UDC 518:519.34:517.948

MATHEMATICS

E. I. POLSKII

ON A GENERALIZATION OF PROJECTION METHODS

(Presented by Academician G. I. Petrov on 19 VI 1965)

1°. Ordinary projection methods consist in the following \((^{1})\). Consider the equation \(Lx=f\). The operator \(L\) acts from the space \(H_1\) into the space \(H_2\). An approximate solution
\[ x_n=\sum_{i=1}^{n} c_i\varphi_i \]
is sought in the subspace \(R_n\subset H_1\) from the condition that \(P_n Lx_n=P_n f\), where \(P_n\) is the projection operator onto the subspace \(M_n\subset H_2\). As was shown in \((^{1})\), for convergence of \(Lx_n\) to \(Lx\) for every \(f\) it is necessary and sufficient that \(\tau=\lim\limits_{n\to\infty}\tau_n>0\); here \(\tau_n=\min |P_n y_n|\), where \(|y_n|=1;\ y_n\in L_n=LR_n\).

We shall call the quantity \(\tau_n\) the \(P_n\)-projection deviation of \(M_n\) from \(L_n\)*. The scheme presented assumes that the subspaces \(R_n\) are contained in the domain of definition of the operator \(L\). In particular, if \(L\) is a differential operator, then the coordinate functions \(\{\varphi_i\}\) must satisfy the boundary conditions of the problem. It is clear that, for a complicated boundary or complicated boundary conditions, the selection of the system \(\{\varphi_i\}\) is very difficult. Such is the situation in all projection methods: Ritz, Galerkin, moments, least squares, etc.

The situation is different in Trefftz’s method \((^{3,4})\). There the coordinate functions \(\{\varphi_i\}\) are chosen to satisfy the differential expression itself. The method consists in selecting such combinations
\[ x_n=\sum_{i=1}^{n} c_i\varphi_i \]
as to satisfy the boundary conditions approximately. The difficulties in applying Trefftz’s method are usually caused by the complexity of the principal differential expression.

Below we consider a generalization of the projection method which does not require the coordinate functions to satisfy either the differential expression or the boundary conditions. This generalization contains, in particular, the projection methods and Trefftz’s method. Apparently, the possibility of generalizing the Ritz–Galerkin method, providing the indicated freedom in the choice of coordinate functions, was first pointed out in \((^{5})\). A justification of such a generalization for second-order elliptic differential equations was given in \((^{6})\). In the present paper a general scheme for the application of such methods is considered.

2°. Let \(D\) be a linear set dense in the Hilbert space \(H_1\). In \(D\) there are given, generally speaking, unbounded operators \(\mathcal L\) and \(\Lambda\) with values in the Hilbert spaces \(H\) and \(G\), respectively. We denote the orthogonal sum of \(H\) and \(G\) by \(H_2=H\oplus G\). In this case, if \(f_i\in H\), \(\varphi_i\in G\), then \(F_i=\{f_i,\varphi_i\}\in H_2\) \((i=1,2)\) and \([F_1,F_2]=(f_1,f_2)+\langle\varphi_1,\varphi_2\rangle\). We consider the problem of finding an element \(x\in D\subset H_1\) which is carried by the operators \(\mathcal L\) and \(\Lambda\) into the prescribed elements \(f\in H\), \(\varphi\in G\), respectively. Denoting the operator \(L=\{\mathcal L,\Lambda\}\), we obtain the equation \(Lx=F\), where \(x\in D\subset H_1,\ F\in H_2\). Assuming that this equation is uniquely solvable—

* If \(P_n\) is the operator of orthogonal projection, then the quantity \(\tau_n\) is connected with the gap \(\theta_n\) between the subspaces \(M_n\) and \(L_n\) \((^{2})\) by the relation \(\tau_n^2+\theta_n^2=1\).

in this way, for any \(F\), we apply the projection method \((^1)\) to its solution.

Recall that a sequence of subspaces \(M_n\) of a Hilbert space \(H\), together with the sequence of operators \(P_n\) projecting onto \(M_n\), is called projection-complete \((^1)\) if \(\lvert f-P_n f\rvert\to 0\) for every \(f\in H\) as \(n\to\infty\).

The approximate solution \(x_n\) is sought, as in the case of ordinary projection methods, in a certain finite-dimensional subspace \(R_n\subset D\subset H_1\). The operator \(L\) maps \(R_n\) into the \(n\)-dimensional space \(L_n\subset H_2=H\oplus G\). Next, in \(H_2\) one chooses a subspace \(M_n\) and an operator \(P_n\) projecting onto \(M_n\); moreover, let \(\Pi_n\) be the projector in \(H_2\) onto \(L_n\). Then \(x_n\) is sought from the condition \(P_n Lx_n=P_n F\). As in (1):

The necessary and sufficient condition for unique solvability of the approximate equations for any \(F\) is the condition \(\tau_n>0\). In this case, in the norm of \(H_2\) the estimate is valid:

\[ \lvert Lx-Lx_n\rvert \le (1+\lVert P_n\rVert/\tau_n)\lvert F-\Pi_n F\rvert . \tag{1} \]

If the sequence \(\{L_n,\Pi_n\}\) is projection-complete in \(H_2\), then convergence of \(Lx_n\) to \(Lx\) for every \(F\) takes place if and only if \(\tau=\lim\limits_{n\to\infty}\tau_n>0\).

Remark. If all \(F\) are considered from some subspace, then the necessary and sufficient condition for convergence is the projection completeness of the system \(\{L_n,\Pi_n\}\) in this subspace and such a choice of the system \(\{M_n,P_n\}\) that \(\tau>0\).

What has been said makes it possible to transfer to the method under consideration all results obtained in various works for projection methods, for example \((^1,^7,^8)\).

\(3^\circ\). Let us note that the subspaces \(U_s=\mathcal L R_n\subset H\) and \(V_t=\Lambda R_n\subset G\) may, generally speaking, have dimensions \(s\) and \(t\), different from \(n\). However, always

\[ \max(s,t)\le n\le s+t . \]

Let \(\varphi_1,\ldots,\varphi_n\) be a complete system of independent elements in \(R_n\), and let \(\Psi_1,\ldots,\Psi_n\) be such a system in \(M_n\), where \(\Psi_i=\{g_i,\psi_i\};\ g_i\in H;\ \psi_i\in G;\ i=1,\ldots,n\). If, in addition, \(P_n\) is the operator of orthogonal projection in \(H_2\), then the approximate system of equations takes the form

\[ [Lx_n,\Psi_i]=(\mathcal L x_n-f,g_i)+\langle \Lambda x_n-\varphi,\psi_i\rangle=0,\quad i=1,\ldots,n. \tag{2} \]

We single out in particular the case when \(n=s+t\). If by \(D\subset H_1\) one understands the set of functions sufficiently smooth in some domain \(\Omega\), including the boundary of the domain \(\sigma\), \(\mathcal L\) is a differential operator in the domain \(\Omega\), and \(\Lambda\) is a collection of boundary differential expressions, then \(n=s+t\) means that \(s\) functions \(\varphi_i\) satisfy the conditions \(\Lambda\varphi_i=0\), while \(t\) functions satisfy the conditions \(\mathcal L\varphi_i=0\).

In this case the subspace \(L_n\subset H_2\) may be regarded as the orthogonal sum of the subspaces \(U_s\subset H\) and \(V_t\subset G\). Let also \(M_n=U_s'\oplus V_t'\), where \(U_s'\subset H;\ V_t'\subset G\). Then, if \(F=\{f,\varphi\}\in H_2\), the operator \(\Pi_n\), projecting \(H_2\) onto \(L_n\), is naturally represented in the form

\[ \Pi_n F=\{S_s f,T_t\varphi\}, \]

where \(S_s\) projects \(H\) onto \(U_s\), and \(T_t\) projects \(G\) onto \(V_t\). It is easy to see that projection completeness of the sequence \(\{L_n,\Pi_n\}\) is equivalent to the simultaneous projection completeness of \(\{U_s,S_s\}\), \(\{V_t,T_t\}\), and

\[ \lvert F-\Pi_n F\rvert_{H_2}=\lvert f-S_s f\rvert_H+\lvert \varphi-T_t\varphi\rvert_G . \]

Analogously, the operator \(P_n\) is represented in the form

\[ P_n F=\{S_s' f,T_t'\varphi\}. \]

Theorem. Let \(\tau_n\) be the projection deviation of \(M_n\) from \(\dot L_n\), let \(\tau'\) be that of \(U_s'\) from \(U_s\), and let \(\tau''\) be that of \(V_t'\) from \(V_t\); then
\[ \tau_n=\min(\tau',\tau''). \]

Proof. By definition,
\[ \begin{aligned} \tau_n&=\min\left(|S_s'g|+|T_t'\psi|\right), &&\text{where } g\in U_s,\ \psi\in V_t,\ |g|+|\psi|=1;\\ \tau'&=\min |S_s'g|, &&\text{where } g\in U_s,\ |g|=1;\\ \tau''&=\min |T_t'\psi|, &&\text{where } \psi\in V_t,\ |\psi|=1. \end{aligned} \]
Let, for definiteness, \(\tau'\le \tau''\). Since \(|S_s'g|+|T_t'\psi|\) for \(\psi=0\) assumes the same values as \(|S_s'g|\), it is evident that \(\tau_n\le \tau'\).

On the other hand, let \(\tau_n\) be attained on the elements \(\bar g\) and \(\bar\psi\), with
\[ |\bar g|=a,\qquad |\bar\psi|=1-a. \]
Then
\[ \begin{aligned} \tau_n &=|S_s'\bar g|+|T_t'\bar\psi| = a\left|S_s'\left(\frac1a\bar g\right)\right| +(1-a)\left|T_t'\left(\frac1{1-a}\bar\psi\right)\right| \ge\\ &\ge a\tau' +(1-a)\tau'' \ge \tau'. \end{aligned} \]
By virtue of the theorem just proved, estimate (1) takes the form
\[ |Lx-Lx_n|\le \left(1+\frac{\|S_s'\|+\|T_t'\|}{\min(\tau',\tau'')}\right) \left(|f-S_sf|+|\varphi-T_t\varphi|\right). \]

If, in addition to the condition \(n=s+t\) adopted above, for example \(t=0\), i.e. \(n=s\), then necessarily \(\varphi=0\) as well, since otherwise the vector \(F=\{f,\varphi\}\) will not be approximated by the spaces \(M_n\). If, however, \(s=0\), then \(f=0\) and \(n=t\).

The case considered makes it possible to reduce the question of convergence of the residual in \(H_2\) to checking the convergence conditions in \(H\) and \(G\) separately.

\(4^\circ\). Let us note also the case when convergence is always guaranteed by just the projection completeness of the sequence \(\{L\varphi_n\}\) in \(H_2\). Namely, let
\[ \Psi_i=\{\mathcal L\varphi_i,\Lambda\varphi_i\}, \]
and let the operator \(P_n\) project orthogonally. Then system (2) takes the form
\[ [Lx_n,L\varphi_i] =(\mathcal Lx_n-f,\mathcal L\varphi_i) +\langle \Lambda x_n-\varphi,\Lambda\varphi_i\rangle=0,\qquad i=1,\ldots,n. \tag{3} \]
Here the subspaces \(L_n\) and \(M_n\) coincide, i.e. \(\tau_n\equiv 1\), and the process always converges, which gives a direct generalization of the least-squares method, whose convergence was first substantiated in \((^9)\).

Remark. As in the general variant of system (2), so also in the case of the least-squares method (3), it is necessary to keep in mind that the process may cease to converge if one of the equations of the system \(\mathcal Lx=f\) or \(\Lambda x=\varphi\) is multiplied by a constant \(\alpha\). In this case the condition \(\tau_n>0\) may be violated. To preserve the convergence of the least-squares method, system (3) should be given the form
\[ (\alpha\mathcal Lx_n-\alpha f,\alpha\mathcal L\varphi_i) +\langle \Lambda x_n-\varphi,\Lambda\varphi_i\rangle=0, \qquad i=1,\ldots,n. \]
Moreover, multiplication of one of the equations by \(\alpha\) will not affect convergence and the estimate if \(n=s+t\). In all cases, multiplication by \(\alpha\) of one of the equations, or even of the functions \(\varphi_i\), may affect the conditioning or stability of the approximate system.

\(5^\circ\). Examples of application of the considerations set forth may be differential equations, where \(\mathcal L\) is a differential operator in a certain domain \(\Omega\), and \(\Lambda\) is a differential operator on the boundary \(\sigma\) of the domain \(\Omega\). It was noted above that convergence always takes place, for example, in the least-squares method, i.e. when
\[ \Psi_i=L\varphi_i=\{\mathcal L\varphi_i,\Lambda\varphi_i\}. \]
In the case of methods of Galerkin type,
\[ \Psi_i=V\varphi_i=\{V\varphi_i,V\varphi_i\}, \]
where \(V\) is the embedding operator from \(H_1\) into \(H_2\). In particular, in \((^6)\)
\[ H_1=W_2^1 \]
with scalar product
\[ (u,v)=\int\left[a_{ik}\frac{\partial u}{\partial x_k}\frac{\partial v}{\partial x_i}+uv\right]d\Omega, \]
and
\[ H_2=L_2(\Omega)\oplus L_2(\sigma). \]
The approximate equations take the form
\[ [Lx_n-F,V\varphi_i]=0. \]
The second-order elliptic differential operator
\[ L=\{\mathcal L,\Lambda\} \]
in \((^6)\) is chosen so that the operator
\[ V^*L=I+A, \]
where \(I\) is the identity, and \(A\) is a completely continuous operator in \(H_1\).

Other examples of applications of the theorems proved may be eigenvalue problems, when the parameter enters not only into the operator $\mathcal L$, but also into the boundary operator $\Lambda$. Important examples of such problems are contained in \({}^{10}\).

The results obtained also make it possible to prove the convergence of various methods of the Trefftz type considered in \({}^{4}\).

Let us also note the case when $\mathcal L$ is an ordinary differential operator of order $n$. Then the space $H$ is Hilbert, while $G$ and the operator $\Lambda$ are finite-dimensional. In this case the Trefftz method degenerates into a trivial problem of linear algebra.

The author expresses sincere gratitude to A. D. Myshkis for a useful discussion of certain questions connected with this work.

Institute of Technical Thermophysics
Academy of Sciences of the Ukrainian SSR

Received
16 VI 1965

CITED LITERATURE

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\({}^{3}\) S. G. Mikhlin, Variational Methods in Mathematical Physics, 1957.
\({}^{4}\) M. Sh. Birman, Vestn. Leningrad Univ., No. 13, ser. math., mech., astr., issue 3 (1956).
\({}^{5}\) Ya. I. Perel’man, Prikl. matem. i mekh., 5, issue 2 (1941).
\({}^{6}\) Yu. I. Trostenberg, DAN, 85, No. 3 (1952).
\({}^{7}\) V. A. Medvedev, DAN, 156, No. 2 (1964).
\({}^{8}\) G. M. Vainikko, Uch. zap. Tartu Univ., issue 150 (IV) (1964).
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\({}^{10}\) S. G. Krein, DAN, 159, No. 2 (1964).