 17, I, 24-41, 1977.
A new truncation procedure in the Barley-Fox method
21
Assume that there is a self-adjoint pdo An in H with known eigenvalues {3.io} and eigenelements {uio},which is semi-similar to the operator A (i.e., the energy spaces HA and HA o consist of the same elements, see [l] ), and is connected with it by the relation (u, v),=(u,
\u,
v)_&-+ (Su, SP),
r=IZ,.
(1-l)
where S is an operator from Hinto a Hilbert space HI with scalar product (.J1 and norm ( l II . Our problem can then be solved by the Bazley-Fox method (see e.g., [2]), which approximates the required eigenvalues from below. For this, we have to choose in H, a sequence of finite-dimensional subspaces {w,} ,asymptotically dense in HI,and after arbitrarily ftig n, we have to evaluate the eigenvalues {ii”) of the pdo A,, generated in H by the closed bilinear form (u, v)aO+(OJu, Su),, where 0, is the orthogonal projector in H, onto W, . Notice that, to evaluate the {hi”} we do not need a knowledge of the explicit form of the operator A, ; the eigenvalues{k”}are fully defined by the above bilinear form, with the aid of which the eigenvalue problem for the operator A, can be written as the identity
(u, v)A,+(o+su, Sv)I=h(u, v),
O+u=H,,,
(l-2)
~vEH,,.
On varying n, we obtain for each eigenvalue Xi a sequence hi”, n=l, 2. . . . , convergent from below to Xi. The rate of this convergence is estimated in [3] . It was shown there #at, for sufficiently large values of n, (l-3) where I, is the identity operator in HI, PC -I1 is the multiplicity of the eigenvalue Xi, s is the least number for which X, = Xi, and C, is a positive constant, independent of n. If we restrict ourselves to the assumptions made above, it becomes necessary to solve complicated transcendental equations in order to determine the eigenvalues hi”, i=l, 2,. . . , The computations can be simplified by imposing extra conditions on the subspaces W, . The condition was originally pointed out by Bazley and Fox [4], and can be stated as whatever the such that
n=l,
SU,~J-.W*,
2,. . . ) for
there exists a number G-N(~).
N(n), (1.4)
In this case, the eigenvalues of the problem (1.2) can be found by solving an algebraic eigenvalue problem. Experience shows that condition (1.4) is extremely rigid, and not many types of problem have as yet been found in which it is satisfied. Moreover, our studies of convergence for some of these problems (see e.g., [5]) h ave revealed that the rate of convergence of hi” to hi may be extremely slow. In addition to the method considered, involving a special choice of test spaces, Bazley and Fox [4,6] found another means for overcoming the difficulties that arise when solving the intermediate problem (1.2). Their new idea was to solve problem (1.2) itself approximately, while still retaining the main aim of obtaining lower bounds for the eigenvalues hi- While realization of this idea demands certain restrictions, these are much less rigid than condition (1.4). The restrictions are as follows:
22
L T. Poznyak
1) D(P) =LHi, 2) W,dl(S'),iz=l, 2,...,where S* is the operator adjoint to S, and the bar denotes the closure operation. Restriction 1) follows from restriction 2) and the condition made at the start, that the sequence {IV,} be asymptotically dense in H, . The need for condition 2) is clear from the type of problem which Bazley and Fox proposed to solve instead of (1.2):
+ (S’O,Su, u) =
Here, utO=(h?)“Q~, H.
i=l,
a(u,
u) (
OZud,
VUEH.
2, . . . , are the eigenelements of the operator A 0, orthonormalized in
Problem (1 S) is obtained from problem (1.2) by replacing the bilinear form (u. Y)~u by the bilinear form, bounded in H, m
c
,
(ajO-hrE+,) (u, u,O)(Q”,u) fd+1
(0,&a, Sv)i
(S'OJu,u),bounded
the
u .
Ha symmetric bounded pdo Au(“):
A;“)
u
(1.6)
H. The second
To the form (1.6) there corresponds in
.. (hj"-hi+i)
=
(u,
u_i0+h~+*u7
j-t
which is called the m-th order truncation of the operator Au. This name originates from the method of obtaining the operator A0(m): in the spectral decomposition
Aou =
E aI0(4 9”) uj
of the operator A, in the space H, the eigenvalues It:+,, hom+Z,. . . have to be replaced by the same eigenvalue X0,+, . The form (1.6), which can now be written as (Afj”‘)~, V) , is called the m-th order truncation of the form (u, Y)Ao, while the method of replacing problem (I .2) by problem (1.5) is known as the truncation method. The condition 2) enables us to write problem (1 S) in the operator form in the space H:
A~m%z+S'O,Su=~u. It can easily be seen, by analyzing the structure of the operator of its eigenvalues amounts to a problem of linear algebra.
Ad""+S*O,S, that determination
While application of the truncation method rarely presents serious difficulties, it does not
A new truncation procedure in the Bazley-Fox method
23
always give good results. Of course the poor results stem from the slow convergence of the method; but it is hard to say what causes this slow convergence, since we know so little as yet about the rate of convergence of the eigenvalues h,anrnof the problem (1 S) to the exact eigenvalues Xi. In [6], Bazley and Fox obtained the estimate 3Lin-)L{“mGC* (n) (c+,)-ie
(1.7)
They made the assumption that A = Au t B, where B is a symmetric positive operator, and they took relation (1.1) in the form (u, v)a=(u,
C)&+(u,
U)B,
so that H, = HB, while S is equal to the identity operator considered as an operator from H into HB. Bazley and Fox did not investigate the dependence of the constant C,(n) on n. Weinelt attempted to explain it in [7]. Making the additional assumption that IV, is the same as the linear hull of the elements ulo, . . . , u,‘, and that the operator B is positive definite and comparable in force with the operator Aob, OGpG1, i.e.,
Weinelt obtained for C,(n) the estimate C,(n)Gc‘(h,0)2P
(1.8)
with a constant C, which is independent of n. This is all that is known about the convergence of Xinm to Ai” for fured n. An idea of the convergence of Xinm to Xi can be gamed by combining (1.7), (1.8) with (1.3). For instance, if B is bounded @ = 0), it follows from (1.3), (1.7), and (1.8) that the double sequence hinm, n, m=l, 2,. . . , is convergent to Xi, at a rate not less than C,(GI+,)-‘+C,
2 I (E-On)&+jlB2, j-0
where E is the identity operator in H. Unfortunately, this case is not typical in practice. And in the caSe of an unbounded operator B, the expressions in question give no satisfactory estimate of the rate of convergence of xinrn to Xi. If the asymptotic behaviour n”, u > 0, of the eigenvalues {LO}, is knmvn, then (1.3), (1.7), and (1.8) can be used to extract from the double sequence h,““, n, m=l , -I***, 7 the ordinary sequences IL?>“‘(~), n=l, 2,. . . ? and to estimate their rate of convergence to Ai; the estimates thereby obtained have a low order. Recall that everything just said about the convergence of Arm to Xi only holds under the above special assumptions about the operators A, Au and the test subspaces W,. In the general situation considered at the start of the section, nothing is known about the convergence of hnm to x, This present state of affairs in the Bazley-Fox method compels us to look for new ways of realizing the basic idea of the method, concerning approximate solution of the intermediate problem (1.2). A new way of simplifying problem (1.2) is described in the present paper. It differs from the method of truncations mainly in the fact that we replace the bilinear form (u, v) in the identity
24
L. T. Poznyak
(1.2) by a larger bilinear form, whereas Bazley and Fox replaced the form (u, v)~ o in it by a smaller bilinear form. The problem that then arises again has the “intermediate” property: its eigenvalues give lower bounds for the eigenvalues Xi. We shall show that the determination of the eigenvalues of the new problem reduces to a problem of linear algebra. Finally, the feature of principal importance is that, under the general assumptions made below, an estimate can be obtained for the rate of convergence of the new approximate eigenvalues to the exact eigenvalues. The efficiency of the estimate is illustrated by an example of a Neumann problem for a two-dimensional secondorder elliptic equation.
2. A new truncation procedure in the Bazley-Fox method Turning to a detailed treatment of the new approximate method for solving problem (1.2), we fust observe that it can be used under the same general assumptions as the Bazley-Fox method itself. The new method amounts to replacing the bilinear form (u, V)in the identity (1.2) by the symmetric bilinear form
cm[
(Ajo)-‘-(~~+l)-‘l(u, qO)
A0
cu;,v>Ao+(hi+i)-l (u, v)
Ao.
(2.1)
j-1
The approximate problem which we propose to solve instead of problem (1.2) then has the form
(4 ~)A,f(~,Jh Sv)*=h{ (hnf+*)-‘(u, v),, (2.2)
It is easily shown that the form (u, u) is less than the form (2.1):
(2.3) :-m+1
ki
The form (2.1) is obviously positive definite in HA ,,, and corresponds in this space to the symmetric bounded pdo (A,-‘) (m):
c[ m
(/lo-‘)
(m)g =
(hjo)-I-
G+d -'I (l-5,
Uj')
&Uj”+
(hri+i)-‘U*
(2.4)
A new truncation procedure in the Bazley-Fox method
25
We aim to emphasize, by the notation (A,- ’ >(m) that this operator is formally obtained from the operator A0 - 1 by means of the same procedure as the operator Ao(“‘l is obtained from the operator A,. In fact, for Au-’ we write the spectral representation A,-5.1
=
c
(5’) --! (72,Uj')
A&j0
j-1
in the space HA O,and then we replace all the eigenvalues of Au- ’ with numbers m t 1, m t 2, . . . , (m) in this representation by the (m + l>th eigenvalue (AR+r) -‘. Hence it is natural to call (A,-') the m-th order truncation of the operator Au- ‘, and calI the present method the method of inverse operator truncations. Let us return to problem (2.2). We shall show that it has eigenvalues representing lower bounds of the eigenvalues of the problem (1.2), and we shall give the method of calculating them. Considering S as an operator from from
HA Ointo
HI, we introduce the adjoint operator s’, acting
HI into HA*: (Su,W)i=(U,S'W)Ao VUEH,,, vw=H,.
It was shown in [3] that the operator s’ is bounded. Using this operator, we write (2.2) as an operator problem in HA0:
(z+S'O,S)u=3L(Ao-')'"'u, O+U~H&,
(2.5)
where I is the identity operator in HAO.We can write the last equation out more fully as d(n) (U, S’W~)&tlijSrWj=h’ (d+i)-‘U usc i,j-i
(2.6) +
c
1=j
[
(ijo>-‘-(Cn+i)-‘I (UTUj”)AoUjO 1t
where LL’~. . . . , Ed(n) iS the basis in w,, matrix
(aij),
D=( (wi? Wj)i),
i, i=i,
2,. . . , a(rz),is the
i, j=1,2,.
. . , d(n).
inVerSe
matrix to the
(2.7)
Symmetric pdo’s in HAM appear in both sides of Eq. (2.5). It is clear from (2.6) that the subspace V, stretched over ulo, . . . , IL,‘, S’wi, . . . , s’wd( ,,), reduces theSt? operators. On solving problem (2.5) in the subspace, we obtain a finite number of eignvalues of finite multiplicity pi”“< . . . G G& ’ where r(n) denotes the dimensional@ of the subspace I’. In the orthogonal complement to V, Eq. (2.6) transforms into the equation u-h (AZ+*) -?.z, so that its spectrum in this complement consists of the unique eigenvalue A, + 1O, whose corresponding eigenelement is the e:exhaust the spectrum of the entire subspace HA06 V. The eigenvalues problem (2.5). We arrange the eigenvalues of the problem (2.5), not exceeding Xo,+1, in increasing order while allowing for their multiplicities: 3L~“m~h~“m~ . . . . Clearly, starting from some number i, not exceeding r(n) + 1, this sequence becomes “stationary”: 3Ljnm=AR+1, Pi. In the case
L. T. Poznyak
26
wfien A,+1o is the least eigenvalue of problem (2.5), the entire
sequence hjnm, j-1,
2,. . . , is
2, . . . .
stationary: h,“m==h~+~,j-1,
On now applying to problem (1.2), (2.2) the familiar comparison theorems, and recahing (2.3), we obtain the estimates of interest: i=l,
~i”“Ghi”Ghi,
2,. . . .
(2.8)
Since Az+1-+m as m-+00, and the right-hand sides of the inequalities (2.8) are independent of m, the number A,,,+1o cannot be the least eigenvalue of the problem (2.2) for sufficiently large m; for such values of m, there must necessarily be eigenvalues of finite multiplicity of problem (2.2) to the left of A,+1 o , the number of which increases without limit as m increases.
In short, we have established that the inverse operator truncation method reduces the determination of lower bounds for the eigenvalues {I.,} to the solution of problem (2.6) in finite-dimensional space V. In turn, this problem is equivalent to the matrix problem Ax=ABx
(2.9)
with symmetric positive matrices A and B of order r(n). The matrices A, B depend on the choice of basis in V. We can assume without loss of generality that uio, . . . , U,‘, SWi, . . . , S’Wen)-m form a basis in V. With this basis,A and B have the block forms E + CD-X / K’ + C’D-ll? -----------K + F’D-‘C 1 M + F’D-‘F
B=
(2.10)
AoK’
-__--------i--
(2.11)
KLoK’ + h0,+,M where
E=(dij), c=
i, j=l,
&=(
2,. . . , m, i=l,
( (S’wt, u4> &)I
2,. . . ,d(n),
K= ( (S’WI, uj”) ao) t
i-l,2
E= ( (S’wi, S’wj) 4,) 7
i-l,
M= ( (S’wiy S’Wj) A) 7
i, j=1,2,.
Lo= ( [ (Ato) --I- od+i) -‘l&d 9
Expressions (2.7), (2.9)-(2.12) operator truncations.
,...,
i, j=1,
(hf")-'6*j),
j=1,2
r(n)-n2,
2,. . .,d(n),
,...,
2,. . . , m, m,
j=l,2,...,m; j=1,
(2.12)
2,. . . ,r(n)-m,
. . ,r(n)-m,
i,j=l,2
,...,
m.
represent the computational formulae for the method of inverse
A new truncation procedure
in the Bazley-Fox
method
27
3. Convergence and convergence rate estimate We shah start our study of the convergence of the approximate eigenvalues hinm, i= 1, 2,. . . , by seeing how they depend on the index m. From the definition of the truncation (Au- l)(m) it is clear that it decreases monotonically with respect to m: (A,-‘) (m)> (Ao-‘) cm+i)aAo-i, m= 1, 2,... . In view of this, for futed n every eigenvalue hinm is monotonically increasing with respect to m: hi”“
m-l,
,
2,. . . .
Let us show that the sequence hiRm, m=l. 2? . . . , converges to Ain. For this, we reduce each of problems (1.2) and (2.2) to an eigenvalue problem for a symmetric bounded operator in HA ,,. We have already taken a step towards this reduction in the case of the problem (2.2), by replacing the identity (2.2) by the equation (2.5). The same step, for the problem (1.2), leads to the following equation in HA o:
(I+S’OJ)
u=l.A,-‘u.
(3.1)
We introduce the notation F,=I+S’O,S. The properties of the operators F, were examined in detail in [3] and they will be used below without reference. If we make the replacement v=F’,o u, in (2.5) and (3.1), it becomes obvious that (Ai”)-‘, i= 1, 2, . . . , are the eigenvalues of the asymmetric completely continuous operator F,?A,-‘Fi’” in f&o, while (hi”“) -I, i=l, 2, are the eigenvalues of the symmetric bounded operator F,” (Ao-‘) cm)F,‘h in the same . *-, space. By a well-known theorem * (see e.g., [l] , p. 258), hymn)
--!_
(hi”)
IF,li’
-‘<
( (&,-l)
(m)-A,-i)F,‘”
1~0.
(3.2)
It is easily shown that 1 (&-‘)
(m)-Ao-il,,o=
(h:+i) -‘.
(3.3)
On further recalling that 1F,-“’ 1Ao< 1, we obtain from (3.2) and (3.3): (Qy
-‘-
(L+l)-‘3
(hi")-'<
or alternatively,
(h?t+i) -ly
,j,in-hinmG?b*nhinm
which shows that Xinm are convergent
to Xin
(3.4)
as m + 00.
On coarsening the inequality (3.4), we obtain an estimate for the rate of convergence of hinm
t0 Ai”
’ h+hr”m~A,z
*The theorem was proved in [l] only one of which is completely
for completely continuous continuous.
(?hn+,)-i,
operators,
but it remains
(3.5)
true for the present
operators,
L. T. Poznyak
28
in which, as distinct from the case of Bazley and Fox’s estimate (1.7), the constant on the right-hand side is independent of n. We can also obtain from (3.4) effective, practically computable estimates for the error introduced by the operation of truncation of the operator Au - l. In fact, an upper bound is easily obtained for the eigenvalue Xi by Ritz’s method; this bound usually holds before application of the Bazley-Fox method (the latter is in fact used to estimate the error of the Ritz method). Denoting by & an upper bound for hi, computed by the Ritz method, we find from (3.4) that ~~n-.liRm~h**m31~(X:+1) -*)
(3.6)
3Lin-hinmGXfz(~+~)-1. The estimate (3.6) is a posteriori
(3.7)
while the estimate (3.7) is a priori
We now turn to a study of the behaviour of the approximate eigenvalue hinm as a function of the two indices n and m. To be more precise, we shah henceforth regard Xi”“’ as a double sequence and examine its convergence regardless of how n and m tend to 00.We shall give the same interpretation to the convergence of other objects (elements, operators) encountered below, dependent on the pair n, m. The convergence of hinm to Xi is proved in an elementary way on the basis of the inequalities (1.3) and (3.5):
At the same time, (3.8) gives an estimate for the rate of convergence of Xinm to hi. This is not a limiting estimate, however. A better estimate, of higher order with respect to m, can be obtained by comparing problem (2.2) directly with the initial problem OklEIZ&
(u, U)A”f(SU. Su),=h(u, v),
VVEHA,
(3.9)
in accordance with the same scheme as was used in [3] for and estimating the error k-hi”” estimating the error of the Bazley-Fox method without truncation. Let us briefly run over the scheme. We first have to reduce the initial problem (3.9) to the equivalent operator problem in the space H,,: (Z+S’S) ~=hA~-‘zz, then the latter, and the approximate equation (2.5), have to be transformed respectively to u=hF-‘Ao-‘1~. n=hF,-‘(A,>-‘) where F=Z+S’S. It is easily seen that F-lA,,-i=A-‘. notation Tnm for the operator F,-’ (ilo-‘) Irn).
(3.10) (m’zz,
(3.11)
For brevity, we introduce the simpler
When estimating the error h~-h,“‘~ an important role is played by the difference R,,,,=T,,,,,-A-!:
(3.12)
and in particular, by its two obvious representations R,,=F-‘S’(Zi-O,)ST,,+F-‘[
(A,q-‘)‘m)-Ao-‘],
(3.13)
A new truncation procedwe in rhe Baziey-Fox merhod
R,,=F,-‘S’(Z,-O,,)SA-lf-F,-‘i
29
(Ao-l)~*~-_40-i],
(3.14)
and the property I%,
as
].4,-0
(3.15)
rl. m-w,
which is proved in the same way as in [3]. The operator R,, characterizes the proximity of the exact problem (3.10) to the approximate problem (3.11); naturally, the error ;*s-)bi”“’ also depends on it. The nature of the latter dependence is proved in the same way as in [3] . In fact, we initially obtain, with the aid of (3.10) and (3.11): (I-&A-l)
uinm= (hi”“) -l (i,,““:-_3,,) ui”x+_thiR,,u,“m,
(3.16)
where uinm IS . the eigenelement of the problem (3.1) corresponding to the eigenvalue Xinm. Then, multiplying (3.16) scalarly in HA by the element Piuinm, we find O= (3uanrr2) -’ (hznm-l~i) ~~~~~~~ Prui71m) A+2vi(R,,uinm, PiuiRrn)_k_
(3.17)
where Pi is the orthogonal projector in HA onto the subspace stretched over uI, u,+~, . . . , usi,. Finally, noting that ( Uinm, Piurnm) _4= 1PJL,“~ 1Az, and for brevity, putting y= 1P+Uinm1a-‘Uinrn! we obtain from (3.17) the required expression 3.~,-~~,nm=3iii,,nm (Rmy, P,y) -4.
(3.18)
Notice that, when obtaining (3.18), we have tacitly assumed that I Piuinm IA>O. We justify this assumption below, for sufficiently large n and m. The next “block” in the scheme of arguments in [3] is to isolate the “unimportant” terms in the scalar product (R,, y, P,y >A. Replacing R,, in accordance with (3.13) and using relation (1 .I), we can write (R,, y, Piy), as the sum (Rnmy, Piy) ,=(O’“‘STnmP
I a,2+ (O’“‘ST,mP”‘y,
0’“‘SPy)
1-k 1 ( (Ao-‘) (m) (3.19)
0’“‘SPiy) *+ ( ( (AO-‘) (m’
-A,-‘) PCi’y, P,y) Ao, PCil=I-Pi.
We obtain the expression for T,,,, from relations (3.12) and (3.14): T,,,,=A-‘+F,,-‘S’O’n’SA-‘+E’,,-‘(
(A,-‘) ‘“;.-Ao-‘),
and we substitute this expression into the first term on the right-hand side of Eq. (3.19). After obvious transformations, we obtain (Rnmy, Piy) A=&-’ IO’“‘SPiy Ii2+ I ( (AG-‘) (m’-Ao-l) ‘“Piy IA02 +I”,-: IF,‘“S’O’“‘SP,y + (O’“‘ST,,P”‘y,
IA?-+ (SF,-’
( (A,-‘! cm’-Ao-l) Piy, 0’“‘SPiy)
0’“‘SPiy) ,+ ( ( (Ao-‘) ‘“‘-A,-1)
i
(3.20)
P”‘y, P*y),,.
This is in fact the required expansion of the scalar product (R,,, y, Piy) A into essential and inessential terms. Let us show that the last three terms in (3.20) are inessential. We shall show, in fact, that each of them has higher order than 8,,,,= I O(“‘SPiy / 1’+ 1( (Ao-‘) (m’-Ao-‘)“P,y 1Ar,2. For the first term, the proof is easy:
L. T. Poznyak
30
I (SF,-’ ( (A,-‘) (n’)-Ao-l) P,y, O’“‘SP,y) i I
~ISlOil (Ao-l)(m’-A,-‘I~oI( (A,-‘)‘“‘--Ao-l)‘~~PiylAoIO(~‘SPiyl~ GISl,,l
(3.21)
(Ao-l)(m’-Ao-lI~~e,,,
where I -lo 1 denotes the norm of the bounded operator acting from H,Q into HI. For the other
two terms, the required bound cannot be obtained directly, since they contain the expression P($, the connection of which with the quantities ( Q’“‘SPjy 11 and 1 ( (A,-‘) (nd’-AO-‘) ‘“P,y 1Ao is as yet unknown. The connection is established by: Lemma The pair n(i), m(i) exists such that, for n > n(i), m > m(i), we have 1PiUintn
1 *>O, (3.22) 1*<(Js ( IO'"'SPiy
Ip(i’y
1I+ 1( (~40~‘)'mi-AO-')'hPiy
1Ao)t
where the constant C, is independent ofn, m, and i. The proof is the same as the proof of Lemma 8 in [3]. We can now easily obtain the required estimates: I ( ((A-‘)
(m’-A,-‘)
P”‘Y, Pi, y)
acI<2CSI (&lo-‘) (m’-Ao-t I :&I,,,
( (09.w,,P(~)y,o(wP,y)
l
1ecs 1oYw,,
(3.23)
loienm.
(3.24)
Notice that the convergence to zero of the quantity ( Ocn)ST,, I o, follows from (3.12), (3.15), and Lemma 1 of [3]. In short, when estimating the rate of convergence Of hinm to Xi we can neglect the last three terms in (3.20). The third term on the right of (3.20) also has no influence on the order of smallness of the error L-k,“” since
A more exact result, which follows from (3.18), (3.20), (3.21), and (3.23)-(3.25), stated as follows.
can be
Theorem 1 If the sequence of subspaces {IV,} is asymptotically dense in HI, then, given any i=l,2,.. .,apairn’(i)> n(i), m’(i)>m(i), canbefoundsuchthat,forn>rz’(i), m>m’(i) we have the two-sided estimate o.5ki
1O(n)Sp,y
<&_hi”“<2CJ,
1 ,2-/-(). jhi2 1 (
(A,-‘)
(n”-~4-1>“‘P$
1AU’
1O’“‘SP{Y l ,2f2h,2 l ( (A,-L)(rn’-&-‘)‘:?p4 I AZ.
(3.26)
where C,=l+13’1,,‘. If we use the same method as in [3] , and expand Pfl in eigenelements u,, . . . , zzScxr corresponding to the eigenvalue Xi, we easily obtain from (3.26) an estimate connecting the error Ac-&“~~ with the errors of approximation of the elements SuJ, . . . , SU~+~.
A new mtncation procedure in fhe Bazley-Fox method
31
Theorem 2 Let the conditions of Theorem 1 hold. Then, for m>m’( i), n>n’( i) (3.27) ( IO(n)SU*+jli2+ I ( (A,-‘) cm’-Ao-‘) ‘ZZ,+jI*.2)7 ?bi-?Linrn
cm)-A,-l)“:=(
(A,-‘) ‘“‘-A,-‘)‘“(&Q,).
In the light of this equation and the estimate (3.3), we have 1((A,-‘)
(A:+,)-‘“I (I-Qm)%lAo.
(m)-Ao-‘)~ujI~,<
If it is assumed that UjED (A0°.5+B), i=l, we can obtain the better estimate
(3.28)
2, . . . , for some 0 > 0, then instead of (3.28)
I _40~(A:+,) -“-‘I (E-Qm) AOp”+’ UjIa I ( (A,-‘) ‘n”-AOwi) “2~~
(3.29)
For, under the condition indicated, 1(z-Q~)B~I~~=IA~-~(E-Q~)A~~~+~IL~I~IA~-~(~-Q~) I
which, in conjunction with (3.28), gives (3.29). We have thus proved:
If the conditions of Theorem 1 hold, r&n’(i), for some /32 0, then
m>m’( i) and l!jED (Ao”~‘+‘), i=l.
h,-h:““‘GCT (Xi) 2 10(n)SU~+jlI’+0 ( (lb,‘) -‘-2b). I=”
2, . . . ,
(3.30)
Note. In the next section we shah consider an example in which the test subspaces form a generalized and not an ordinary sequence {W,}. The extension of the earlier results to this case is trivial: we simply have to replace the index n in all the expressions by the symbolo.
32
L. T. Poznyak
4. Application
the Neumann
The main practical difficulties that arise when using the new truncation operation are connected with the operators’. We previously had to deal with this operator in [3,5], though there it played an auxiliary role. Now, the operator s’ occurs in the computational expressions (2.12). It is only rarely that s’ can be found in explicit, closed form. It is fortunately not essential to be able to do this. The practical problem concerned with s’ may be stated alternatively as: to select the test subspaces W, (or IVat)in such a way that the operator s’ can be simply evaluated on the elements of W, (War).While this last problem is again difficult, the important example given below shows that a solution is possible. Let fi be a circle or rectangle, and aa the boundary of Q, ~=QUaQ. functions.a(s,, z2), aij(Zt, ZZ), i, j=1, 2, satisfying the conditions: k>l,
fZEP(a),
Uijd”+i(52)T
Uij=CZji,
Given in 3 the
(4.1)
ah, 4 >I.
T=COnSt>l, i,j-1
In the space H= L2(s2) we defme the self-adjoint pdo A by the relations
addrv=en~=o
D(A)={u]u.=W,~(Q),
on
anj,
Au=-div PVufau, where 9” (a+<), i, j-1, 2; Yis the unit vector (written in the column form, as are all vectors) of the inward normal to X2. Hence A is the operator of the Neumann problem for a second-order two-dimensional elliptic equation; and its properties are well known. We specify the operator A0 by the expressions D(A,)={~~uEW~~(Q),
au/&=0
Aeu=-4u+u.
on&?},
We have N_4=W2’ (Q),
(u, II),
= JJ (9Vu
Vv+auu)dB,
P H ib=wz*
(S-J),
(u, v> Ao=
(Vu VVSUV)dSZ, JJ 0
so that (u,v)za=(u,u).ko+
(4.2)
JJ[(sa)vuvu+(a-I)WI~R, 0
where8=(6il),
i,i=l,2. Ifweput H,=Lz(~)XL?(Q)XL2(~), where T denotes transposition, u,,==acz/axi, (u*,, uz*, u)T, ~ =
(9-8)‘h
II
0
0 (a-l)‘”
D(S) =Wz’(Q),
II ’
then Eq. (4.2) can be written in the form (I .l) and all the conditions for application of the Barley-Fox method to the problem Au = )uc are satisfied.
su=sc
A new truncation procedure in the Bazley-Fox method
33
In order to find the test subspaces in which we are able to compute the values of the operator s’, we have to analyze in more detail the structure of the space H, . Theorem 3 The
can be written as the orthogonal sum of the three
space H,=L2(Q)XL2(Q)XL2(S2)
subspaces H,=X@Y@Z, where x={w~U’=(Lz,,,
ux2,U)T, u=IVw,l(fi)},
Y={w~w=(u,2.
-t&O)‘,
Z={r.+=@,,,
us,, Au)~, u~W,~ (S2), du/~?v=O on &2}.
U~IT,~(Q)},
Proof. It is easily shown that X, Y are subspaces. Their orthogonality follows from the well-known theorem on the decomposition of the space Lf (52) XL,(Q) into an orthogonal sum ofsubspacesG={gIg= (u%,,u,)*, u~IJV~~(Q)} and i={gjg=(zz,, -uxJT, ZZEW~~(Q)}. For, given any pair ‘p=X, $EY we have (p=(u.,, (9,
cl
i =
r&c,,U)r, JJ
uf=W*‘(Q),
( wx,-wx,)
$= (v-9 --vx,, 0) 3
VEW,’
(s-2)
)
dQ,
G
and since (u.?, uxl) ‘=G, ( r;,, -vZJT~l,
we have (cp, $) 1~0.
It remains to show that (X@ Y) l=Z. We take an arbitrary element w= ( w1 (z,, zZ) , ~22 (x,,L_), IL’~(z~,z~))~E(X@Y)~. Wehave
ss ss
(u,,w~+u,,w~+uw~) dQ=0
vu= Wz’ (s-2) )
(4.3)
P
(v,,w~-vI,w2)
vvdk2’(8).
cm=0
P
By the theorem just mentioned, on decomposition of the space L, (Q) XL, (Q) ,it follows from (4.4) that w,=z=,, w2=zx,, where ZEWZ’ (i-2). On using this fact in the identity (4.3), we get u,,Z,,+U&,+Uw~) as-J=0 rr.Z=w, (Q) . SS( This last identity implies &at 2(x,, x2) is the generalized solution of the Neumann problem 4z=u,3
in S-2,
dz/&=O
on CP2.
(4.5)
Since 52 is a circle or rectangle, Eqs. (4.5) will hold almost everywhere, in 52 and on 22 respectively (see [8] ). The theorem is proved. The set {U/ UEW,~ (S-2)) c%/&=O on d0) brevity we shall henceforth denote it by %.
is a subspace of the space WZ2(R). For
Let us turn to the construction of the test subspaces. They will now depend on the three integer indices Z,p, 4. The set B of all triples a= (2, p, q) will be assumed to be partially ordered
34
L, T. Poznyak
in the natural way: o.Ccc’, if ZGl’, p
-fm
o>=, fafp,
zg=(z,,, z+,, AZ)‘,
ZdY,.
Let us find s’w for ZUEW,. We have
i = JJ (VuVL+UU)ds2 P + J J (ux,j.rz-ux~,) cm+ J J (vuvz+uAz) dW Vu=D (8) , (Su, w) I= (9-‘Su,
rp+l#+Q
l-4
Q
and since the last two integrals vanish in accordance with Theorem 3, we have (Sa, w) I= (zz, v) + But this implies that 5”~ = v. In short, the difficulty arising in applying the inverse operator truncation method can be overcome in practice in the present example. In order for the approximate to converge to the exact eigenvalues, the generalized sequence (IV%} has to be asymptotically dense in HI. Sufficient conditions for this are given by: Theorem 4
If the sequences {L,) , {Ni}, {Cr,} are asymptotically dense in FV2’(Q) , W,’ (‘2) and R, respectively, then the generalized sequence {w,} is asymptotically dense in HI. Roo.J Let II, and QP be the orthogonal projectors in W, l(n) onto Ll and Mp respectively, and let r4 be the orthogonal projector in W-J~(G!)onto U4. We take an arbitrary element wHYi, multiply it on the left by the matrix 8, and on the basis of Theorem 3, decompose%!w into a sum of three mutually orthogonal terms:
~w=(~w),+(~w).f(~w),,
(4.6)
where(%u),=(v,,,
uxl, ujT, v~FVwz’(Q), (9w) Y=(L _fEl, O)T, f=Wz’(Q) ,(sew),= Using the decomposition (4.6), the orthogonality of the elements (5&u) =, (5%~) y, ( %LJ) z and the minimal property of the orthogonal projector Ua, we fmd that ( z=,, z,, AZ) =, z&l.
IW--O~WI,‘9l5e-~l,~(IIV-~,VIIr~,~(P)+llf-~Pfll~~~~!o, (4.7) -+/jA(s-r,s) Since IIAull~~~~,+lli ~1zl!1:,(Q)~211ull~,z(Q), hypotheses.
Iht(Q)
+(I
1v
b--r,z)
1 lk(Q,).
the theorem now follows from (4.7) and the
Given a concrete choice of the subspaces {L,} , {MC}, {Ui) we can estimate the rate of convergence of the approximate to the exact eigenvalues. We shall confine ourselves to the case when 52 is a circle. It can be assumed without loss of generality that the center of the circle is the origin. As Li we take the set of polynomials of degree not higher than i with respect to each of the variables ~1, ~2, while as Vi we take the linear hull of the fust i eigenfunctions of the operator Ao, and we define Mi by the expression I~i={fIf=(51’+5?‘-P3)U,
OEL,}*
A new truncation procedure in the Bazley-Fox mefhod
35
where p is the radius of the circle 52. To estimate the rate of convergence, we use the corollary to Theorem 2, after first replacing the index n in it by the multi-index (Y= (I, p, 4) (see note on the corollary), and replacing the eigenvahres Am0 by their asy m ptotic form in 112.The inequality (3.30) then takes the form Y hi-?L,a’nGC7(Ai)
c
I (Ii--0,)
su s+jl i*+O
(.VZ-i-ze).
(4.8)
j-0
inequality (4.8) holds for the values of fl for which uj=D (Aoo.5+B), j=i, 2, . . . . In the case of a circle, it follows from the assumptions (4.1) that ~~jEJVz”“(Q)(see e.g., [9]). Noting this, and the results of [lo], we can assert that p=0.25-okwhere E > 0 and is arbitrarily small, so that the second term on the right of (4.8) has order o (m-'.'+'). The
Let us find estimates for the quantities 1(Ii-O,) SUj1,‘, j=l, element %S’Uj as the sum of its projections onto XBY, Z: ~suj=(9SUj)
I.+
X+(%eSUj)
2, . , . . We write the
(4.9)
z.
(%!!sUj)
Let Vj, fjy Zj be elements of W,’ (Q) , W2’ (3)
and R, respectively, realizing the projections in question, i.e.,(%%j)x=(V~,.,, ah, Vj)‘, (S!SUj)~~=(f~, -fir,, 0)‘, (SeSUj) z=(~p,, IZ~,Asj)‘. Proceeding in the same way as when obtaining (4.7), we obtain I (Z~~0~)S~jl~2~219e~il~2(llUj~~lUj~l$~~(0)
(4.10) +llfj~~~fjll~~~~~~+ll~j~~*~~ll~*~~~~~~ Estimation of the quantities II~.,-ILu,ll lrtlca). II.fj-Q)pfjIIu*,s(p, and I/sj-T+;I]~~-~~n) is a familiar problem in approximation theory and has in fact been solved. The answer given by this theory depends on the differential and integral properties of the functions Vi,fi, and Zj. Using the decomposition (4.9), the conditions (4.1) and the relation UjEWyk (‘i-2)) it is easily shown that Uj, fjy ZjEVPk f Q). We then obtain directly from the results of [ 11, 121 the estimates
IIVj-lTVjll
Wz’(Q)
=0(1--k--1)
Ilf,-Q)Pf,llw,~c~,=O(p-R-l)
,’
*
(4.11)
(4.12)
We shall initially estimate the quantity II~;--r& IITV2'(Q) by using the minimal property of the orthogonal projector r, and the equivalence of the norms IIuIIw~vQ)ad llAoull,,cP, (see PI >:
Ii+- r 41 P
2
W~z~Q~~~~zj-f:(~j~~~).~~~w,;,., 1-1
(4.13)
then we apply Theorem 2 of [13] to the term closing the chain of inequalities (4.13). The result thus obtained will depend on the value of the parameter u, for which the relation AaZjED (-40"). holds. We proved above that cj=W yk (a). In accordance with [ lo] , this ensures that A OsjED (A o”.5) for k = 1, and .40s,E D (A;O75-o.‘sjfor k> 1. For these cases, Theorem 2 of [ 131 gives respectively (4.14)
L. T. Poznyak
36
Combining the estimates (4.8), (4.10)-(4.15), ;.,-;*,“” = where a(l) =O, a(k) =0.5
we fmally get
~(~-‘A-?)+~(p--?A--3)+o((I--L--n(k~*e)+o(m-I.5+e),
(4.16)
for k> 1.
It should be mentioned that the estimate (4.16) is better than the estimate obtained for the same problem in [S] , where the Bazley-Fox method is used with a special choice of test spaces. Translated by D. E. Brown REFERENCES 1.
MIKHLIN, S. G., Numerical realizationof variationalmethods (Chislennaya realizatsiya variatsionnykh metodov), Nat&a, Moscow, 1966.
2.
GOULD, S. H., Vmiational methods for eigenvalueproblems:Ai1int&uctin intermediate problems, U of Toronto Press, 1966.
3.
POZNYAK, L. T., On the convergence of the Bazley-Fox method in the problem of the eigenvalues of one bilinear form with respect to another, Zh. v*hisL Mat. mat. Fiz., 13, No. 4, 839-853, 1973.
4.
BAZLEY, N. W., and FOX, D. W., Lower bounds to eigenvalues using operator decompositions B*B, Arch Ration Mech Analysts, 10, No. 4,352-360,1962.
5.
POZNYAK, L. T., Application of the Bazley-Fox method to two-dimensional equations.,?%. vFht&! Mat. mat. Fiz., 16, No. 1,83-101, 1976.
6.
BAZLEY, N. W., and FOX, D. W., Truncations in the method of intermediate to eigenvalues,J. Res. NBS, 65B, No. 2,105-111,196l.
I.
WEINELT, W., Uber apriore Fehlerabschatzungen der Eigenwertnaherungen Beitr. Numer. Math, No. 1,195-236, 1974.
8.
LADYZHENSKAYA, 0. A., and URAL’TSEVA, N. N., Linear and quasi-linearequationsof elliptic type (Lineinye i kvazilineinye uravneniya ellipticheskogo tipa), Nauka, Moscow, 1964.
9.
AGMON, S., Lectures on elliptic boundary value problems, Princeton, Van Nostrand Math. Studies, 1965.
10. GRISVARD, P., Caracterisation No. 1,40-63, 1967.
de quelques espaces d’interpohtion,
to the Weinstein method of
secondorder
of the form
elliptic
problems for lower bounds
beim Bazley-Fox-Verfahren,
Arch Ration. Mech. Analysis, 25,
11. IL’IN, V. P., Some inequalities in functional spaces and their application to investigation of the convergence of variational processes, Tr. Matem. in-taAkad. Nauk SSSR, 53,64-127, 1959. 12. SHAPOSHNIKOVA, T. O., Asymptotic estimates for convergence of Ritz’s method in eigenvalue problems, Izv. vuzov. Matematika, No. 6 (121),86-91, 1972. 13. POZNYAK, L. T., Evaluation of lower bounds for eigenvalues of some ordinary differential equations by the Bazley-Fox method, Zh v.%hisLMat. mat. Fiz., 14, No. 4,873-890,1974.