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UDC 518:517.944/.947
MATHEMATICS
M. N. YAKOVLEV
ON THE THEORY OF THE METHOD OF FINITE DIFFERENCES FOR SOLVING NONLINEAR BOUNDARY-VALUE PROBLEMS
(Presented by Academician L. V. Kantorovich on 2 IV 1966)
I. Consider the following Dirichlet problem. Find a function \(u(x,y)\), twice continuously differentiable in the closed domain \(\overline D\), satisfying the differential equation
\[ F(x,y,u,u_x,u_y,u_{xx},u_{yy})=0 \tag{1} \]
and assuming continuous values on the piecewise smooth boundary
\[ u|_{\Gamma}=\varphi(x,y). \tag{2} \]
Here the function \(F(x,y,u,p,q,r,t)\) is continuously differentiable with respect to all its arguments and satisfies the inequalities
\[ F_r(x,y,u,p,q,r,t)\ge 0,\qquad F_t(x,y,u,p,q,r,t)\ge 0, \]
\[ F_r(x,y,u,p,q,r,t)+F_t(x,y,u,p,q,r,t)\ge m>0, \]
\[ F_u(x,y,u,p,q,r,t)\le 0 \]
for \(x,y\in \overline D\), \(-\infty<u,p,q,r,t<+\infty\).
Under these conditions, for the solution of the Dirichlet problem (1), (2), a uniqueness theorem is valid; the existence of the solution is assumed by us.
In the usual way we introduce a square mesh with step \(h\), the mesh domain \(D_h\), and its boundary in the sense of the five-point star \(\Gamma_h\). We divide the boundary points into two groups \(\Gamma_h^{(1)}\) and \(\Gamma_h^{(2)}\), \(\Gamma_h=\Gamma_h^{(1)}+\Gamma_h^{(2)}\). To \(\Gamma_h^{(1)}\) we assign all points of \(\Gamma_h\) that belong to \(\Gamma\), and to \(\Gamma_h^{(2)}\) all the remaining points of \(\Gamma_h\).
The boundary conditions are approximated in two ways:
A. Simple transfer. We set
\[ u_{ij}=\varphi(x_i,y_j),\quad \text{if } x_i,y_j\in \Gamma_h^{(1)}, \]
\[ u_{ij}=\varphi(\bar x_i,\bar y_j),\quad \text{if } x_i,y_j\in \Gamma_h^{(2)}. \tag{3} \]
Here \(\bar x_i,\bar y_j\) is the point of \(\Gamma\) nearest to \(x_i,y_j\).
B. Collatz interpolation. We set
\[ u_{i,j}=\varphi(x_i,y_j), \qquad \text{if } x_i,y_j\in \Gamma_h^1, \]
\[ u_{i,j}=\frac{\delta}{\delta+h}\,u_{i,j-1} +\frac{h}{\delta+h}\,\varphi(\bar x_i,\bar y_j), \quad \text{if } x_j y_j\in \Gamma_h^{(2)}. \tag{4} \]
(For more details, see, for example, \((^1)\).)
- As the difference equation approximating the differential equation (1), consider the equation
\[ \Gamma\left( x_i,y_j,u_{i,j}, \frac{u_{i+1,j}-u_{i-1,j}}{2h}, \frac{u_{i,j+1}-u_{i,j-1}}{2h}, \frac{u_{i+1,j}-2u_{i,j}+u_{i-1,j}}{h^2}, \right. \]
\[ \left. \frac{u_{i,j+1}-2u_{i,j}+u_{i,j-1}}{h^2} \right)=0 \tag{5} \]
for \(x_i y_j\in D_h\).
Theorem 1. Let
\[ \left|F_p(x,y,u,p,q,r,t)\right|\leq M F_r(x,y,u,p,q,r,t), \]
\[ \left|F_q(x,y,u,p,q,r,t)\right|\leq M F_t(x,y,u,p,q,r,t) \]
for \(x,y\in \overline D\), \(-\infty<u,p,q,r,t<+\infty\).
Then, for \(h<1/M\), the systems of difference equations (5), (3) and (5), (4) have solutions. These solutions are unique and
\[ \lim_{h\to 0}\ \max_{x_i,y_j\in \overline D_h}\left|u(x_i,y_j)-u_{ij}\right|=0. \]
If, in addition, \(u\in C^{(3)}(\overline D)\), then
\[ \max_{x_i,y_j\in \overline D_h}\left|u(x_i,y_j)-u_{ij}\right|\leq Ch. \]
If \(u\in C^{(4)}(\overline D)\) and Kolmat interpolation is used, then
\[ \max_{x_i,y_j\in \overline D_h}\left|u(x_i,y_j)-u_{ij}\right|\leq C_1h^2. \]
- We now consider other ways of approximating the first derivatives \(u_x\) and \(u_y\). Namely, if \(F_p(\ldots)\geq 0\), we replace \(u_x\) by \((u_{i,j}-u_{i-1,j})/h\); if \(F_p(\ldots)\leq 0\), we replace \(u_x\) by \((u_{i+1,j}-u_{i,j})/h\). The treatment is analogous for \(u_y\). In all, four cases occur. We state the result for one of them.
Consider the following difference equation approximating the differential equation (1):
\[ F\left( x_i,y_j,u_{ij}, \frac{u_{i,j}-u_{i-1,j}}{h}, \frac{u_{i,j}-u_{i,j-1}}{h}, \frac{u_{i+1,j}-2u_{i,j}+u_{i-1,j}}{h^2}, \frac{u_{i,j+1}-2u_{i,j}+u_{i,j-1}}{h^2} \right)=0. \tag{6} \]
Theorem 2. Let
\[ F_p(x,y,u,p,q,r,t)\geq 0,\qquad F_q(x,y,u,p,q,r,t)\geq 0, \]
\[ F_u(x,y,u,p,q,r,t)\leq q_1<0 \]
for \(x,y\in \overline D\), \(-\infty<u,p,q,r,t<+\infty\).
Then the systems of difference equations (6), (4) and (6), (3) have solutions. These solutions are unique and
\[ \lim_{h\to 0}\ \max_{x_i,y_j\in \overline D_h}\left|u(x_i,y_j)-u_{ij}\right|=0. \]
If, moreover, \(u\in C^{(3)}(\overline D)\), then
\[ \max_{x_i,y_j\in \overline D_h}\left|u(x_i,y_j)-u_{ij}\right|\leq C_3h. \]
II. We now consider a more general boundary-value problem. Find a function \(u\in C^{(2)}(\overline D)\) satisfying equation (1) and the boundary condition
\[ f(x,y,u,u_x,u_y)\big|_{\Gamma}=0. \tag{7} \]
Here the function \(f(x,y,u,p,q)\) is continuously differentiable with respect to all its arguments and satisfies the inequalities
\[ -f_u(x,y,u,p,q)\geq c>0, \]
\[ f_p(x,y,u,p,q)\cos(\mathbf n,x)\geq 0,\qquad f_q(x,y,u,p,q)\cos(\mathbf n,y)\geq 0 \]
for \(x,y\in \Gamma\), \(-\infty<u,p,q<\infty\); \(\mathbf n\) is the inward normal to the boundary \(\Gamma\). We assume that the solution of problem (1), (7) exists and is unique.
We approximate the boundary conditions (7) as follows. Let the point \(x_i,y_j\in \Gamma_h\). Draw the inward normal to the boundary \(\Gamma\) passing through the point \(x_i,y_j\). We assume that the domain \(D\) is such that, for \(h<h_0\)
(\(h_0\) does not depend on \(x_i, y_j\)) there exists a grid square with vertex at \(x_i, y_j\) such that the two vertices of this square adjacent to the vertex \(x_i, y_j\) belong to \(\bar D_h\), and the normal constructed above passes inside or along the boundary of this square. For definiteness, suppose that the vertices indicated above have coordinates \(x_{i-1}, y_j\) and \(x_i, y_{j-1}\). Denote by \(\bar x_i, \bar y_j\) the point of intersection of the normal drawn at \(x_i, y_j\) with the boundary \(\Gamma\). Then we set
\[ f\left(\bar x_i,\bar y_j,u_{i,j},\frac{u_{i,j}-u_{i-1,j}}{h},\frac{u_{i,j}-u_{i,j-1}}{h}\right)=0 \tag{8} \]
for \(x_i, y_j \in \Gamma_h\).
Theorem 3. Suppose
\[
F_u(x,y,u,p,q,r,t)\le q_1<0,
\]
\[
|F_p(x,y,u,p,q,r,t)|\le M F_r(x,y,u,p,q,r,t),
\]
\[
|F_q(x,y,u,p,q,r,t)|\le M F_t(x,y,u,p,q,r,t)
\]
for \(x,y\in \bar D\), \(-\infty<u,p,q,r,t<+\infty\).
Then, for sufficiently small \(h\), the system of difference equations (5), (8) has a solution. This solution is unique and
\[ \lim_{h\to 0}\ \max_{x_i,y_j\in \bar D_h}|u(x_i,y_j)-u_{i,j}|=0. \]
If, moreover, \(u\in C^{(3)}(\bar D)\), then
\[ \max_{x_i,y_j\in \bar D_h}|u(x_i,y_j)-u_{i,j}|\le C_4 h. \]
If \(u\in C^{(4)}(\bar D)\), \(f_p=f_q=0\), \(\Gamma_h=\Gamma_h^{(1)}\), then
\[ \max_{x_i,y_j\in \bar D_h}|u(x_i,y_j)-u_{i,j}|\le C_5 h^2. \]
In the case of sign-definiteness of the derivatives of the function \(F\) with respect to the variables \(u_x\) and \(u_y\), and approximation of the differential equation in the form (6), theorems analogous to Theorem 2 hold. We state one of them.
Theorem 4. Under the assumptions of Theorem 2, the system of difference equations (6), (8) is uniquely solvable and
\[ \lim_{h\to 0}\ \max_{x_i,y_j\in \bar D_h}|u(x_i,y_j)-u_{i,j}|=0. \]
If, moreover, \(u\in C^{(3)}(\bar D)\), then
\[ \max_{x_i,y_j\in \bar D_h}|u(x_i,y_j)-u_{i,j}|\le C_6 h. \]
Remark. All results carry over to the case of any number of dimensions.
Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
2.IV.1966
References
- L. Collatz, Numerical Methods for Solving Differential Equations, Moscow, 1953.