On the rational choice of coordinate functions in Ritz's method

ON

THE

RATIONAL CHOICE OF COORDINATE FUNCTIONS IN RITZ’S METHOD* S. G. MIKHLIN (Leningad) (Received 2 June 1961) 0 1. FORMULATION

OF THE PROBLEM

The terminology and the results of the monograph qualifications and references. 1. Let the following equation be given Au

=f,

[l] are used here without

(1)

where A is a self-conjugate positive definite operator acting in a separable Hilbert space H, f is the given and u the required elements of the same space. Equation (1) can be solved by Ritz’s method. An essential part of this method is the choice of the system of coordinate elements (we shall simply call this the coordinate system), which in the most general case are subject to the following three conditions: 1) they belong to the space HA ; 2) taken in any finite number they are linearly independent ; 3) the coordin$e system is complete in HA. Two cases should be distinguished. If a rough approximation is to be constructed using a small number of coordinate elements, then. any particular choice of the coordinate system has little effect on the process of forming the approximate solution under the condition that the coordinate system satisfies the requirements mentioned above. The picture changes greatly if, in order to obtain a more accurate approximation, a large number of coordinate elements are used. In this case an unsuitable choice of the coordinate system can lead (see [2]-[4]) to instability of the Ritz process, i.e. to such an accumulation of errors that the advantage of the high order of approximation is cancelled. On the other hand, the convergence of the Ritz method can be improved with a suitable choice of the coordinate system [S]. In the present paper we shall give some coordinate systems which make solutions with Ritz approximation stable for very simple boundary problems associated with one or two independent variables. In some cases these systems are such that the error Au, -f tends to zero, u, being the approximate solution of equation(1) obtained by using the first n coordinate functions. 2. Let H be a separable Hilbert space, A and B self-conjugate and positive definite operators in this space. We shall call the operators A and B similar if D(A) = D(B), * Zh. vych. mat. 2: No. 3, 437-444, 1962. 461

462

S.G. MIKHLIN

and semi-similar, if D(l/z) = D(JB), or what is the same thing, if the spaces HA and HB consist of the same elements. It follows from a theorem of Heinz [6] that similar operators are at the same time semi-similar. We shall select the coordinate systems on the basis of two theorems, the first of which was proved in [4], and the second in [5]. THEOREM 1. Let A and B be semi-similur operators, and let the sequence of elements pk E HB, k = 1,2, . . . , be orthonormalized and complete in HB. Then this sequence is also complete in H A ; .if it is used as the coordinate system for the solution of equation (I), then the approximate Ritz solution of this equation is stable in the metric of the space HA with respect to small variations in the matrix of Ritz’s system and in the column of its free terms. THEOREM 2. Let A and B be similar operutors and the spectrum of operator B a pure point ;pectrum. If the eigenelements of the operator B are used as coordinate elements for the solution of equation (I), then the error Au,, -f, where u, is the n-th Ritz approximation, tends to zero. We note the following. If in the conditions of Theorem 1 the operator B has a pure point spectrum, then the eigenelements of this operator normalized in the metric Hs can be used as coordinate elements. Also, in the conditions of Theorem 2 the operators A and B are semi-similar too. Hence, normalizing in the metric HB the system of eigenelements of the operator B we shall obtain a coordinate system giving, in virtue of Theorem 1, also the stability of the approximate Ritz solution. Thus we have reduced the rationa selection of the coordinate system to the construction of the operator B, which is similar or at least semi-similar to the operator A and has a pure point spectrum. We assume, of course, that the eigenelements of the operator B are known. In some cases, when the operators A and B are semisimilar but the eigenfunctions of the operator B are in fact unknown, or do not form a complete system (in which case the spectrum of the operator B is not a pure point spectrum), it is possible to choose a simple system of elements that is complete and orthonormalized in the metric HB. According to Theorem 1, such a system gives the stability of the approximate Ritz solution of equation (1). § 2. ONE-DIMENSIONAL PROBLEMS In this section H = L,(O, 1) throughout. 1. Let the operator A be defined by the relations Au=

-

$(Pwg)+q(x)u,

O
(2)

u (0) = u(l) = 0. (3) Let us assume that the functions p(x), p’(x), q(x) are continuous in the segment 0 p,,, where p,, is a positive constant, and q(x) > - 1,) where 1, is the minimum eigenvalue of the operator -

$-(p(x)

g)

(4)

Rational choice of coordinate functions in Ritz’s method

463

under boundary conditions (3). In the given case the operator B = - d2/dx2 under boundary conditions (3) is similar to operator A. For, as can be easily seen, both the regions of definition D(A) and D(B) consist of functions that are continuous together with their first derivatives in the segment 0 < x < 1, become zero at the ends of it, and finally have second derivatives quadratically summable in the same segment. The eigenfunctions of the operator B normalized in the metric HB are c q,.(x) = nn

sinnnx,

n = 1,2,...

If these functions are taken as coordinate functions in the solution of equation (l), the approximate Ritz solution is stable in the sense of theorem 1. Further, if u,(x) is the nth Ritz approximate solution, then llAun -flI + 0, whence it is easy to conclude that 11~;- uri / + 0 where u,,(x) is the exact solution of the problem. It is known from general considerations regarding the convergence of Ritz’s method that j1~;- u:\/ + 0, then from the convergence of the second derivatives it is easy to see that U;(X) --) u:(x) uniformly. Similar conclusions can be drawn in some other examples of this section. 2. Now let the operator A be defined by the differential expression (2) under the boundary conditions U(0) = 0,

V(l) = 0;

(6)

We retain the assumptions of section 1, but 1, will denote the minimum eigenvalue of the operator (4) under the boundary conditions (6). In this case the operator B = - d2/dx2 will be similar to the operator A under these boundary conditions; its eigenfunctions normalized in Hs are 21/F p”(X) = GnT

sin +

3. We retain all the assumptions

7dx,

n = 1,2,...

of section 1 except one. We shall consider A be defined by the differential expression

q(x) 3 qO= const. > 0. Let the operator

(2) and the boundary conditions U’(0) = U’(1) = 0.

(8)

In this case the operator B, defined by the differential expression Bu = - d2U/dx2 + u and the boundary conditions (8), is similar to A. Its eigenfunctions normalized in Hs are %(X) = 17

V”(X) = d

riw 2+ 1

cosmx,

n = 1,2,...

(9)

4. We now consider the case when the operator A is defined by the differential expression (2) and the boundary conditions u’(0) - c%(O) = 0,

U’(1) + @U(l) = 0;

c(,B>O.

(10)

We assume that p(x) and q(x) satisfy the conditions of section 1, where this time A1is the minimum eigenvalue of the operator (4) under the boundary conditions (10). In the case considered the similar operator will be B defined by the differential

S.G. MIKHLIN

464

expression Bu = - d2u/dx2 and the boundary conditions (10). But the construction of the eigenfunctions of this operator is difficult, since it is connected with the solution of a transcendental equation. But in this case it is easy to construct an operator semi-similar to A and a complete system orthonormalized in the metric of this operaor. Thus the operator B of the example in section 3 is semi-similar to the operator A of the present example. The corresponding spaces HA and HB consist of functions absolutely continuous in the segment 0 < x < 1, the derivatives of which are quadratically summable in this segment. In solving problem (l), in which A is the operator of the present example, the functions of (9) can be used as coordinate functions. The approximate solution will be stable in the sense of Theorem 1. The ogerator i defined by the relations j.l(=-X!Y

u’(0) - u(0) = 0,

u’(l) = 0;

dx= ’ can also be taken as a semi-similar operator. The corresponding and norm are

[u, v]~ =

u(O)*)

(11)

energy derivative

+ ju’(x)s’(x)dx, 0

lujjj

/u(0)la + 5 Iu’(x)12dx.

=

0

It can be seen at once from (12) that the spaces HA and H; consist of the same elements. It is easy to construct a system of functions complete and orthonormalized in HE, for example like the system l,x,$sinnx&sin27zx

,..., gsinnnx

,...

(13)

It is obvious that in the metric (12) this system is orthonormalized; only its completeness has to be proved. For this we determine what functions are orthogonal in the metric (12) to the functions

1/T .

nrr sinnnx,

n = 1,2,...

Let o(x) be such a function. Then u,(x),gsinnnxlB

= I,6iw’(x)cosnnxdx=O,

n= 1,2,...

0

Hence o’(x) = c and w(x) = cx + cl, where c, c1 = const., and the completeness of system (13) is proved. The use of system (13) as the coordinate system leads to a stable approximate solution of equation (1). 5. Let us consider the case of a degenerating differential operator. Suppose the function p(x) in expression (2) has the form P (x> = X”Pl(X) 3

(14)

Rational choice of coordinate

functions in Ritz’s method

465

where pi(x) is continuously differentiable in the segment 0 2 the spectra of the operators considered below become continuous and lose their point nature. We confine ourselves to the simplest boundary conditions

U(0) = U(1) = 0,

zf(1) = 0,

O
l
(15)

As is known, under the boundary conditions (15) the operator (4) is positively definite and has a discrete spectrum. Let 2, be the minimum eigenvalue of this operator. We shall now require that in expression (2) the coefficient q(x) be continuous in the segment 0 < x < 1 and satisfy the condition q(x) > --A,.Let A denote an operator initially defined by the relations (2) and (15) on smooth functions and then extended by Friedrichs’s method to a self-conjugate operator. The region of definition of this operator consists of the functions U(X) subjected to the following requirements: 1) u(x) is continuous in the segment 0
and the functions UE D(B) satisfy the three conditions the operator A. The eigenfunctions of the operator B are f&(x) = C”X(l_=)‘V”(y”,“Xl-=‘2);

y=

stated above, is similar to

l-a 2--G(, I

I

n =

1,2,...,

(16)

yy,. is the nth positive root of the Bessel function J,(x); the coefficient c, is chosen so that

where

IP)$ = i x’[v;(x)]“dx = 1. 0 5

3. TWO-DIMENSIONAL

PROBLEMS

In this section H = L,(Q), where Q is a finite plane region, the boundary which will be denoted by 5’. 1. We consider the non-degenerate elliptical equation -

2 &(A,k$j +

Cu=f(X,,X,);

x1 =x,

x2 =

y

of

(17)

ilk=1

with the boundary condition +=o.

(18)

.& will denote the minimum eigenvalue of the operator

-

2 &(Ajkej

i,k=l

(19)

S.G.

466

MIKHLIN

with the boundary condition (18), and consider that C > -A,. We also assume that Al,, aAjk/axi, C are continuous in the closed region a = 12 U S. Let us assume that the boundary S is sufficiently smooth and that we have found a transformation x’ = qJ(X,Y),

Y’ = y(x, u),

(20)

which gives a one-to-one mapping of the closed region a on the circle xQ + ~‘2 < 1, and the functions v and y are twice continuously differentiable and the Jacobian

is positively bounded above and below. The transformation (17)-(18) to the following:

4=x’,

-j$l&(A;&)+CJu=/,;

(20) converts the problem

x;=Y’>

uIp=O.

(21)

(22)

Here r is the circle xf2+ y’2 = 1 and 2'

Aik = J c r,s=1

w.

A,, +-=. I

ax; s

Let A denote an operator defined by the left hand side of equation (21) and the boundary condition (22). It follows from the results of a number of investigations* that in the definition of the operator A the region D(A) consists of functions belonging to the class Wi2) of Sobolev in the circle x’~ + y’2 < 1, and become zero on the circumference of this circle. It is clear from this that the operator similar to A in this case is B=-A=_&&

with the boundary condition (22). The eigenfunctions of the operator B are vk,

n@‘,

Y’>

=

ck, n

Jkb’k,n ‘-i;;,tke

>

k,n=0,1,2

,...,

(23)

where x’ = r cos 8, y’ = r sin 8 and the coefficient ck,, is such that (p)k,nja= 1. These functions, taken as coordinate functions in the solution of the problem (21)-(22), give stability to the approximate Ritz solution and the convergence of the error to zero. The latter fact implies the convergence of the second derivatives in L,. This in turn means that the first derivatives converge in LP for any p > 1, and the approximate solutions themselves converge uniformly. Example. Let the region Q be included between the straight lines y = f b touching the boundary S, and let any straight line y = const. situated between them * A review of these can be found in [7].

Rational choice of coordinate functions in Ritz’s method

467

S only at two points, the abscissae of which will be denoted by ab)

intersect

B(y). The transform

(20) is given by x’ = p(y)x

where P(Y) =

and

Y’ = $Y,

+ Y(Y),

-~ 2 I/bZ-y2 b[B(n

-

B(y)+

a091 ’

a09

y(y) = B(y)-a(y)

l/b2-P

b-’

This transformation

is suitable for our purpose if the functions ,&y) and y(y) have derivatives in the segment lyl
second

vk,

n(X’

) Y’>

=

ck,

n Jknip(YknjS,

n r,

sin

‘F

?

k,n=

1,2 ,...,

(24)

for the segment 0 < Y < 1, 0 < 0 < /3 and k,n = 1,2 ,...,

(25)

for the rectangle 0 < x’ < a, 0 < y’ < b. The coefficient ck,n in formula (24) is determined from the condition

If the functions (24) or (25) are used as coordinate functions in the solution of problem (21)-(22), the approximate Ritz solution will be stable and the error will tend to zero. Example. Let Sz be a right-angled triangle with an acute angle b. The dimensions and the position of th: triangle are shown in the Figure. The mapping of this triangle on the circular sector 0 < r < 1, 0 < 0 < ,5 is given by the equations x’

=

1

I

,pcot/?

+

bsin b a

(x

-ycot/?)

y’ = +y.

It can be easily seen that this transform is infinitely differentiable and its Jacobian has a positive upper and lower bound. 3. We shall briefly consider a third boundary problem; its limiting case-the

468

S. G. MIKHLIN

second boundary problem-is studied almost in the same way. Let it be required to integrate equation (17) under the boundary condition 2

[C

Aj&COS(V~X&)+~U

1s

i

j,k=l

=O,

(26)

where v is the external normal to S and o is a non-negative function not identically equal to zero. As above, we use a transform of the type (20) to map, if possible, the region Q on a circle, circular sector or rectangle. As a rule it is difficult in this case to construct a similar operator with known eigenfunctions, but a semi-similar Y

l/r\ b

B

a

X

operator can be easily constructed. Such an operator, for example, will be B (r is the boundary of the region on which Q is mapped) (27) Its eigenfunctions vk,n

yk, n is the

have the following form; for a circle =

ck,

n Jk(?k,

k=0,1,2

n r)s’W’~

n = 1,2,...,

,...,

cw

nth positive root of the derivative J;(X); for the sector 0 < P < 1,O -=z0 < ,8 kn0 Qik,n

=

Ck,nJkn,S(Y~,urlB,nT)COS-;

B

k=0,1,2

,...,

n=

1,2,...

(29)

for the rectangle 0
ck,ncos

Q

cos -3kny’ b

k,n=0,1,2

,..,.

The coefficient c&,nin formulae (28)-(30) is chosen from the condition lq&,nlB= 1. Translated by

PRASENJIT BASU

REFERENCES 1. MIIUUIN, S. G., Variatsionnye metody v matematicheskoi fizike (Variational mathematical physics.) Fizmatgiz, Moscow, 1957. 2. MIKHLIN, S. G., Izv. vyssh. ucheb. zav., Matematika, No. 5(6), 91-94, 1958. 3. MIKHLIN, S. G., Dokl. Akad. Nauk, SSSR 135: No. 1, 16-19, 1960. 4. WIN, S. G., Vestnik Leningr. Univ., No. 13, 40-51, 1961. 5. MIKHLIN, S. G., Dokl. Akad. Nauk, SSSR 106: No. 3, 391-394, 1956. 6. HEINZ, E., Math. Ann. 123: No. 4, 1951. 7. KOSHELEV, A. I., Usp. mat. nauk 13: No. 4 (82), 29-88, 1958.

methods

in