On the reasonable choice oî the coordinate functions in the Bubnov-Galerkin method

ANNALES SOCIETATIS MAT H E MAT IC AE POLONAE Series I: COMMENTATIONES MATHEMATICAE X V II (1973) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE X V II (1973)

J

(Krakôw)

On the reasonable choice oî the coordinate functions in the Bubnov-Galerkin method

In the Mikhlin’s monograph [5] conditions of the reasonable choice of coordinate functions for the approximate solution of the operator equation

were given with the use of the Bitz method. It is shown in [5] that on the choice of these coordinate functions the following properties depend:

1° convergence of the approximate solution and, possibly, conver­

gence of the “residuum Aun — to zero, 2° stability of the method,

3° rate of convergence of an approximating sequence.

It is known, however (see [3], p 23), that for the application of the Bitz method to equation (1) it is needed that the operator A in (1) was selfadjoint and positive-definite. If A is not such, then for the solution of (1) with some additional assumptions, more general Bubnov-Galerkin method can be applied.

Let H be a separable Hilbert space. We assume that the domain of A , D{A) is a dense subset of H , A is a linear operator. The Bubnov-Galerkin method as applied to (1) goes as follows: we choose the sequence of elements (2 )_ <Pl, <p

, <Pn, •••

satisfying conditions :

2° for every n the elements <рг , <p2, are linearly independent^

3° the sequence (2) is a complete set in H.

We want to obtain the approximate solution of equation (1) in

(1)

Au = f

1° (pne

D(A) (n = 1 , 2 , . . . ) ,

the form

ft

(3)

where the ak s fk = 1, 2, ..., n) satisfy the system of equations:

The %’s in (3) depend on n and should be denoted aff\ but we feel free to neglect such pedantry and shall simply write them ak.

The problem of convergence of the Bubnov-Galerkin method was dealt by many authors, but the most general sufficient condition of its convergence were given by Mikhlin (see [1], [2] or [3], chapter У).

The aim of this paper is to give a certain fashion of the reasonable choice of the sequence (2). This system (2) will be hereinafter called the system of coordinate functions. We shall show that, under some restrictions imposed on the operator A in (1), the system of coordinate functions can be chosen in such a manner that conditions 1° and 2° mentioned earlier will be satisfied. In this paper we shall not consider the rate of conver­

gence of an approximating sequence, although it will be the subject of another paper.

First we shall prove some lemmas.

Lemma

1. I f В is self adjoint, positive-definite operator, A — a linear operator such that I)(A) ~ B(B) <= I)(A*), then the operator B~l A is bounded.

Proof. We shall show first that A *B~X is bounded. We have B(A*B~1) = H. Indeed, B (B ~X) = B(B) = H, since В is selfadjoint and positive-definite. On the other hand -#(B-1) = В (В) с: В (A*), hence, if / is an arbitrary element, then the equality A*B

f = A*(B~xf) makes sense. We shall now show that A*B~l is closed. Let f n - » / and A*B~

f n -^g. We designate B~

f n = hn. B~x is bounded, hence B~lf n B~xf. Let B~lf = h, therefore hn — > h and A*hn -> g. It is known ([7], p. 657) that A* is closed, thus he В (A) and A*h = g, that is A*B~xf = g, what means that A*B~l is closed, since operator A*B~x is defined in the whole space В and closed it is then bounded (see [7], p. 560). Bounded­

ness of B~lA follows from the equality B~lA = (A*B~x)*.

Lemma

2. If: 1° operators A and В are such as in Lemma 1, 2° A~l exists, 3° the inequality

(5) \(Au, B u ) \^ с^Ви\\2, c > 0, u e B (B ), holds, then the operator BA~l is bounded.

Proof. Let us put in (5) v = Bu. Then и = B~xv and so we have П

(

)

c|NI2 < |(AB xv, ®)J < \\AB h?! *||«U or, in other words

(6)

\\AB lv\\^c\\v\\.

Bubnov-GalerTcin method

From (6) it follows the existence and boundedness of the reciprocal operator for the operator AB~l. In view of the assumptions made on A and В it is evident that (AZ?-1)-1 = ВА~г, and hence, Lemma 2 is proved.

Le m m a 3.

I f operators A and В satisfy conditions

and

of Lemma

and if <Px, <Pzi - i <Pn is an orthonormal sequence of elements belonging to the domain of A*, then the matrix

(7) Vn = \\{<Pj , B - 1A*<Pk)\\lk=l

posesses the reciprocal matrix and, moreover, ЦупЧ ^ счч where c2 is a constant which does not depend on n.

Proof. Let t denote any vector of the form t = (tlt t2, ..., tn). Let us denote, for brevity B~

A*pm = we have

n n n

(8) WvJW* = £ I ^ toki Tm)tm\Z = l(.Vb (pk)?i

k—\ m=1 fc=l

n n

where \p = ^ îmWm — В A tmcpm.

' m=l m=1

W e'now apply inequality (5), putting there Bu = v. We obtain

\{AB~lv, ®)| > c|H|2,

where v is an arbitrary element of H. Further, we have

\(AB~

v , v )I = \{v, B~l A*v)\ = |(Б-1 A*v, ®)| > c ||v|j2.

П

In this last inequality we now replace v by the sum tm(pm and this

yields m=1

\{Ч>>*)\>Ф\\% =c\\t\\K On the other hand

П П

l(vb v)l2< ( 2 \ШУ>,<Рт)\У<\\П* £

\(V,<Pm)\2-

m=l m=

From (8) and this above inequality we obtain П

( 9 ) Ш Г = y, \ ( V, <P m ) \ * > \\t\r* №, ■>} ) ?> C\\t\\K

m=l

Now, from (9) it follows that the matrix y~l exists and also the inequality:

(

10

Now we shall prove the following theorem, whose proof is based

on Lemmas

2 and 3.

T

О

1. If:

1° operators A and В satisfy hypotheses 1° and 3° of Lemma

, 2° equation (1) has the unique solution u0,

3° operator В posesses the discrete spectrum 4° the term

П

(11) Un = У ak<pk

k= 1

is a n-th successive approximation in the sense of Bubnov-GalerMn of the solution of equation (1); where {yn} is an orthonormal sequence of eigenvec­

tors of В corresponding to the eigenvalues {/!,„}, then un -> u

and Aun—f -> О when n -> oo, in the metric of the space H.

P ro o f. Let us denote w

= BuQ and wn = Bun; then we can write (I) in the form

(12) w

= BA~xf

and П

(13) wn = ] £ c k<pk, ck — Akak.

k = l

Coordinates ak (1c = 1, 2, .. . , n) in (11) satisfy the system of equation (4).

This system can be transformed in the following manner:

(Atpt , <pj) = (<pk, A*<pj) = Xk(B~l<fk, A*<pf) =

(/, 9t) = (A B - 'B A - 'f , 9j) = (B A -'f, В -Ы *й ) = (w«, B - lA"b ).

Then the system (4) takes the form П

(II) ck{<Pki B~lA*<pf) = (w0, B~l A*(pf)-, j = 1, 2 , . . . , n.

&=i

But the system (14) may we write in the following form

(15) P nwn = P nw0,

where P n is a projection operator on the space K n spanned by the vectors ys = В~гА*щ, j = 1 , 2 ,

From a theorem of 1ST. I. Polskii we know that for the convergence wn wQ it suffices that the inequality

(16) М < С ||Р Я*||,

is satisfied. In this inequality L n denotes the space spanned by the vectors

<Pi, y 2, where {<pn} is a complete system in H , and the constant G

does not depend on n (see [6] or [5], p. 122).

Bubnov-GalerTcin method

Operator P n can be defined as follows: we find such constants jujk, that П

<17) = min, for j = 1, 2, ..., n,

k =i

then for arbitrary v e Ln we have

П П

(18) v = У у к(рк, P nv = У ySfAjkB - lA*(pk.

fe=i j,fc=i

Inequality (16) takes then the following form

Î1 П

|| ^У*9>*||2< 0 а|| ,

k = 1 ?',*= 1

or it can be written:

n n n

( 1 9 ) ^ ! n - l 2 < ^ 2 У/У * £ f ^ j r ^ k s { B ~ l A * c p r 1 B - l A * ( p s ) .

k= 1 7,fc=l r , s — 1

From (19) it follows, that to show that inequality (16) is true it suffices to show that the minimal value of the quadratic form

П П

(20) Г == JT' ytf>k £ pJr~fika{B -

A*<pr, B -

A*<p8),

i , k — 1 r ,s = I

is bounded from below by the non-negative constant independent of n.

We denote

( 21)

then

Let

hence

y ^ j Уз H' jr ? 3=1

г = y Ô M B - W q , , , B - ' A ^ , , ) = ! B - ' A * 2 ô r4,r

r=l

r , s —l

2ôrfr = S, В - Ы * { = 4)

f = ( A * r 1S 4 and Ull < ||(V) - 1 B U M = ЦВЛ-ЧичИ.

We deduce from Lemma 2 that the operator BA~l is bounded, thus

and so we have

П П

(22) « В А - Ч т у Ч м 2 = l| B A '4 r 2- y W 2-

Let, further, Mn denote the matrix of transformation (21) Mn - I M £ =1.

We shall show that there exists the reciprocal matrix M~l and l i ^ l l Ojy where С

is constant wliicli does not depend on ть* Indeed^

from (17) follows, that the constants jujk satisfy the system of equations:

n

(23) ^ (B ~1A *(pk , B ~l A *(pm)fAjk = (<pj, B ~1A *q> m), j , m = l , . . . f n . fc= 1

Denote now by Фп the matrix, whose elements are (B~lA*(pk,

B-1 A*<pm).

Then the system (23) can be written in the form:

(24) МпФп = уя,

where the matrix y n is defined in (7).

Let us observe, that the matrix Фп in (21) is bounded.

In fact, similarly as in Lemma 3, let t = (tx, t2, .. . , tn) be an arbitrary vector, hence

11

n

\\

j r = У = IIB~lA ’

, t k9kr

k,m~\ k~l

n n

< p r ' ^ Y l l у Ч -л И 2 = HAB-’f £ 11,1- = ||4B -4|S||«||2.

A=1 A=1

From Lemma 1 we deduce that the operator AB~X is bounded, thus, from the latter inequality is follows that

(25)

Ц

- 1!.

From (24) in view of (10) and (25) we obtain 11^41 <<?, = И В-Ч |е-*.

Observe, that the system (21) can we writ

in the form Л/* y = à, where У = (Yi, ■■■, Уп) and Ô = (<5i, ..., dn). Hence ||5|| > C'a-1 1|УII, and so

(26) Г ^ \ \ В А -

\\-*С^\\у\\\

We have already mentioned that inequality (26) implies inequality (16).

From this, in view of the quoted theorem of Polskii, follows that wn -> wor or Bun Bu

in the metric of space H. But the operators B~l and A B~X are both bounded and so

B~

(Bun) = un B~x(Bu0) — u

as well as

Aun—f = AB~

(Bun — B-uu) -> 0,

and this is what had to be proved.

Bubnov-GalerMn method 16 B em ark 1. Operator В in (5) can be replaced by the operator В -f kEr where E is the identity operator and Jc some non-negative constant (see

[5], p. 129).

Stability of the Bubnov-Galerkin method.

Suppose, that in a certain computational process we deal with the system of equations:

(27) 4.®»*» =**»>, « = 1 , 2 , 3 , . . . ,

where A^ is an operator from one Banach space X n to another Banach space Y n. We assume also that, for every n, A ~l exists and is defined in the whole space Yn. Simultanously with (27) we consider the sequence of equations

(28) (An+ r n)z ^ ,= yW + ôW.

According to the definition, given by Mikhlin in [4] or [5], p. 70, we say that this computational process is stable, if there exist constants p, q, r independent on n and such that for \\Гп\| < r and arbitrary

equations (28) have solutions and there holds the inequality

(29) ||*w - < V Ю + . .

We say that the computational process (27) is Conyergent if there exists a limit æ0 — limæ(n) in the norm of a space X, where X n are the subspaces

of X ^

We shall now prove the following

Th e o r e m- 2 .

I f the sequence {yn\ of coordinate functions is chosen etc- cording to Theorem, 1, then the Bubnôv-Oalerlcin method for equation (1) is stable.

P roof. Evidently, it suffices to show that the computational process for the solution of the sequence of equations

(30) A„a™ = /"*>, » = 1 , 2 , . . . ,

where A n = U M k r t M l j - i i «!*’ - (“i, •••, «„) a n d /1"* = ((/, f , ) , . . . , ( / , <pj) is stable.

Indeed, in the situation we are considering X n — Yn are both w-di- mensional euclidean spaces, and X is a Z2 space. Since the sequence {<pn}

is orthonormal, we have

n n n

( 3 1 ) I K P = ( 2 Ч<Рк) = = \\a{n)\\2n.

к- l k=l k=. 1

Prom (31), in view of Theorem 1, it follows the convergence of the process

(30). This in turn implies, by Theorem 13.3 Of [5], p. 74, that the process

(30) is stable if and only if \\An *|| < C, where C is a constant not depending on n. We observe that

(32) A , = 1(4%,

= WhWktB

= A„ tfn, where An is a diagonal matrix

~ (^1

^n)l

;and matrix xpn is defined in (7).

From (32), by Lemma 3 we conclude that A"1 exists.

Let

= (tn

.., <n) be an arbitrary vector. We have

n n » n

11АГ = £ j

**(%> B~lA*9t)bII2 > д‘ IH B - ' A ' v j t t f

k= 1 j = l k= l j = l

= Я?||Л «Р.

From this, by inequality (9), we obtain

<Ô3) I! A j r > X \ c m r . So the proof of Theorem 2 is completed.

R em ark 2. If we assume that the operator A in (1) is self-adjoint and positive-definite then, as is well-known the Bubnov-Galerkin method is equivalent to Ritz method. In this case, obviously, Theorem 1 of this paper is the same as Theorem 23.1 of [5], p. 124.

R em ark 3. It was mentioned in the introduction that S. G. Mikhlin has given the sufficient conditions for the convergence of Bubnov-Galerkin method, when appropriate assumptions about operator A in (1) were made.

We consider important to emphasize that these conditions imposed on A by Mikhlin do not overlap with the conditions of Theorem 1, hence Mikhlin’s theorem does not imply the convergence of Bubnov-Galerkin method for equation (1) with the hypotheses of Theorem 1, neither the conver­

gence to zero of “residuum Aun—f ”.

References

[1] С. Г. М ихлин, О сходимости метода Галеркина, ДАН, т. 61, № 2 (1948).

[2] — Некоторые достаточные условия сходимости метода Галеркина, Уч. зап. ЛГУ,№ 135, сер. матем. наук, № 18 (1950).

f3] — Прямые методы в математической физике, Москва 1950.

[4] — Об устойчивости некоторых вычислительных процессов, ДАН 157, № 2 (1964).

[5] — Численная реализация вариационных методов, Москва 1966.

[6] Н. И. П ольский, О сходимости некоторых приближенных метод''анализа, Укр. Мат.

Журн. № I (1955), р. 5 6 -7 0 .

17] В. И. С мирнов, Курс высшей математики, т. 5, Москва 1957.