MATHEMATICS

O. K. BOGARYAN

ON THE CONVERGENCE OF THE RESIDUAL OF THE BUBNOV–GALERKIN AND RITZ METHODS

(Presented by Academician V. I. Smirnov on 20 VI 1961)

In the paper \((^{1})\) it was proved that the residual of the Ritz method for the equation \(Au=f\), where \(A\) is a positive-definite, self-adjoint operator, converges to zero if the operator has a purely point spectrum and the system of eigen-elements of a convergent operator is taken as the coordinate elements.

In the present paper this result is generalized.

\(1^\circ\). In a separable Hilbert space \(H\) consider the linear equation

\[ Lu \equiv Au+Ku=f . \tag{1} \]

The operator \(A\), with domain of definition \(D(A)\), is assumed to be positive-definite and self-adjoint. We shall suppose that \(D(A)\subset D(K)\).

Let \(A\) and \(B\) be convergent \((^{1})\) operators, i.e., both are positive-definite, self-adjoint, and \(D(A)=D(B)\). Then \(D(A^\alpha)=D(B^\alpha)\) \((0\leq \alpha \leq 1)\) \((^{2})\), and, in particular, from the condition that the domains of definition coincide it follows that \(H_A=H_B\) \((^{3})\). If the operators \(A\) and \(B\) are convergent, then the operators \(AB^{-1}\), \(BA^{-1}\), \(\sqrt{A}\sqrt{B^{-1}}\), \(\sqrt{A^{-1}}\), \(\sqrt{B}\), and the operators obtained from them by permuting the factors are bounded; hence the inequality immediately follows

\[ d_1 \leq \frac{\|u\|_A}{\|u\|_B} \leq d_2,\qquad d_1,d_2>0. \tag{2} \]

Let \(\{\lambda_n\}\) be a sequence of positive numbers, and let \(\{\varphi_n\}\) and \(\{\psi_n\}\) be sequences of elements such that

\[ B\varphi_n=\lambda_n\varphi_n+\psi_n,\qquad n=1,2,\ldots \]

We assume that \(\{\varphi_n\}\) is a system of elements in \(H_B\) that is orthonormal in the metric of the space \(H\) and complete (from inequality (2) its completeness in the space \(H_A\) follows).

Next suppose that:

1. \[ \sum_{i,j=1}^{n}(B\varphi_i,\varphi_j)t_i\bar t_j \geq \frac{1}{2}\sum_{i=1}^{n}\lambda_i |t_i|^2 \quad \text{for any } n\ (n=1,2,\ldots). \]

2. \[ \sum_{i=1}^{\infty}\left\{\frac{\|\psi_i\|}{\lambda_i}\right\}^2 < \frac{d_1^4}{4\|AB^{-1}\|^2}. \]

3. The operator \(KA^{-1}\) is completely continuous.

For solving equation (1) we shall use the Bubnov–Galerkin method, choosing the elements \(\{\varphi_n\}\) as coordinate elements.

The aim of the present note is to prove that, under the stated assumptions, the relation

\[ \|Lu_n-f\|\to 0 \quad \text{as } n\to\infty, \]

holds, where \(u_n\) are the approximations constructed by the Bubnov–Galerkin method. We have

\[ u_n=\sum_{k=1}^{n} a_k\varphi_k . \]

The coefficients \(a_k\) are determined from the algebraic system

\[ \sum_{k=1}^{n} a_k(A\varphi_k,\varphi_m) + \sum_{k=1}^{n} a_k(K\varphi_k,\varphi_m) = (f,\varphi_m), \quad m=1,2,\ldots,n, \]

which is easily reduced to the form

\[ \sum_{k=1}^{n} b_k(\varphi_k,B^{-1}A\varphi_m) + \sum_{k=1}^{n} b_k\left(\frac{\psi_k}{\lambda_k},B^{-1}A\varphi_m\right) + \]

\[ + \sum_{k=1}^{n} b_k(BA^{-1}KB^{-1}\varphi_k,B^{-1}A\varphi_m) + \sum_{k=1}^{n} b_k\left(BA^{-1}KB^{-1}\frac{\psi_k}{\lambda_k},B^{-1}A\varphi_m\right) = \]

\[ = (BA^{-1}f,B^{-1}A\varphi_m), \quad m=1,2,\ldots,n, \quad \text{where } b_k=\lambda_k a_k . \tag{3} \]

Put \(v_n=\sum_{k=1}^{n} b_k\varphi_k\) and consider the equation

\[ v+BA^{-1}KB^{-1}v+T_1v+BA^{-1}KB^{-1}T_1v=BA^{-1}f, \tag{4} \]

where the operator

\[ T_1v=\sum_{k=1}^{\infty}(v,\varphi_k)\frac{\psi_k}{\lambda_k} \]

is completely continuous by virtue of condition 2.

The identity \(KB^{-1}=KA^{-1}AB^{-1}\) shows that the operator \(KB^{-1}\) is completely continuous. Hence the operator
\(BA^{-1}KB^{-1}+T_1+BA^{-1}KB^{-1}T_1\) is also completely continuous.

As in paper \((^1)\), it is proved that the relation \(v_n\to v\) holds as \(n\to\infty\), where \(v\) is the exact solution of equation (4), and the coefficients \(b_k\) in the representation \(v_n=\sum b_k\varphi_k\) are determined from the algebraic system (3).

It is not difficult to see that the equality

\[ Lu_n=AB^{-1}v_n+AB^{-1}\bigl(BA^{-1}KB^{-1}v_n+T_1v_n+BA^{-1}KB^{-1}T_1v_n\bigr), \]

holds, whose right-hand side tends, as \(n\to\infty\), to the right-hand side of equation (1)—to the element \(f\), and, consequently,

\[ \|Lu_n-f\|\to 0 \quad \text{as } n\to\infty . \]

\(2^\circ\). In equation (1) put \(K=0\). Then the result obtained in item \(1^\circ\) shows that the residual \(Au_n-f\) of the Ritz method for the equation \(Au=f\) converges to zero, where \(A\) is a positive-definite self-adjoint operator and \(u_n\) are the Ritz approximations, provided only that the system of elements \(\{\varphi_n\}\) of item \(1^\circ\) is taken as the coordinate system. The case when all \(\psi_k=0,\ k=1,2,\ldots\), was considered by S. G. Mikhlin \((^1)\): the residual of the Ritz method converges to zero if the operator has a purely point spectrum and the system of eigenfunctions of the convergent operator is taken as the coordinate system.

\(3^\circ\). In a separable Hilbert space \(H\) consider the equation

\[ Lu\equiv Au+Ku=f, \tag{5} \]

without assuming that \(A\) is a positive-definite operator; we shall assume only that \(D(A)\) is dense in \(H\).

We shall suppose that \(D(A) \subset D(K)\).

Let there exist sequences of positive numbers \(\{\lambda_n\}\) and of elements \(\{\varphi_n\}\) and \(\{\psi_n\}\) such that

\[ A\varphi_n = \lambda_n\varphi_n + \psi_n, \qquad n = 1, 2, \ldots, \]

where the system \(\{\varphi_n\}\) is complete and orthonormal in the space \(H\).

Next, suppose that the operator \(A^{-1}\) exists and that the operators

\[ KA^{-1} \quad \text{and} \quad T_1 v = \sum_{k=1}^{\infty} (v,\varphi_k)\frac{\psi_k}{\lambda_k} \]

are completely continuous in the space \(H\). Finally, assume that equation (5) is uniquely solvable.

We shall construct approximate solutions of equation (5) in the form \(u_n=\sum_{k=1}^{n} a_k\varphi_k\), and determine the coefficients \(a_k\) from the system of algebraic equations

\[ (Lu_n,\varphi_m) = (f,\varphi_m), \qquad m = 1, 2, \ldots, n. \tag{6} \]

We shall prove that, for sufficiently large \(n\), system (6) is uniquely solvable and that the relation \(\|Lu_n-f\|\to 0\) holds as \(n\to\infty\). Denote

\[ v_n = \sum_{k=1}^{n} b_k\varphi_k, \qquad b_k=\lambda_k a_k. \]

The system of equations (6) can be rewritten as

\[ (v_n+Tv_n,\varphi_m)=(f,\varphi_m), \qquad m=1,2,\ldots,n, \tag{7} \]

where \(T = KA^{-1}+T_1+KA^{-1}T_1\) is a completely continuous operator. Consider the equation

\[ v + Tv = f. \tag{8} \]

It can be proved that, for sufficiently large \(n\), system (7) is uniquely solvable and that the relation \(\|v_n-v\|\to 0\) holds as \(n\to\infty\), where \(v\) satisfies equation (8) \({}^{4}\).

It can also be shown that the equality \(Lu_n=v_n+Tv_n\) holds, whose right-hand side tends to the element \(f\), the right-hand side of equations (5) and (8), and, consequently, \(\|Lu_n-f\|\to 0\) as \(n\to\infty\).

Consider the equation

\[ v + Tv = f. \tag{9} \]

It can be proved that, for sufficiently large \(n\), system (7) is uniquely solvable and that the relation \(v_n\to v\) holds as \(n\to\infty\), where \(v\) satisfies equation (8) \({}^{4}\).

It can also be shown that the equality \(Lu_n=v_n+Tv_n\) holds, whose right-hand side tends to the element \(f\), the right-hand side of equations (5) and (8), and, consequently, \(\|Lu_n-f\|\to 0\) as \(n\to\infty\).

Remark. If the operator \((A+K)^{-1}\) or \(A^{-1}\) is bounded, then also \(\|u_n-u_0\|\to 0\) as \(n\to\infty\), where \(u_0=(A+K)^{-1}f\).

Computing Center
Academy of Sciences of the Armenian SSR

Received
2 VI 1961

REFERENCES

\({}^{1}\) S. G. Mikhlin, DAN, 106, No. 3 (1956).
\({}^{2}\) E. Heinz, Math. Ann., 123, H. 4 (1951).
\({}^{3}\) S. G. Mikhlin, The Problem of the Minimum of a Quadratic Functional, 1950.
\({}^{4}\) S. G. Mikhlin, Variational Methods in Mathematical Physics, 1957.