UDC 517.949.8

MATHEMATICS

E. G. DYAKONOV

ON THE SOLUTION OF CERTAIN NONLINEAR SYSTEMS OF DIFFERENCE EQUATIONS

(Presented by Academician A. N. Tikhonov on 6 III 1969)

The present paper is devoted to the construction of rapidly convergent iterative processes for solving nonlinear systems of difference equations and is conceptually connected with the works \((^{1-8})\), in which a more detailed bibliography can also be found. One of the main results of this paper is the derivation, for difference analogues of the first boundary-value problem in a parallelepiped for strongly elliptic systems in partial derivatives with power nonlinearity, of an iterative method that makes it possible to find the solution of the difference analogue with accuracy \(\varepsilon\) in \(O(N\ln N|\ln\varepsilon|)\) arithmetic operations (\(N\) is the total number of unknowns).

1. Let \(\omega \equiv \{h\}\) be some set in a Banach space, containing a sequence converging to 0, but \(0\notin\omega\). For each \(h\in\omega\) we define: \(\dot H\) is a finite-dimensional real Hilbert space of dimension \(\dot N\) with scalar product \((\dot u,\dot v)\) and \(\|\dot u\|\equiv(\dot u,\dot u)^{1/2}\); \(\dot B\) is a linear self-adjoint and positive operator mapping \(\dot H\) into \(\dot H\); \(\dot H_B\) is a Hilbert space differing from \(\dot H\) by the definition of the scalar product:
   \[ (\dot u,\dot v)_{\dot H_B}\equiv(\dot u,\dot v)_{\dot B}\equiv(\dot B\dot u,\dot v), \]
   \[ \|\dot u\|_B^2\equiv(\dot u,\dot u)_B; \]
   \(\dot L\) is some nonlinear operator mapping \(\dot H\) into \(\dot H\); \(\dot S(\dot u;r)\equiv\{\dot v:\|\dot v-\dot u\|_B\le r\}\), \(r>0\); here and in what follows we indicate the dependence on \(h\in\omega\) of the concepts used by means of a dot over the corresponding symbols.

Definition 1. We shall say that \(\omega,\dot H,\dot L,\dot B\) are connected by the relation \(C^0(\dot u,r)\) if on \([0,r]\) there exist bounded functions \(\delta_0(t)\), \(-\delta_1(t)\), nondecreasing as \(t\) decreases, such that \(\delta_0>0\) and, for all \(\|\dot z\|_B\le r\), the inequalities
\[ \delta_0(\|\dot z\|_B)\|\dot z\|_B^2 \le (\dot L(\dot u+\dot z)-\dot L(\dot u),\dot z), \tag{1} \]
\[ \|\dot B^{-1}(\dot L(\dot u+\dot z)-\dot L(\dot u))\|_B^2 \le \delta_1(\|\dot z\|_B)\|\dot z\|_B^2. \tag{2} \]

Definition 2. We shall say that \(\omega,\dot H,\dot B,\dot L\) are connected by the relation \(C^1(\dot u,r)\) if for any \(\|\dot z\|_B\le r\) there exists
\[ \lim_{\alpha\to0}\alpha^{-1}\bigl(\dot L(\dot u+(1+\alpha)\dot z)-\dot L(\dot u+\dot z)\bigr) \equiv \dot L'(\dot u+\dot z)\dot z, \]
\(\dot L'(\dot u+\dot z)\) is a linear operator continuous with respect to \(\dot z\), and there exist on \([0,r]\) bounded functions \(\sigma_0(t)\), \(-\sigma_1(t)\), \(-\sigma_2(t)\), nondecreasing as \(t\) decreases, such that \(\sigma_0(r)>0\) and the inequalities hold:
\[ \sigma_0(\|\dot z\|_B)\|\dot z\|_B^2 \le (\dot L'_c(\dot u+\dot z)\dot z,\dot z) \le \sigma_1(\|\dot z\|_B)\|\dot z\|_B^2, \tag{3} \]
\[ \|\dot B^{-1}\dot L'_k(\dot u+\dot z)\dot z\|_B^2 \le \sigma_2(\|\dot z\|_B)\|\dot z\|_B^2, \tag{4} \]
where \(\dot L'_c(\dot u+\dot z)\), \(\dot L'_k(\dot u+\dot z)\) are, respectively, the symmetric and skew-symmetric parts of the operator \(\dot L'(\dot u+\dot z)\) in \(\dot H\).

Theorem 1. Suppose that \(\omega,\dot H,\dot B,\dot L\) are connected by the relation \(C^0(\dot u,r)\) or \(C^1(\dot u,r)\) with functions \(\tilde\delta_0(t)\), \(\tilde\delta_1(t)\), \(\tilde\sigma_0(t)\), \(\tilde\sigma_1(t)\), \(\tilde\sigma_2(t)\), and
\[ \dot B\equiv\tilde{\dot B}(\dot E-\dot T_M)^{-1}, \tag{5} \]
where \(\dot T_M\) is a linear self-adjoint operator in the space \(\dot H_{\tilde{\dot B}}\), and \(\|\dot T_M\|_{\tilde{\dot B}}\le q<1\), \(\dot E\) is the identity operator. Then \(\omega,\dot H,\dot B,\dot L\) are connected

with respect to, respectively, \(C^0(\dot u,r')\) or \(C^1(\dot u,r')\), with the functions
\(\delta_0(t)=\bar\delta_0(t')(1-q)\), \(\delta_1(t)=\bar\delta_1(t')(1+q)^2\),
\(\sigma_0(t)=\bar\sigma_0(t')(1-q)\), \(\sigma_1(t)=\bar\sigma_1(t')(1+q)\),
\(\sigma_2(t)=\bar\sigma_2(t')(1+q)^2\), \(r'\equiv r(1-q)^{-1/2}\), \(t'\equiv t(1+q)^{1/2}\).

Theorem 2. Let \(\omega\), \(\dot H\), \(B\), \(\dot L\) be related by the relation \(C^0(\dot u,r)\) or \(C^1(\dot u,r)\), \(\|\dot z\|_B\le r\), \(\gamma>0\), and \(\gamma\le 2(\sigma_0(r)+\sigma_1(r))^{-1}\) in the case \(C^1(\dot u,r)\). Then

\[ \left\|\dot z-\gamma\dot B^{-1}\bigl(\dot L(\dot u+\dot z)-\dot L(\dot u)\bigr)\right\|_B \le \rho(\gamma)\|\dot z\|_B, \tag{6} \]

where \(\rho(\gamma)\) for the case \(C^0(\dot u,r)\) or \(C^1(\dot u,r)\) has, respectively, the form

\[ \rho(\gamma)\equiv \rho_I(\gamma)=\bigl(1-2\gamma\delta_0(r)+\gamma^2\delta_1(r)\bigr)^{1/2}, \tag{7} \]

\[ \rho(\gamma)\equiv \rho_{II}(\gamma)= \left\{1-2\gamma[\sigma_0(r)+\sigma_1(r)] +\gamma^2\left[\sigma_2(r)+\left(\frac{\sigma_0(r)+\sigma_1(r)}{2}\right)^2\right]\right\}^{1/2} +\frac{\sigma_1(r)-\sigma_0(r)}{2}\gamma; \tag{8} \]

\[ \min_\gamma \rho_I(\gamma)=\rho_I(\gamma_I)<1,\qquad \min_\gamma \rho_{II}(\gamma)=\rho_{II}(\gamma_{II})<1. \tag{9} \]

It follows from Theorem 2 that, for \(\dot u^0\in S(\dot u,r)\) and a suitable choice of \(\gamma_k\), the functions \(\dot u^{k+1}\) obtained from the iterative process

\[ \dot B\dot u^{k+1}=\dot B\dot u^k-\gamma_k\bigl(\dot L(\dot u^k)-\dot L(\dot u)\bigr) \tag{10} \]

converge to \(\dot u\).

2. We present some of the studied types of nonlinear systems of difference equations for which the results of Sec. 1 are applicable. Let \(\Omega\) be a bounded domain in the space \(x\equiv(x_1,x_2,\ldots,x_n)\); \(h_s>0\) the mesh step in \(x_s\); \(h\equiv(h_1,h_2,\ldots,h_n)\); \(\omega=\{h:\max_s h_s\le h_0,\ h_0>0\}\); \(i\equiv(i_1,i_2,\ldots,i_n)\) a vector with integer components; \(x_i\equiv(i_1h_1,i_2h_2,\ldots,i_nh_n)\); \(\dot\Omega\equiv\{x_i:x_i\in\Omega\}\); \(\dot H\) a Hilbert space of mesh vector-functions \(\dot u\equiv(\dot u_1,\dot u_2,\ldots,\dot u_N)\), defined on \(\dot\Omega\) and equal to zero outside \(\dot\Omega\);

\[ (\dot u,\dot v)=h_1h_2\cdots h_n \sum_{r=1}^{N}\sum_{x_i\in\dot\Omega}\dot u_r(x_i)\dot v_r(x_i), \qquad \|\dot u\|\equiv(\dot u,\dot u)^{1/2}. \]

Let \(\{\alpha\}\) be the space of vectors \(\alpha\equiv(\alpha_1,\ldots,\alpha_n)\), where \(\alpha_s\) are natural numbers, \(|\alpha|\equiv\sum_{s=1}^{n}\alpha_s\), and let the operators \(\dot B\) and \(\dot P\) be defined in \(\dot H\) by

\[ \dot B\dot u\equiv(\dot B_1\dot u_1,\ldots,\dot B_N\dot u_N), \qquad \dot B_r\dot u_r\equiv(-1)^{m_r}\sum_{s=1}^{n}\bar\partial_s^{\,m_r}\partial_s^{\,m_r}\dot u_r, \tag{11} \]

\[ \dot P(\dot u)\equiv(\dot P_1(\dot u),\ldots,\dot P_N(\dot u)), \]

\[ \dot P_r(\dot u)\equiv \lambda\sum_{|\alpha|\le m_r}(-1)^{|\alpha|} \left[ \bar\partial^{\,\alpha}a_{r,\alpha}(x,\partial^\beta\dot u_l) + \partial^\alpha a_{r,\alpha}(x,\bar\partial^\beta\dot u_l) \right], \tag{12} \]

where \(\bar\partial_s,\partial_s\) are the left and right differences with respect to \(x_s\); \(\bar\partial^\alpha\equiv\bar\partial_1^{\alpha_1}\cdots\bar\partial_n^{\alpha_n}\); \(m_r\ge1\); \(\beta\in\{\alpha\}\); \(a_{r,\alpha}\) depend, generally speaking, on \(x\) and all \(\partial^\beta\dot u_l(\bar\partial^\beta\dot u_l)\), \(|\beta|\le m_l\), \(1\le l\le N\); \(\lambda\) is a number; let
\(\psi\equiv\{(r,\alpha):m_r-|\alpha|>n/2,\ 1\le r\le N\}\),
\(\psi^*\equiv\{(r,\alpha):|\alpha|\le m_r,\ (r,\alpha)\notin\psi,\ 1\le r\le N\}\), and
\(d_{r,\alpha,l,\beta}(|t^*_{k,\gamma}|,|\tau^*_{k,\gamma}|)\) denote the sum of terms of the form
\(\prod_{k,\gamma}|t^*_{k,\gamma}|^{q_{k,\gamma}}|\tau^*_{k,\gamma}|^{p_{k,\gamma}}\), where \(\gamma\in\{\alpha\}\), \((k,\gamma)\in\psi^*\), and \(q_{k,\gamma}\ge0\), \(p_{k,\gamma}\ge0\) are such that, for \(u_k,v_k,z_k\in W_2^{m_k}\), \(1\le k\le N\), the functions
\(\left(\prod_{k,\gamma}|D^\gamma u_k|^{q_{k,\gamma}}|D^\gamma v_k|^{p_{k,\gamma}}\right)|D^\alpha z_r|\in L_1(\Omega)\), by virtue of the inequalities-

of Hölder and embedding theorems of S. L. Sobolev. Continuous functions of \(t,\tau\), monotonically increasing in \(t,\tau\), will be denoted by \(\chi(t,\tau)\); \(\chi(0,0)=0\); \(K_0,K\) are constants.

Theorem 3. Let all \(a_{r,\alpha}\in \dot P\) be such that for any \(K_0>0\), when
\[ \max_{(l,\beta)\in\Psi}\max\bigl(|t_{l,\beta}|,|\tau_{l,\beta}|\bigr)\le K_0 \]
there is a \(K(K_0)\) such that
\[ \bigl|a_{r,\alpha}(x,t_{l,\beta}+\tau_{l,\beta})-a_{r,\alpha}(x,t_{l,\beta})\bigr| \le K(K_0)\sum_{l,\beta}|\tau_{l,\beta}|\, d_{r,\alpha,l,\beta}\bigl(|t^*_{l,\beta}|,|\tau^*_{l,\beta}|\bigr). \tag{13} \]
Then there exist \(\chi_1(t,\tau),\chi_2(t,\tau)\) such that, for \(u\in S(v,r)\), \(u+z\in S(v,r)\),
\[ |(\dot P(u+z)-\dot P(u),z)| \le |\lambda|\chi_1\bigl(\|u\|_{\widetilde B},\|z\|_{\widetilde B}\bigr)\|z\|_{\widetilde B}^{2}, \tag{14} \]
\[ (\dot{\widetilde B}^{-1}(\dot P(u+z)-\dot P(u)),\ \dot P(u+z)-\dot P(u)) \le |\lambda|^2\chi_2\bigl(\|u\|_{\widetilde B},\|z\|_{\widetilde B}\bigr)\|z\|_{\widetilde B}^{2}. \tag{15} \]

Theorem 4. Let all derivatives \(\partial a_{r,\alpha}(x,t_{l,\beta})/\partial t_{l,\beta}\) be continuous in the space \((x,t_{l,\beta})\) and, when
\[ \max_{(l,\beta)\in\Psi}\max\bigl(|t_{l,\beta}|,|\tau_{l,\beta}|\bigr)\le K_0, \]
\[ \left| \frac{\partial a_{r,\alpha}}{\partial t_{l,\beta}}(x,t_{l,\beta}+\tau_{l,\beta}) \right| \le K(K_0)d_{r,\alpha,l,\beta}\bigl(|t^*_{l,\beta}|,|\tau^*_{l,\beta}|\bigr). \tag{16} \]
Then there exist \(\chi_3(t,\tau),\chi_4(t,\tau)\) such that, for \(u\in S(v,r)\), \(u+z\in S(v,r)\),
\[ |(\dot P'_c(u+z)z,z)| \le \chi_3\bigl(\|u\|_{\widetilde B},\|z\|_{\widetilde B}\bigr)\|z\|_{\widetilde B}^{2}; \]
\[ (\dot{\widetilde B}^{-1}\dot P'_k(u+z)z,\ \dot P'_k(u+z)z) \le \chi_4\bigl(\|u\|_{\widetilde B},\|z\|_{\widetilde B}\bigr)\|z\|_{\widetilde B}^{2}, \tag{17} \]
where \(\dot P'(u+z)z\equiv(\dot P'_1z,\ldots,\dot P'_Nz)\),
\[ \dot P'_rz\equiv \lambda\sum_{|\alpha|\le m_r}(-1)^{|\alpha|} \left\{ \bar\partial^\alpha \sum_{l=1}^{N}\sum_{|\beta|\le m_l} \frac{\partial a_{r,\alpha}}{\partial t_{l,\beta}}(x,\partial^\beta u_l+\partial^\beta z_l)\partial^\beta z_l +\right. \]
\[ \left. +\partial^\alpha \sum_{l=1}^{N}\sum_{|\beta|\le m_l} \frac{\partial a_{r,\alpha}}{\partial t_{l,\beta}}(x,\bar\partial^\beta u_l+\bar\partial^\beta z_l)\bar\partial^\beta z_l \right\}, \tag{18} \]
\(\dot P'(u+z)z\) is the Gâteaux differential for \(\dot P(u+z)\), while \(\dot P'_c(u+z)\), \(\dot P'_k(u+z)\) are the symmetric and skew-symmetric parts in \(\dot H\) of the operator \(\dot P'(u+z)\).

Theorem 5. Let
\[ \dot\Lambda u\equiv(\Lambda_1u,\ldots,\Lambda_Nu),\qquad \dot L(u)\equiv \dot\Lambda u+\dot P(u), \tag{19} \]
where
\[ \Lambda_ru\equiv \sum_{l=1}^{N}\sum_{|\alpha|\le m_r}(-1)^{|\alpha|} \left[ \bar\partial^\alpha \sum_{|\beta|\le m_l} b_{r,l,\alpha,\beta}(x)\partial^\beta u_l+ \partial^\alpha \sum_{|\beta|\le m_l} b_{r,l,\alpha,\beta}(x)\bar\partial^\beta u_l \right]; \]
\(\omega,\dot H,\dot{\widetilde B},\dot\Lambda\) are related by the relation \(C^0(u,r)\) \((C^1(u,r))\), and the conditions of Theorem 3 (4) are fulfilled.*

Then, for any \(r>0\), there is a \(\lambda_0(r)>0\) such that, for \(|\lambda|\le\lambda_0\), \(\omega,\dot H,\dot{\widetilde B},\dot L\) are also related by the relation \(C^0(u,r)\) \((C^1(u,r))\).

Theorem 6. Let the conditions of Theorem 5 be fulfilled, \(|\lambda|\le\lambda_0\), \(\Omega\) be an \(n\)-dimensional parallelepiped, and for solving the equation
\[ L(u)=f \tag{20} \]
the iterative process (10) with
\[ \dot Bu\equiv(\dot B_1u_1,\ldots,\dot B_Nu_N),\qquad \dot B_l\equiv \dot{\widetilde B}_l(E-\dot T_{M,l})^{-1},\qquad \|\dot T_{M,l}\|_{\widetilde B}\le q<1, \tag{21} \]
is used

where \(E\) is the identity operator, \(\dot T_{M,l}\) is the error-reduction operator in the variable-directions method (see \((^{2,3})\)), \(u^0\in S(\dot u,r)\), and \(\gamma\) is defined from (9).

Then, in order to obtain the estimate

\[ \|u^{n+1}-\dot u\|_{\widetilde B}\leqslant \varepsilon\|u^0-\dot u\|_{\widetilde B} \tag{22} \]

it is sufficient to expend

\[ O\left(|\ln\varepsilon|\frac{|\ln\min_s h_s|}{h_1h_2\ldots h_n}\right) \]

arithmetic operations; if \(n=2\), all \(m_l=1\), and \(T_{M,l}\) corresponds to the iterative process \((^8)\), then, with the corresponding choice of \(\gamma\), to obtain (22) it is sufficient to expend

\[ O\left(|\ln\varepsilon|\frac{1}{h_1h_2}\right) \]

operations.

3. The types of nonlinear systems presented in Sec. 2 occur in the numerical solution of many important applied problems, for example nonlinear problems of the theory of elasticity (see \((^9)\)). Difference schemes for them can be chosen so that the potentiality of the differential operator is preserved, and therefore in (4) one may take \(\sigma_2(t)=0\). For such problems \(\gamma\) in Theorem 6 may be chosen on the basis of the method of steepest descent (see \((^5)\)).

Nonpotential difference operators arise, for example, in the approximation of nonlinear problems in the theory of thin plates (see \((^{10})\)). Difference schemes for them can be chosen so that \((\dot P(u),u)=0\). This makes it possible to obtain a priori estimates and existence theorems (see, for example, \((^{11})\)); for the problem from \((^{10})\) the following system may serve as such a scheme:

\[ c_1(\Lambda_1+\Lambda_2)^2w_i-\Lambda_1w_i\Lambda_2F_i-\Lambda_2w_i\Lambda_1F_i+ \bar\partial_1\bar\partial_2w_i\bar\partial_1\bar\partial_2F_i+ \partial_1\partial_2w_i\partial_1\partial_2F_i=f_i, \tag{23} \]

\[ \begin{aligned} &c_2(\Lambda_1+\Lambda_2)^2F_i-\widetilde\partial_1\widetilde\partial_2w_i (\bar\partial_1\bar\partial_2w_i+\partial_1\partial_2w_i) +\Lambda_1w_iT_1T_{-1}\Lambda_2w_i+{}\\ &\quad+\Lambda_2w_iT_2T_{-2}\Lambda_1w_i -\frac12(\partial_1\partial_2w_i-\bar\partial_1\bar\partial_2w_i) (h_2\widetilde\partial_1\Lambda_2w_i+h_1\widetilde\partial_2\Lambda_1w_i)-{}\\ &\quad-\frac{h_1^2+h_2^2}{2}\Lambda_{1/2}w_i\Lambda_1\Lambda_2w_i=0, \qquad x_i\in\dot\Omega, \end{aligned} \tag{24} \]

where \(w_i\equiv w(x_i)\), \(\Lambda_s\equiv \bar\partial_s\partial_s\), \(\widetilde\partial_s\equiv {}^1\!/_{2}(\partial_s+\bar\partial_s)\), \(T_sw_i\equiv {}^1\!/_{2}(w(x_i)+w(x_i+h_se_s))\), \(T_{-s}w_i\equiv {}^1\!/_{2}(w(x_i)+w(x_i-h_se_s))\), \(e_1\equiv(1,0)\), \(e_2\equiv(0,1)\),

\[ \Lambda_{1/2}w_i\equiv \frac{1}{h_1^2+h_2^2} \bigl(w(x_i+h_1e_1+h_2e_2)-2w(x_i)+w(x_i-h_1e_1-h_2e_2)\bigr). \]

Equations (23), (24) approximate the equations from \((^{10})\) with accuracy \(O(h_1^2+h_2^2)\). The constructions presented are applicable also to more general boundary-value problems.

Moscow State University
named after M. V. Lomonosov

Received
17 II 1969

REFERENCES

\(^{1}\) E. G. Dyakonov, DAN, 163, No. 6 (1965).
\(^{2}\) E. G. Dyakonov, Zhurn. vychisl. matem. i matem. fiz., 6, No. 1, No. 4 (1966).
\(^{3}\) E. G. Dyakonov, Iterative Methods for Solving Difference Analogues of Boundary-Value Problems for Equations in Partial Derivatives of Elliptic Type, Kiev, 1969.
\(^{4}\) J. E. Gunn, J. Soc. Ind. Appl. Math., 2, No. 1 (1965).
\(^{5}\) M. N. Yakovlev, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 84 (1965).
\(^{6}\) A. I. Koshelev, DAN, 172, No. 2 (1967).
\(^{7}\) W. V. Petryshyn, Illinois J. Math., 10, No. 2 (1966).
\(^{8}\) N. S. Bakhvalov, Zhurn. vychisl. matem. i matem. fiz., 6, No. 4 (1966).
\(^{9}\) A. Lyangenbakh, Vestn. LGU, No. 1 (1961).
\(^{10}\) N. F. Morozov, Vestn. LGU, No. 19 (1958).
\(^{11}\) Yu. A. Dubinskii, DAN, 176, No. 3 (1967).