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UDC 517.949.8
MATHEMATICS
Yu. K. DEM’YANOVICH
ON ESTIMATES OF THE RATE OF CONVERGENCE OF SOME PROJECTION METHODS FOR SOLVING ELLIPTIC EQUATIONS
(Presented by Academician V. I. Smirnov on 13 VII 1966)
The purpose of the present work is to obtain new estimates of the rate of convergence for certain schemes of the method of nets, the method of lines, and the method of integral relations in the case of elliptic problems with a positive definite self-adjoint differential operator of order \(2k\). For a one-dimensional fourth-order equation with discontinuous coefficients, estimates of the rate of convergence of the method of nets were obtained by A. A. Samarskii \((^{1})\) and Hao Shou \((^{2})\). V. I. Lebedev \((^{3})\) gave estimates, in the metric of the space \(W_2^k\), for the rate of convergence of the method of nets for the polyharmonic equation with general boundary conditions, depending on the smoothness of the boundary of the domain.
The variational approach used here previously led \((^{4-9})\) to new estimates of the rate of convergence of the method of nets in the case of Dirichlet and Neumann problems for a second-order elliptic-type equation with measurable and bounded coefficients and with a square-summable right-hand side.
\(1^\circ\). Suppose that in the \(n\)-dimensional Euclidean space \(R_n\) there is a certain regular cellular subdivision, and let \(\{\Omega^\nu\}\) be the collection of its \(n\)-dimensional cells, \(\nu \in \mathfrak N\), where \(\mathfrak N\) is some set of indices. Let \(e_1,\ldots,e_n\) be an orthonormal frame in \(R_n\); \(k\) an integer; \(D_{\rho_i}^{k e_i}u(x)\) the \(k\)-th right difference quotient in the direction of the \(i\)-th coordinate axis with step \(\rho_i\) \((\rho_i>0,\ i=1,\ldots,n)\):
\[ D_{\rho_i}^{k e_i}u(x) = \underbrace{D_{\rho_i}^{e_i}\ldots D_{\rho_i}^{e_i}}_{k}u(x), \qquad D_{\rho_i}^{e_i}u(x)=\frac{u(x+\rho_i e_i)-u(x)}{\rho_i}. \tag{1} \]
The set of points at which the function \(u(x)\) must be evaluated in order to construct the difference derivative (1) will be called the scheme of \(D_{\rho_i}^{k e_i}(x)\). By \(u_{\rho^p}(x)\) we shall denote the \(p\)-fold averaging of the function \(u\) over the parallelepiped \(0<\xi_i<\rho_i,\ i=1,\ldots,n\):
\[ u_{\rho^p}(x)=(u_{\rho^{p-1}})_\rho(x),\qquad p=2,\ldots,k;\qquad \rho=(\rho_1,\ldots,\rho_n), \tag{2} \]
\[ u_\rho(x)=(\ldots((u_{\rho_1})_{\rho_2}\ldots)_{\rho_n})(x),\qquad u_{\rho_i}(x)=\frac{1}{\rho_i}\int_0^{\rho_i}u(x+\xi e_i)\,d\xi . \]
We shall call the subdivision \(\{\Omega^\nu\}\) regular if, for each interior point of any \(n\)-dimensional cell \(\Omega^\nu\) \((\nu\in\mathfrak N)\), all points of the scheme \(D_{\rho_i}^{k e_i}(x)\) \((i=1,\ldots,n)\) are interior points of some cells of the subdivision \(\{\Omega^\nu\}\). In each \(n\)-dimensional cell \(\Omega^\nu\) we select a point \(x_0^\nu\) so that the scheme \(D_{\rho_i}^{k e_i}(x_0^\nu)\) consists only of points of the system \(\{x_0^\nu\}\); the system of points \(\{x_0^\nu\}\) will be called a net, and each point \(x_0^\nu\) a node of the net. Let
\[ h=\max_{\nu\in\mathfrak N}\operatorname{diam}\Omega^\nu . \]
In what follows we shall consider functions \(u(x,\tau)\) defined in the direct product \(R_n\times R_s\), \(x\in R_n,\ \tau\in R_s\). Retaining the preceding constructions in the pro-
in the space \(R_n\), consider the functions \(v(x,\tau)\) that are piecewise constant with respect to \(R_n\):
\[ v(x,\tau)=v(x_0^\nu,\tau),\qquad x\in \Omega^\nu \subset R_n,\quad \nu\in \mathfrak N. \]
Let us average them by formula 2, regarding \(\tau\) as a parameter:
\[ \widetilde u(x,\tau)=(v_{\rho}^{k})(x,\tau),\qquad \rho=(\rho_1,\ldots,\rho_n). \tag{3} \]
By \(Q_{(\varepsilon)}\) we denote the \(\varepsilon\)-neighborhood of the domain \(Q\subset R_n\times R_s\), and by \(Q^h\) the set of points of the domain \(Q\) whose distance to its boundary \(\partial Q\) is greater than \(h\). For functions \(u\in \dot W_2^k(Q_{(\varepsilon)})\) we define the modulus of continuity by the formula
\[ \omega_{W_2^k(Q)}(u,h)=\sup_{\|t\|<h}\|u(x+t)-u(x)\|_{W_2^k(Q)},\qquad h<\varepsilon/\sqrt{n+s}. \]
Theorem 1. Let the function \(u\in W_2^k(Q_{(\varepsilon)})\), \(Q=\Omega\times T\) (\(\Omega\) and \(T\) are finite domains in the spaces \(R_n\) and \(R_s\), respectively), and suppose that the partition \(\{\Omega^\nu\}\) in \(R_n\) is regular. Then there exists a function \(\widetilde u(x,\tau)\) of the form (3) such that
\[ \|u-\widetilde u\|_{W_2^k(Q)} \leq C_1\omega_{W_2^k(Q_{(\varepsilon/2)})}(u,a_1\rho_0), \tag{4} \]
where \(\rho_0=\max_{i=1,\ldots,n}\rho_i\), and the constants \(C_1,a_1\) do not depend on \(u,\rho_0\).
Consider the space \(\bar X\) of functions \(v(x_0^\nu,\tau)\in W_2^k(T)\), \(\nu\in\mathfrak N\), continuous with respect to \(\tau\), defined on the direct product \(\{x_0^\nu\}\times T\) and equal to zero for \(x_0^\nu\in \Omega^q\), \(q=k/\sqrt{n+s}\rho_0\), and also vanishing inside some neighborhood of the complement \(CT\) of the domain \(T\) to the whole space \(R_s\). Let \(X\) be the isomorphic space of functions of the form (3), \(v\in X\). By \(Q_t\) we denote the set \(\{y,y-t\in Q\}\), \(t\in R_n\times R_s\).
Suppose that the domain \(Q\) satisfies the following condition:
\((*)\) There exist \(\widetilde N\) unit vectors \(m_1,\ldots,m_{\widetilde N}\) belonging to the \((n+s)\)-dimensional space \(R_n\times R_s\), as well as a positive number \(h_0\) and a function \(\varphi(h)>0\), \(\varphi(h)\to 0\) as \(h\to0\), such that for \(h<h_0\) the boundary strip of width \(h\) lies in the union
\[ \bigcup_{i=1}^{\widetilde N}(Q\setminus Q_{\varphi(h)m_i}). \]
Remark 1. Let \(C^2\) be the class of piecewise twice continuously differentiable surfaces without zero angles. If \(\partial Q\in C^2\), then condition \((*)\) is fulfilled for \(\varphi(h)=k'h\), \(k'=\mathrm{const}>0\).
Theorem 2. Let the function \(u\in \dot W_2^k(Q)\), and let the domain \(Q\) satisfy condition \((*)\). Then there exists in the space \(\bar X\) a function \(v(x_0^\nu,\tau)\) such that the function \(\widetilde u(x,\tau)\) obtained by formula (3) belongs to the class \(\dot W_2^k(Q)\) and the inequality
\[ \|u-\widetilde u\|_{W_2^k(Q)} \leq C_0\omega_{W_2^k(Q)}(u',\varphi(a_0\rho_0)), \tag{5} \]
holds, where \(u'\) is the extension by zero of the function \(u\) outside the domain \(Q\).
Theorem 3. Let \(\widetilde u\in \dot W_2^k(Q)\), \(u(x,\tau)\in W_2^k(\Omega)\), \(\tau\in T\). Then, if condition \((*)\) is fulfilled with respect to the domain \(\Omega\) in the space \(\bar X\), there exists a function \(v(x_0,\tau)\) such that the function \(\widetilde u(x,\tau)\) obtained by formula (3) belongs to \(\dot W_2^k(\Omega)\), \(\tau\in T\), and the inequality
\[ \|u(\tau)-\widetilde u(\tau)\|_{W_2^k(\Omega)} \leq C_0\omega_{W_2^k(\Omega)}(u'(\tau),\varphi(a_0\rho_0)), \]
holds, where \(u'(\tau)\) is the extension by zero of the function \(u(\tau)\) outside the domain \(\Omega\).
\(2^\circ\). The method of integral relations has been used repeatedly in practice \((^{7-11})\) and has yielded positive results, a detailed survey of which is contained in \((^{12})\). Some estimates of the rate of convergence of this method are given in \((^{13})\). Here, for brevity, we shall restrict ourselves only to the first boundary-value problem.
Let \(\alpha\) and \(\beta\) be \((n+s)\)-dimensional vectors:
\(\alpha=(\alpha_1,\ldots,\alpha_{n+s})\), \(\beta=(\beta_1,\ldots,\beta_{n+s})\). Introduce the symbols
\[ \alpha\beta=(\alpha_1\beta_1,\ldots,\alpha_{n+s}\beta_{n+s}); \]
\[ e_0=(\underbrace{1,\ldots,1}_{n},\underbrace{0,\ldots,0}_{s}); \qquad \hat e_0=(\underbrace{0,\ldots,0}_{n},\underbrace{1,\ldots,1}_{s}); \]
\[ y=(x_1,\ldots,x_n,\tau_1,\ldots,\tau_s),\qquad x\in R_n,\quad \tau\in R_s;\qquad |\alpha|=\sum_{i=1}^{n+s}|\alpha_i|, \]
as well as the notation for derivatives, difference ratios, and averages:
\[ D^\alpha=D^{\alpha_1 e_1}\ldots D^{\alpha_{n+s}e_{n+s}}, \qquad D^{\alpha_i e_i}=\frac{\partial^{\alpha_i}}{\partial y_i^{\alpha_i}}, \]
\[ \overline D_{\rho_i}^{-e_i}u(y)=\bigl[u(y)-u(y-\rho_i e_i)\bigr]/\rho_i, \]
\[ \overline D_{\rho_i}^{-\alpha_i e_i} =\underbrace{\overline D_{\rho_i}^{-e_i}\ldots \overline D_{\rho_i}^{-e_i}}_{\alpha_i}, \qquad u_\rho^\alpha=(\ldots(u_{\rho_1}^{\alpha_1})_{\rho_2}^{\alpha_2}\ldots)_{\rho_{n+s}}^{\alpha_{n+s}}, \qquad y=(y_1,\ldots,y_{n+s}), \]
\[ u_{\hat\rho}^{\alpha}=(\ldots(u_{\hat\rho_1}^{\alpha_1})_{\hat\rho_2}^{\alpha_2}\ldots)_{\hat\rho_{n+s}}^{\alpha_{n+s}}, \]
\[ u_{\hat\rho_i}(y)=\frac{1}{\rho_i}\int_0^{\rho_i}u(y-\xi e_i)\,d\xi, \qquad i=1,\ldots,n+s. \]
Consider, in a bounded domain \(Q=\Omega\times T\) of the space \(R_n\times R_s\), the self-adjoint equation
\[ A_0u\equiv \sum_{i=0}^{k}(-1)^i \sum_{|\alpha|=|\beta|=i} D^\alpha a_\alpha^\beta D^\beta u =f(y),\qquad y\in Q. \tag{6} \]
Assume that the coefficients \(a_\alpha^\beta(y)\) are measurable and bounded in the domain \(Q\), and that \(a_\alpha^\beta=a_\beta^\alpha\). Assume also that, for any numbers \(\xi_\alpha\), \(|\alpha|=k\), the inequality
\[ \sum_{|\alpha|=|\beta|=k} a_\alpha^\beta(y)\xi_\alpha\xi_\beta \ge \gamma\sum_{|\alpha|=k}\xi_\alpha^2, \qquad \gamma=\mathrm{const}>0 \tag{7} \]
holds. Let the quadratic form \([u,u]\),
\[ [u,u_1]= \int_Q \sum_{i=0}^{k} \sum_{|\alpha|=|\beta|=i} a_\alpha^\beta D^\alpha uD^\beta u_1\,dy, \tag{8} \]
be positive definite, \([u,u]\ge \mu_0(u,u)\), where
\((u,u_1)=\int_Q uu_1\,dy\). By a solution of the first boundary-value problem for equation (6) we shall mean [14] the solution of the problem of minimizing the quadratic functional
\(F(u)=[u,u]-2(u,f)\) on the space \(W_2^k(Q)\). As an approximate problem we consider the problem of minimizing the aforementioned functional on the space \(\overline X\). Sufficient conditions for the minimum of \(F(u)\) on \(\overline X\) have the form
\[ \overline A_0v\equiv \sum_{i=0}^{k} \sum_{|\alpha|=|\beta|=i} (-1)^{|\alpha|} \overline D_{\rho\alpha}^{\alpha e_0} D^{\alpha\hat e_0} \left( \int_{\Omega^\nu} d_\alpha^\beta(x,\tau) D_{\rho\beta}^{\beta e_0} D^{\beta\hat e_0} v_{\rho k e_0-\alpha e_0}\,dx \right)_{\hat\rho k e_0-\alpha e_0}(x_0^\nu,\tau) = f_{\hat\rho k e_0}(x_0^\nu,\tau), \qquad v(x_0^\nu,\tau)\in \overline X. \tag{9} \]
The operator \(\overline A_0\) is symmetric positive definite and can be extended to a self-adjoint operator in the sense of Friedrichs (we assume that the sta-
the scalar product in the space \(\overline X\) induced by the space \(\mathring X \subset L_2(Q)\), so that the isomorphism (3) is an isometry).
Theorem 4. Under the assumptions formulated with respect to the first boundary-value problem for equation (6), the method of integral relations (9) converges, and the difference between the solution \(u^*\) of the exact problem (6) and the solution \(v_*\) of the approximate problem (9) is estimated by the inequality
\[ \|\widetilde u_*-u^*\|_{W_2^k(Q)} \leq C_0' \,\omega_{W_2^k(Q)}(u^*, a_0\varphi(\rho_0)), \tag{10} \]
where \(\widetilde u_*=(v_*)^{\rho_k}\), \(u^{*'}\) is the extension by zero of the function \(u^*\) to the exterior of the domain \(Q\), and the constants \(C_0'\), \(a_0\) do not depend on \(u^*\), \(\rho_0\).
Corollary. If \(u^* \in W_2^{(k+1)}(Q)\), \(\partial Q \in C^2\), then
\[ \|\widetilde u_*-u^*\|_{W_2^k(Q)} \leq C_0'' \rho_0^{1/2}, \tag{11} \]
where \(C_0''\) does not depend on the parameter \(\rho_0\).
Remark 2. The results extend to the Neumann problem, to the corresponding problems for a strongly elliptic system of differential equations, to certain non-self-adjoint problems (see, for example, (7)), and also to certain nonlinear problems of the form \(Pu=f\); here, with respect to the nonlinear operator \(P\), it is necessary to require that the Gâteaux derivative \(P_u'\) satisfy the inequality
\[ \nu_0\|g\|_{W_2^k(Q)}^2 \leq (P_u'g,g)<\nu_1\|g\|_{W_2^k(Q)}^2, \qquad g\in W_2^k(Q), \]
uniformly in \(u\in D(A)\); \(\nu_i\) are constants independent of \(u\) (see (15), p. 317). In the case of the Neumann problem, for \(u^*\in W_2^{(k+1)}(Q)\) the convergence has order \(\rho_0\). For \(s=0\), system (9) is the method of nets for problem (6).
In conclusion the author expresses his deep gratitude to Prof. S. G. Mikhlin for his attention to this work.
Leningrad State University
named after A. A. Zhdanov
Received
10 VII 1966
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