Abstract
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MATHEMATICS
V. I. LEBEDEV
ON THE METHOD OF NETS FOR THE THIRD BOUNDARY-VALUE PROBLEM
(Presented by Academician S. L. Sobolev on 3 V 1960)
In the present paper the error arising in the application of difference schemes considered in (4) is estimated; for this purpose a difference Green’s function is constructed and its properties on the boundary are investigated; at the end of the paper some difference schemes approximating the third boundary condition are given.
Let the function \(U(x_1, x_2)\) be a harmonic function in the domain \(\Omega(x_1, x_2)\), whose boundary \(S\) consists of a finite number of twice continuously differentiable curves without points of return, and at each point of it one can touch the vertex of a sector of radius \(r_0\) and angle \(\pi\theta\), \(|\theta| < 1\), lying entirely outside \(\Omega\). For simplicity we shall assume that the domain \(\Omega\) is simply connected. In addition, let \(U \in H(1+\rho, A, r)\) (3), and on the boundary \(S\) let the condition be satisfied
\[ \frac{\partial U}{\partial n} + aU = \varphi, \tag{1} \]
where \(n\) is the exterior normal; \(a\) and \(\varphi\) are piecewise smooth functions of the parameter \(s\) (the length of the contour), measured from the point \(s_0\) in the direction of positive traversal, and \(a \ge c_0 > 0\).
We briefly reproduce the constructions of papers (4–6). Construct, with step \(h\), the net \(\Omega_h\); denote the boundary points of this set by \(S_h\), denote the set of centers of squares of the net domain \(\Omega_h\) by \(\Omega'_h\), and the boundary points of this set by \(S'_h\); let this be the set of points \(s'_k\), \(k = 0, 1, \ldots, N\), renumbered in the direction of positive traversal of the contour; we put them in correspondence with points \(s_k \in S\) so that \(\Delta s_k = O(h)\) (1). In what follows, by
\[ \frac{\Delta}{\Delta s}, \quad \frac{\Delta}{\Delta s'} \]
we shall denote differences divided by \(h\), respectively along \(S_h\) and \(S'_h\). Let \(V(x_1, x_2)\) be the function conjugate to \(U\); \(\Omega_1\) a strictly interior subdomain of the domain \(\Omega\); \(c_i\) positive constants independent of \(h\); \(\bar a_k = \frac{\Delta s_k}{h} a_k\); for simplicity the indices \(k, h\) are sometimes omitted.
On the net \(\Omega_h\) construct an \(h\)-harmonic function \(u\), satisfying on the set \(S_h\) the condition (4–6)
\[ \frac{\Delta u}{\Delta n} + \bar a u = \bar \varphi. \tag{2} \]
By a simple transfer to the discrete case of methods for estimating harmonic functions satisfying condition (1) (7), one can obtain that in \(\Omega_h\)
\[ |u| \le \frac{\max |\varphi|}{\min a}. \tag{3} \]
Difference schemes for which the function \(u\) satisfies inequality (3) we shall conditionally call positive.
I. To estimate the error, we construct the difference Green’s function
\(G(x_1,x_2,y_1,y_2)\) of our problem:
\[ \Delta_h G = \begin{cases} -h^{-2}, & \text{for } x_1=y_1,\ x_2=y_2,\\ 0, & \text{for } x_1\ne y_1,\ x_2\ne y_2 \end{cases} \quad \text{on } \Omega_h; \tag{4} \]
\[ \frac{\Delta G}{\Delta n}+\overline{a}G=0 \]
on \(S_h\), as a function of \((y_1,y_2)\). Then on \(\Omega_h\), \(G \ge 0\).
Let \(G=W+w\), where \(W\) is the \(h\)-singular solution (4), studied by S. L. Sobolev \({}^{(8)}\) and A. Štrom \({}^{(11)}\), and \(w\) is an \(h\)-harmonic function with the following boundary conditions on \(S_h\):
\[ \frac{\Delta w}{\Delta n}+\overline{a}w = -\left(\frac{\Delta W}{\Delta n}+\overline{a}W\right) = \varphi_1. \]
Then, if \(r\) is the distance between \((x_1,x_2)\) and \((y_1,y_2)\), and \((x_1,x_2)\in\Omega_1\), then \(\varphi_1\) is uniformly bounded as \(h\to0\), since, according to the results of \({}^{(8)}\), up to a constant,
\[ \left|W-\frac{1}{2\pi}\ln r\right|\le c_1hr^{-1}, \]
and, consequently, \(w\) is also uniformly bounded. From the function \(w\) on the set \(\Omega'_h\) we construct an \(h\)-conjugate function \(z\):
\[ w_{x_1}=z_{x_2},\qquad w_{x_2}=z_{x_1}; \tag{5} \]
then on \(S'_h\) the following equalities hold for it:
\[ \frac{\Delta z}{\Delta s'}=-\overline{a}w+\varphi_1. \]
We choose the function \(z\) so that
\[ \sum_{S'_h} z=0, \]
then the function \(z\), together with the first difference quotients taken along \(S'_h\), is uniformly bounded on \(S'_h\). We now estimate the other difference quotients of the function \(z\) in the boundary layer of width \(c_2h\). To this end we construct a majorant for \(z\), estimating \(z\) near the point \(s_{i_0}\) lying on the boundary \(S\). Construct a sector \(D_1\), of radius \(r_0\) and with angle \(\pi\theta\), with vertex at this point and lying outside \(\Omega\); take the pole of the polar coordinate system \((r,\varphi)\) at the point \(s_{i_0}\). Suppose that the sector \(D_1\) in this coordinate system is given by the equations \(0\le r\le r_0\), \(-\pi\theta\le\varphi\le0\). Denote the sector \(0\le r\le r_0\), \(0\le\varphi\le\pi(2-\theta)\) by \(D_2\), and \(D_2\Omega\) by \(D_3\), the boundary of the domain \(D_3\) by \(S_3\); it consists of two parts: \(S_1\), a part of the boundary \(S\), and \(S_2\), a part of the circle \(r=r_0\). We assume that \(S_2\) consists of one component; this can be achieved by choosing \(r_0\) sufficiently small. Denote \(D_3S_h\) by \(S_{2h}\); \(D_3S'_h\) by \(S'_{2h}\), and denote by \(S_{1h}\) the set of points of \(\Omega_h\) at distance up to \(2h\) from the line \(r=r_0\). At the point \(B\) construct a barrier. Let \(0\le\theta<1\); \(M=2\max\limits_{S_{1h}+S_{2h}} r^{-1}|z-z(s'_i)|\); note that on \(S_{1h}\)
\[ \bigl|z(s'_i)-z(s_{i_0})\bigr| \le h\sum_{i_0}^{i}\left|\frac{\Delta z}{\Delta s}\right| \le c'\max_{S_h}\left(|\overline{a}w|+|\varphi_1|\right)|s_i-s_{i_0}|. \]
Then the subharmonic function \(Mr\) will majorize the function \(|z-z_{i_0}|\) on the set \(S_{1h}+S_{2h}\), and consequently the function \(q\), harmonic in \(D_2\) and coinciding on the boundary of the sector \(D_2\) with the function \(Mr\), will be a majorant for \(|z-z_{i_0}|\), but in a neighborhood of \(r=0\)
\[ |q|\le c_3 r^d+O(r), \]
where \(d=(2-\theta)^{-1}\), i.e., if \((x_1,x_2)\in\Omega_1\), and the points \((y'_1,y'_2)\) and \((y''_1,y''_2)\) are at a distance less than
\(c_1h\) from the point \(s_i\), then, using equations (5), we find that
\[ \left|w(x_1,x_2,y'_1,y'_2)-w(x_1,x_2,y''_1,y''_2)\right|\leq c_4h^d. \]
It is clear from the proof that the constants \(M, c_3, c_4\) can be chosen independently of \(h\) and of the position of the point \(s_i\) on the boundary \(S\); then on \(S_h\)
\[ \left|\frac{\Delta G}{\Delta s}\right|\leq c_5h^{d-1}. \]
II. We now proceed to estimate the error of the method presented in (4); the error \(\eta=U-u\) satisfies the following equations:
\[ \Delta_h\eta=h^2\psi_1 \quad \text{on } \Omega_h; \]
\[ \frac{\Delta \eta}{\Delta n}+\bar a\eta=\psi_2 \quad \text{on } S_h. \tag{6} \]
The function \(\psi_2\) can be estimated as follows: when replacing the equality
\[ V(s_m)-V(s_0)=\int_{s_0}^{s_m}(\varphi-aU)\,dS \tag{7} \]
by the rectangle formula for the approximate boundary values of the function \(u\) and of the \(h\)-conjugate function \(v\) corresponding to it,
\[ v(s'_m)-v(s'_0)=\sum_{S_h,\ k=0}^{m}(\varphi-au)\Delta s_k, \tag{8} \]
we make an error of order \(h\). Comparing (6) and (8), we see that, although \(\psi_2=O(1)\),
\[ b(s'_m)=h\sum_{S_h,\ k=0}^{m}\psi_2(s'_k)=O(h),\quad m=0,1,\ldots,N. \tag{9} \]
The quantity \(\psi_1\) can be estimated \((^5,^6)\) as follows:
\[ |\psi_1|\leq c_6h^2\omega_\alpha(\rho), \tag{10} \]
where \(\alpha=p+\gamma-3\); \(\rho\) is the distance to the boundary; \(\omega_\alpha(\rho)\) is a function equal to \(\min(h^\alpha,\rho^\alpha)\) for \(\alpha\leq 0\) and equal to 1 for \(\alpha\geq 0\).
Using the Green function constructed above, from the difference analogue of Green’s formula we obtain
\[ \eta(x_1,x_2)=-h^2\left(h^2\sum_{\Omega_h}G\psi_1\right)+h\sum_{S_h}G\psi_2. \]
Let \((x_1,x_2)\subset \Omega_1\). To estimate the first term we use the representation of the Green function \(G=W+w\) (the nature of the singularity of \(W\) was investigated in \((^8)\)), as well as inequality (10); then
\[ h^2\left|h^2\sum_{\Omega_h}G\psi_1\right|\leq c_7h^2\omega_{\alpha+1}(0). \]
We estimate the second term using (9):
\[ \left|h\sum_{S_h}G\psi_2\right|\leq c_8\left(\left|Gb\right|_{k=0}+\left|Gb\right|_{k=N}+h\sum_{S_h}\left|\frac{\Delta G}{\Delta n}b\right|\right)\leq c_9h^d. \]
Thus, if \(d_1=\min((2-\theta)^{-1},\,p+\gamma)\) and \(0\leq\theta<1\), then in any strictly interior subdomain \(\Omega_1\) of the domain \(\Omega\)
\[ |\eta|\leq c_{10}h^{d_1}. \]
III. We note that the difference scheme considered is the only positive scheme known to the author that is suitable for solving both the second and the third boundary-value problems (4); the schemes in the papers \((^2,^9,^{10})\) do not possess this property.
Let us present a three-point positive scheme of accuracy \(h\), suitable for solving the third boundary-value problem. Let \(u_i\) be the values of \(U\), respectively, at the points \(A_i,\ i=0,1,2\), with coordinates \((0,0)\), \((0,-h_0)\), \((-h_0,0)\) in the local coordinate system, and let \(A_0\in S_h\). Construct a linear function \(u\) coinciding with \(U\) at the points \(A_i\). Let \([s^1,s^2]\) be a segment of the boundary \(S\) near the point \(A_0\), of length \(\Delta s\) of order \(O(h)\); let \(\Delta x_1,\Delta x_2\) be its projections onto the axes \(x_1,x_2\), and let \((x_1^1,x_2^1)\), \((x_1^2,x_2^2)\) be the coordinates of its endpoints. Then
\[ \int_{s^1}^{s^2}\frac{\partial u}{\partial n}\,dS = \frac{u_0-u_2}{h_0}\Delta x_2 - \frac{u_0-u_1}{h_0}\Delta x_1 . \]
Applying an approximate formula to the integral and discarding terms of order \(O(h^2)\), we obtain
\[ \frac{u_0-u_2}{h_0}\alpha_2 - \frac{u_0-u_1}{h_0}\alpha_1 + a u_0 = \varphi, \]
where \(\alpha_i=\dfrac{\Delta x_i}{\Delta s}\); by a suitable choice of the endpoints of the segment \([s^1,s^2]\) and of the points \(A_1,A_2\) (\(h_0=h,\ h\sqrt{2},\ h\sqrt{5}\), etc.) we ensure that \(\alpha_2\geqslant 0,\ \alpha_1\leqslant 0\). Analogous constructions can be carried out in the \(n\)-dimensional case, and also for nonharmonic functions.
Finally, consider one four-point positive scheme of accuracy \(h^2\). To this end consider one more point \(A_3(0,h_0)\in\Omega_h\), and take the points \(s^1,s^2\) on the hyperbola \(x_1^2-x_2^2=c^2\) and such that \(x_1^1,x_1^2,x_2^1\geqslant 0,\ x_2^2\leqslant 0\). Construct a harmonic polynomial \(u\) of degree 2, coinciding at the points \(A_i\) with the values \(u_i,\ i=0,1,2,3\). Then, repeating the preceding arguments, we obtain
\[ \frac{a_1+a_2}{2}u_0 + \frac{u_0-u_2}{h_0} \left( \frac{a_1x_1^1+a_2x_1^2}{2}+\alpha_2 \right) + \frac{u_3+u_1}{2h_0} \left( \frac{a_1x_2^1+a_2x_2^2}{2}-\alpha_1 \right) + \]
\[ + \frac{2u_0-u_1-u_3}{h_0} \left( \frac{x_1^1x_2^1-x_1^2x_2^2}{h_0\Delta s} + \frac{a_1x_1^1+a_2x_1^2}{2} + \frac{1}{2}\alpha_1 \right) = \frac{\varphi_1+\varphi_2}{2}, \]
where \(a_j=a(s^j)\), \(\varphi_j=\varphi(s^j)\), \(j=1,2\).
Received
28 III 1960
CITED LITERATURE
- E. A. Volkov, Vychislit. matem., 1, 33 (1957).
- E. A. Volkov, DAN, 102, No. 3 (1955).
- N. M. Günter, The Theory of Potential, 1953.
- V. I. Lebedev, DAN, 127, No. 4, 742 (1959).
- V. I. Lebedev, DAN, 128, No. 4, 665 (1959).
- V. I. Lebedev, DAN, 132, No. 5 (1960).
- O. A. Oleinik, Matem. sborn., 30, 3, 695 (1952).
- S. L. Sobolev, DAN 87, No. 3, 341 (1952).
- Batschelett, ZAMP, 3, No. 3, 156 (1952).
- P. V. Viswanathan, Math. Tables and other Aids Comp., 10, No. 58 (1957).
- A. Stöhr, Math. Nachr., 3, No. 4–6 (1950).