MATHEMATICS

E. G. DYAKONOV

ON AN ITERATIVE METHOD FOR SOLVING SYSTEMS OF FINITE-DIFFERENCE EQUATIONS

(Presented by Academician S. L. Sobolev on 29 XII 1960)

A number of works \((^{1-6})\) have been devoted to the study of the asymptotics of the number of arithmetic operations needed to find solutions of problems of mathematical physics. In the present paper an iterative method is constructed for solving systems of finite-difference equations approximating elliptic equations of order \(2m\), which requires, in order to find the solution of the system with accuracy \(\varepsilon\), a number of arithmetic operations

\[ \asymp \frac{\ln^2 h}{h^2}\ln \varepsilon, \]

where \(h\) is the mesh size. A better asymptotic estimate had previously been obtained only for narrower classes of problems.

1. In the square \(D: 0 \le x \le 1,\ 0 \le y \le 1\), we seek the solution of the self-adjoint elliptic equation (1), satisfying the boundary conditions (2):

\[ Lu = (-1)^m \sum_{|\alpha|=m} D_\alpha (a_\alpha D_\alpha u) + (-1)^{|\beta|} \sum_{|\beta|=m} D_\beta (b_\beta, D_\beta u) = f; \tag{1} \]

\[ \left(u,\ \frac{\partial u}{\partial \nu},\ \ldots,\ \frac{\partial^{m-1}u}{\partial \nu^{m-1}}\right)_S = (0,0,\ldots,0). \tag{2} \]

Here \(\alpha,\beta\) are two-dimensional differentiation vectors; \(a_\alpha, b_\beta, f\) are functions of \(x\) and \(y\); \(\nu\) is the normal to \(S\), the boundary of the domain \(D\);

\[ a_\alpha \in C^{(m+1)};\qquad b_\beta \in C^{(|\beta|+1)};\qquad a_\alpha \ge 0;\qquad b_\beta \ge 0;\qquad a_{(m,0)} > 0;\qquad a_{(0,m)} > 0. \]

We construct a difference approximation of our problem. Let \(D_h\) be the set of points

\[ x_i = ih,\quad y_j = jh,\quad \text{where } h=1/N,\quad 0 \le i \le N,\quad 0 \le j \le N. \tag{3} \]

The set of points for which \(0 \le i \le m-1\) or \(N-m+1 \le i \le N\), or the same condition holds for \(j\), will be denoted by \(S_h\).

\[ u_{ij}=0,\qquad \text{if } i,j \in S_h. \tag{4} \]

For those \(u_{ij}\) for which \(i,j \in D_h \setminus S_h\), we obtain the system of finite-difference equations

\[ L_h u = (-1)^m \sum_{|\alpha|=m} D_{\bar{\alpha}}^h (a_\alpha D_\alpha^h u) + (-1)^{|\beta|} \sum_{|\beta|<m} D_{\bar{\beta}}^h (b_\beta D_\beta^h u) = f; \tag{5} \]

\(D_\alpha^h\) denotes the ordinary “right” differences in the vector \(\alpha\), \(D_{\bar{\alpha}}^h\) the “left” differences; \(u\) is an \((N+1)^2\)-dimensional vector satisfying (4). Such a difference approximation was considered in \((^7)\), where convergence to the solution of (1), (2) was established for it.

Theorem 1. The difference approximation (4)—(5) preserves the properties of self-adjointness and positivity of the differential operators.

The proof is based on the fact that (5) is obtained from the condition of the minimum of
\[ \sum_{ij} h^2 \sum_\alpha (a_\alpha D_\alpha^h u_{ij})^2 + \sum_{ij} h^2 \sum_\beta (b_\beta D_\beta^h u_{ij})^2 - \sum_{ij} h^2 f_{ij} u_{ij} \]
on the class of functions (4), as was established in (8) for equations with constant coefficients.

The various transformations of sums are carried out by means of the method of (9). Let us now consider the question of finding the solution of (4), (5).

1. Let
   \[ M_h u = (-1)^m \left(D_{(m,0)}^h D_{\overline{(m,0)}}^h + D_{(0,m)}^h D_{\overline{(0,m)}}^h\right)u . \]

Theorem 2. If \(\psi\) satisfies (4) and \((M_h\psi,\psi)=1\), then there exist constants \(m_0\) and \(M_0\), independent of \(h\), such that
\[ 0<m_0 \le (L_h\psi,\psi)\le M_0 . \tag{6} \]

Here \((\varphi,\psi)\) denotes
\[ \sum_{i=0}^{N}\sum_{j=0}^{N}\varphi_{ij}\psi_{ij}h^2 . \]

Proof.
\[ (L_h\psi,\psi)\ge (-1)^m\left[(D_{(m,0)}^h a_{(m,0)}D_{(m,0)}^h\psi,\psi) + (D_{(0,m)}^h a_{(0,m)}D_{(0,m)}^h\psi,\psi)\right] \ge \]
\[ \ge \min\{a_{(m,0)},a_{(0,m)}\} \left[\sum_{ij}h^2\left[(D_{(m,0)}^h\psi_{ij})^2+(D_{(0,m)}^h\psi_{ij})^2\right]\right] = m_0(M_h\psi,\psi). \tag{7} \]

To obtain the upper estimate in (6), it is sufficient to obtain the estimate
\[ (-1)^m \sum_{|\alpha|=m}(D_\alpha^h D_\alpha^h\psi,\psi) \le k(M_h\psi,\psi), \tag{8} \]
since
\[ (-1)^m \sum_{|\alpha|=m}(D_\alpha^h a_\alpha D_\alpha^h\psi,\psi) = \sum_{ij}h^2\sum_{|\alpha|=m}a_\alpha(D_\alpha^h\psi_{ij})^2 \le \]
\[ \le \max_{i,j,\alpha} a_\alpha \sum_{ij}h^2\sum_{|\alpha|=m}(D_\alpha^h\psi_{ij})^2 \le (-1)^m \max_{ij\alpha} a_\alpha \sum_{|\alpha|=m}(D_\alpha^h D_\alpha^h\psi,\psi), \]
and for the other expressions of the form
\[ (-1)^{|\beta|}(D_{\overline{\beta}}^h b_\beta D_\beta^h\psi,\psi) \]
an analogous estimate is obtained through
\[ (-1)^{|\beta|}(D_{\overline{\beta}}^hD_\beta^h\psi,\psi) \]
and, consequently, through
\[ (-1)^m(D_{\overline{\alpha}}^h\psi,\psi). \]

We proceed to the proof of (8). Consider the expansion in
\[ v_{kl}(i,j)=c\sin k\pi hi\,\sin l\pi hj, \]
where \(1\le k,l\le N-1\), and
\[ (v_{kl},v_{k'l'})= \begin{cases} 1, & (k-k')^2+(l-l')^2=0,\\ 0, & (k-k')^2+(l-l')^2\ne 0; \end{cases} \tag{9} \]
\[ \Delta_{\overline{x}x}v_{kl}=-\lambda_{kl}^2v_{kl}; \qquad \Delta_{\overline{y}y}v_{kl}=-\mu_{kl}^2v_{kl}; \qquad \psi=\sum_{k,l}d_{kl}v_{kl}. \]

Then, by virtue of the boundary conditions (4), if \(2\alpha=(2m_1,2m-2m_1)\), then
\[ (-1)^m(D_\alpha^h\psi,\psi) = \sum_{i,j=0}^{N}h^2 \left[ \sum_{k,l=0}^{N-1} d_{kl}\lambda_{kl}^{2m_1}\mu_{kl}^{2m-2m_1}v_{kl}(i,j) \right] \left[ \sum_{k,l}d_{kl}v_{kl}(i,j) \right] = \]
\[ = \sum_{k,l=1}^{N-1}d_{kl}^2\lambda_{kl}^{2m_1}\mu_{kl}^{2m-2m_1} \le c_1 \sum_{k,l=1}^{N-1}d_{kl}^2(\lambda_{kl}^{2m}+\mu_{kl}^{2m}), \tag{10} \]
since \((\lambda^2+\mu^2)^m<c_1(\lambda^{2m}+\mu^{2m})\), and \(c_1>0\) does not depend on \(h\).

On the other hand, in an analogous way we verify that

\[ (M_h\psi,\psi)=\sum_{k,l=1}^{N-1} d_{kl}^{2}\left(\lambda_{kl}^{2m}+\mu_{kl}^{2m}\right). \tag{11} \]

Comparing (10) and (11), we establish the validity of (8), and hence of the theorem.

3. Let us construct an iterative process for finding the solution of (4), (5). Suppose that some approximation \(v^{(n)}\) is given. We shall obtain the next approximation from

\[ M_h u^{(n+1)}=M_h v^{(n)}-\tau\left(L_h v^{(n)}-f\right), \tag{12} \]

where \(\tau\) is the iteration parameter. For the errors \(e^{(n)}=u-v^{(n)}\), \(e^{(n+1)}=u-u^{(n+1)}\), we have

\[ M_h e^{(n+1)}=(M_h-\tau L_h)e^{(n)};\qquad e^{(n)}=0 \text{ on } S_h. \tag{13} \]

Since \(M_h=M_h^{*}\), \((M_h-\tau L_h)^{*}=M_h-\tau L_h\), and \(M_h\) is a positive definite operator, there exists a system of functions \(\{\psi_i\}\), satisfying (4) and forming an orthonormal basis in the metric \(\|\psi\|_{M}=(M_h\psi,\psi)\), such that

\[ \{M_h-\tau L_h\}\psi_i=\lambda_i M\psi_i;\qquad (M\psi_i,\psi_j)=\delta_{ij}= \begin{cases} 0, & i\ne j,\\ 1, & i=j. \end{cases} \tag{14} \]

Since

\[ (\psi,\psi)\le k_1(M\psi,\psi), \tag{15} \]

where \(k_1\) does not depend on \(h\), the system \(\{\psi_i\}\) is complete also in the ordinary metric.

\[ e^{(n+1)}=\sum_i d_i^{(n+1)}\psi_i;\qquad e^{(n)}=\sum_i d_i^{(n)}\psi_i. \]

Substituting these expansions into (13), we obtain \(d_i^{(n+1)}=d_i^{(n)}\lambda_i\). From (14),

\[ \lambda_i=\left[(M_h-\tau L_h)\psi_i,\psi_i\right] =1-\tau(L_h\psi_i,\psi_i). \tag{16} \]

By Theorem 2, \(m_0\le (L_h\psi_i,\psi_i)\le M_0\); therefore one can find \(\tau>0\) and \(0<q<1\), independent of \(h\) and \(\psi_i\), such that
\(-q\le 1-\tau M_0\le 1-\tau(L_h\psi_i,\psi_i)\le 1-\tau m_0\le q\), or else

\[ \frac{1-q}{m_0}\le \tau\le \frac{1+q}{M_0}. \tag{17} \]

Then

\[ \left|d_i^{(n+1)}\right|\le q\left|d_i^{(n)}\right|;\qquad \left\|u-u^{(n+1)}\right\|_M\le q\left\|u-v^{(n)}\right\|_M. \tag{18} \]

But we are also forced to solve equation (12) approximately, and therefore we shall always obtain some \(v^{(n+1)}\), and not \(u^{(n+1)}\).

We shall solve equation (12) by the method of variable directions, taking \(v^{(n)}\) as the initial approximation for \(u^{(n+1)}\).* Then, as established in (10), in order that

\[ \left\|u^{(n+1)}-v^{(n+1)}\right\| \le \varepsilon_1\left\|v^{(n)}-u^{(n+1)}\right\|, \tag{19} \]

it is sufficient to perform \(\sim \ln\varepsilon_1 \ln h\) iterations; the number of arithmetic operations performed thereby will be of order \(h^{-2}\ln h\,\ln\varepsilon_1\).

* The advisability of choosing such an initial approximation was pointed out to me by N. S. Bakhvalov.

Since \(\|\psi\|_{M}\le \dfrac{c_2}{h^{2m}}\|\psi\|\), where \(c_2\) does not depend on \(h\), it follows from (19) that

\[ \|u^{(n+1)}-v^{(n+1)}\|_{M}\le \frac{c_2}{h^{2m}}\|u^{(n+1)}-v^{(n+1)}\| \le \frac{c_2\varepsilon_1}{h^{2m}}\|v^{(n)}-u^{(n+1)}\|\le \]
\[ \le \frac{\varepsilon_1 c_3}{h^{2m}}\|v^{(n)}-u^{(n+1)}\|_{M}, \tag{20} \]

i.e.,

\[ \|u^{(n+1)}-v^{(n+1)}\|_{M}\le \frac{\varepsilon_1 c_3}{h^{2m}}\|v^{(n)}-u^{(n+1)}\|_{M} \]

(\(c_3\) does not depend on \(h\)).

Take \(\varepsilon_1=h^{2m}q_1/c_3\), where \(q_1>0\) is a number independent of \(h\) such that \(q+q_1q+q_1=r<1\). Then (20) can be rewritten as

\[ \|u^{(n+1)}-v^{(n+1)}\|_{M}\le q_1\|v^{(n)}-u^{(n+1)}\|_{M}. \tag{21} \]

Using (18) and (21):

\[ \|u-v^{(n+1)}\|_{M}\le \|u-u^{(n+1)}\|_{M}+\|u^{(n+1)}-v^{(n+1)}\|_{M}\le \]
\[ \le q\|u-v^{(n)}\|_{M}+q_1\|v^{(n)}-u^{(n+1)}\|_{M}\le \]
\[ \le q\|u-v^{(n)}\|_{M}+q_1\|u-u^{(n+1)}\|_{M}+q_1\|u-v^{(n)}\|_{M}\le \]
\[ \le q\|u-v^{(n)}\|_{M}+qq_1\|u-v^{(n)}\|_{M}+q_1\|u-v^{(n)}\|_{M}= \]
\[ =(q+q_1q+q_1)\|u-v^{(n)}\|_{M} =r\|u-v^{(n)}\|_{M}. \]

Thus, although we do not find \(u^{(n+1)}\) exactly, but obtain only \(v^{(n+1)}\)—some approximation to \(u^{(n+1)}\), convergence in the metric \(M\) is nevertheless obtained with the rate of convergence of a geometric progression with ratio \(r<1\), and in order to obtain \(\|v^{(n+1)}-u\|_{M}\le \varepsilon\|v^{(0)}-u\|_{M}\), it suffices to perform \(\sim \ln \varepsilon\) iterations of the form (12). Since each such iteration requires \(\sim \ln h\,\ln\varepsilon_1 \sim \ln^2 h\) iterations by the method of alternating directions, the total will be \(\sim \ln^2 h\,\ln\varepsilon\) iterations with the number of arithmetic operations \(\sim h^{-2}\ln^2 h\,\ln\varepsilon\).

Taking (15) into account, we are convinced of the validity of Theorem 3.

Theorem 3. The iterative process (12), in which at each step \(v^{(n)}\) is taken as the initial approximation for \(u^{(n+1)}\) and the error is reduced by the method of alternating directions by a factor \(\varepsilon_1\), requires, for finding the solution of (4), (5) with accuracy \(\varepsilon\), \(\sim h^{-2}\ln^2 h\,\ln\varepsilon\) arithmetic operations.

Remark. The main results of the work can be extended to the multidimensional case.

Moscow State University
named after M. V. Lomonosov

Received
27 XII 1960

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