Abstract
Full Text
V. Ya. Rivkind
ON ESTIMATES OF THE RATE OF CONVERGENCE OF SOLUTIONS OF DIFFERENCE EQUATIONS TO SOLUTIONS OF ELLIPTIC EQUATIONS WITH DISCONTINUOUS COEFFICIENTS AND ON ONE NUMERICAL METHOD FOR SOLVING THE DIRICHLET PROBLEM
(Presented by Academician V. I. Smirnov on 14 XI 1962)
In the paper (¹) (see also (²)) it is shown that diffraction problems can be considered as problems of determining generalized solutions with finite energy integral for equations with discontinuous coefficients, and methods for their solution are given. From (¹,²), in particular, there follows the existence and uniqueness of the generalized solution of the following problem:
\[ -\frac{\partial}{\partial x_1} a' \frac{\partial u^a}{\partial x_1} + \frac{\partial}{\partial x_2} a' \frac{\partial u^a}{\partial x_2} = f'(x_1,x_2), \]
\[ a'= \begin{cases} 1, & x\in \Omega_1,\\ a, & x\in \Omega_2; \end{cases} \qquad f'= \begin{cases} f(x_1,x_2), & x\in \Omega_1,\\ 0, & x\in \Omega_2; \end{cases} \tag{1} \]
\[ [u^a]_{S_1}=0,\qquad \left[a'\frac{\partial u^a}{\partial n}\right]_{S_1}=0,\qquad u^a\big|_{S_2}=0. \]
Here \(a\) is a constant \((a\gg 1)\); \(S_1\) is a sufficiently smooth (¹,²) boundary of an arbitrary domain \(\Omega_1\); \(S_2\) is the boundary of a rectangle \(\Omega\) containing the domain \(\Omega_1\); \(\Omega_2\) is the domain enclosed between \(S_1\) and \(S_2\); \(n\) is the normal to \(S_1\); \([\ ]\) denotes the jump of a function on the line.
For the solution of problem (1), as follows from (¹,²), the estimate
\[ \|u^a\|_{W_2^1(\Omega_1)}^2 + a\|u^a\|_{W_2^1(\Omega_2)}^2 \le C_1\|f(x)\|_{L_2(\Omega_1)}^2, \tag{2} \]
holds, where \(C_1\) depends only on the form of the domains \(\Omega_1,\Omega_2\).
From (²) one easily derives the known fact that \(u^a(x)\), as \(a\to+\infty\), tends in the norm \(W_2^1\) to zero in the domain \(\Omega_2\) and to the solution of the Dirichlet problem
\[ \Delta u=f(x_1,x_2),\qquad u\big|_{S_1}=0 \tag{3} \]
in the domain \(\Omega_1\).
On the other hand, for problem (1) it is shown in (¹,²) how to construct convergent finite-difference schemes. Since the domain \(\Omega\) is a rectangle, there exist comparatively effective methods for solving the algebraic systems that arise in this connection. In view of this, for problem (3) in an arbitrary domain it was natural to analyze the following method of solution: enclose the domain \(\Omega_1\) in the rectangle \(\Omega\) most closely adjacent to it, and in this rectangle solve problem (1) (taking \(\Omega_2=\Omega-\Omega_1\)) by the finite-difference method with sufficiently small mesh size \(h\) and sufficiently large \(a\).
From what has been said above it is clear that if one takes the approximate solution \(u_h^a(x)\), obtained for problem (1) by the finite-difference method (from (¹,²)), and passes to the limit first as \(h\to0\), and then as \(a\to+\infty\), then in the limit one obtains the solution \(u(x)\) of problem (3). However, in practice this is not feasible (passing to the limit \(h\to0\)). In order that the method under discussion be suitable for
of the real solution of problem (3), one must investigate for what positive \(h\) and finite \(a\) the solution \(u_h^a(x)\) differs from the exact solution \(u(x)\) by an arbitrarily prescribed quantity \(\varepsilon\) (in one norm or another). For this it is necessary to obtain estimates of the deviations of \(u_h^a(x)\) from \(u^a(x)\) and of \(u^a(x)\) from \(u(x)\). The present work is devoted to this.
By the method indicated in \((^3)\), for the solution of problem (1) the inequality is established
\[ \|u^a\|^2_{W^1_2(\Omega_1)}+a\|u^a\|^2_{W^1_2(\Omega_2)}\leq C_2\|f\|^2_{L_2(\Omega_1)}, \tag{4} \]
where \(C_2\) depends only on the form of the domains \(\Omega_1\) and \(\Omega_2\).
From inequality (4) and the maximum principle it follows that \(u^a(x)\) tends to \(u(x)\) as \(a\to\infty\) in such a way that
\[ \|u(x)-u^a(x)\|_{C_{0,\alpha}(\Omega_1)} \leq \frac{C_3\|f\|_{L_2(\Omega_1)}}{\sqrt a}, \tag{5} \]
where \(C_{0,\alpha}(\Omega_1)\) is the Hölder space with arbitrary exponent \(\alpha<1\); \(C_3\) depends only on the form of the domains \(\Omega_1,\Omega_2\) and on \(\alpha\).
For problem (1) the following difference scheme is constructed (see \((^1)\)) with step \(\Delta x_1=\Delta x_2=h\):
\[ (a_h \bar u^a_{h x_1})_{\bar x_1}+(a_h \bar u^a_{h x_2})_{\bar x_2}=f_h(x_{ij}); \]
\[ a_h= \begin{cases} 1, & x_{ij}\in\Omega_1,\\[2mm] \dfrac{a+1}{2}, & x_{ij}\in S_1,\\[2mm] a, & x_{ij}\in\Omega_2; \end{cases} \qquad f_h(x_{ij})=\frac{1}{h^2} \int_{x_1^i}^{x_1^i+h}\int_{x_2^j}^{x_2^j+h} f'(x_1,x_2)\,dx; \tag{6} \]
\[ u_h^a\big|_{x_{ij}\in S_2}=0. \]
Here \(x_{ij}\) denotes a grid node with coordinates \(x_1=x_1^i,\ x_2=x_2^j\); \(u^a_{h x_1},\ u^a_{\bar h x_2},\ u^a_{h x_2}\) are difference quotients of the sought grid function \(u_h^a\) \((^1)\).
It follows from \((^2)\) that the solution of problem (6) exists, is unique, and its multilinear interpolations \((u_h^a)'\) (see \((^1)\)) converge weakly in the norm \(W^1_2\) to the exact solution of problem (1).
In \((^2)\) the following estimate is also derived for \(u_h^a\):
\[ \int_{\Omega}\sum_{i=1}^{2}(\widetilde u^a_{h x_i})^2\,dx \leq C_4\int_{\Omega} f^2(x_1,x_2)\,dx, \tag{7} \]
where \(\widetilde u^a_{h x_i}\) are the piecewise-constant interpolations \((^2)\) of the function \(u^a_{h x_i}\); \(C_4\) depends only on the dimensions of the rectangle \(\Omega\).
For the solution of problem (6) one succeeds in establishing that its piecewise-constant interpolations \(\widetilde u_h^a\) converge to the exact solution of problem (1) as \(h\to0\) in such a way that
\[ \int_{\Omega}\left[ (u-\widetilde u_h^a)^2+ \sum_{i=1}^{2}\left( \frac{\partial u^a}{\partial x_i}-\widetilde u^a_{h x_i} \right)^2 \right]dx \leq C_5\int_{\Omega}(f-\widetilde f_h)^2\,dx + C_6 a h^{(1-\delta)/2}\|f\|^2_{L_2(\Omega_1)}, \tag{8} \]
where \(C_5\) and \(C_6\) depend only on the form of the domains \(\Omega_1,\Omega_2\) and \(0<\delta<1\).
Estimate (8) is obtained by comparing the integral identities for the solutions of problem (1), \(u^a(x)\), and problem (6), \(u_h^a\) (see \((^1)\)):
\[ \int_{\Omega} a'\sum_{i=1}^{2} \frac{\partial u^a}{\partial x_i}\frac{\partial \Phi}{\partial x_i}\,dx = \int_{\Omega} f'\Phi\,dx, \tag{9} \]
where \(\Phi\in \overset{0}{D}(\Omega)\) (see (9)) and
\[ \int_{\Omega}\tilde a_h\sum_{i=1}^2 \tilde u_{h x_i}^{\,a}\tilde\Phi_{h x_i}\,dx = \int_{\Omega}\tilde f_h\tilde\Phi_h\,dx, \tag{10} \]
where \(\Phi_h\) is a grid function equal to zero at the nodes lying on \(S_2\). In (9) it is assumed that \(\Phi\equiv u^a-(u_h^a)'\). The values \(u^a-(u_h^a)'\), \((u^a)'\) are fixed at the grid nodes. The resulting grid functions, whose piecewise-constant completions are denoted respectively by \(\tilde u^a-(\tilde u_h^a)'\), \(\tilde u^a\). Obviously,
\(\tilde u^a-(\tilde u_h^a)'\equiv \tilde u^a-\tilde u_h^a\). In (9), \(a'\), \(\partial u^a/\partial x_i\), \(u^a-(u_h^a)'\), \(\partial(u^a-(u_h^a)')/\partial x_i\), \(f'(x)\) are replaced respectively by \(\tilde a_h\), \(\tilde u_{x_i}^{\,a}\), \(\tilde u^a-\tilde u_h^a\), \(\tilde u_{x_i}^{\,a}-\tilde u_{h x_i}^{\,a}\), \(\tilde f_h\). The arising residuals \(R_h\) are transferred to the right-hand side.
If in (10) we put \(\tilde\Phi_h\equiv \tilde u^a-\tilde u_h^a\) and subtract (10) from the transformed identity (9), then one can easily derive
\[ \int_{\Omega}\sum_{i=1}^2\left(\tilde u_{x_i}^{\,a}-\tilde u_{h x_i}^{\,a}\right)^2\,dx\leq |R_h|. \tag{11} \]
Here the following expressions will enter \(R_h\) as summands:
\[ I_1\equiv \int_{\Omega}\tilde a_h\sum_{i=1}^2 \tilde u_{x_i}^{\,a} \left(\tilde u_{x_i}^{\,a}-\frac{\partial u^a}{\partial x_i}\right)\,dx, \]
\[ I_2\equiv -\int_{\Omega}\tilde a_h\sum_{i=1}^2 \tilde u_{x_i}^{\,a} \left(\tilde u_{h x_i}^{\,a}-\frac{\partial (u_h^a)'}{\partial x_i}\right)\,dx, \]
\[ I_3\equiv \int_{\Omega}\tilde a_h\sum_{i=1}^2 \left(\tilde u_{x_i}^{\,a}-\frac{\partial u^a}{\partial x_i}\right) \left(\frac{\partial u^a}{\partial x_i}-\frac{\partial (u_h^a)'}{\partial x_i}\right)\,dx, \]
\[ I_4\equiv \int_{\Omega_{he}}(a'-\tilde a_h)\sum_{i=1}^2 \frac{\partial u^a}{\partial x_i} \left(\frac{\partial u^a}{\partial x_i}-\frac{\partial (u_h^a)'}{\partial x_i}\right)\,dx \]
(\(\Omega_{he}\) is the aggregate of grid squares through which the line of discontinuity of \(a'\) passes).
By applying the Cauchy inequality, the embedding theorems, and estimates (4), (7), it is shown that \(I_1\), \(I_3\), \(I_4\) can be estimated from above by the quantity
\[ aC_7 \left( \int_{\Omega}\sum_{i=1}^2 \left(\frac{\partial u^a}{\partial x_i}-\tilde u_{x_i}^{\,a}\right)^2\,dx \right)^{1/2} \|f\|_{L_2(\Omega_1)}, \tag{12} \]
where \(C_7\) is a constant depending only on the form of the domains \(\Omega_1\) and \(\Omega_2\).
One can arrive at the same conclusion also with respect to \(I_2\), after first representing \(\partial (u_h^a)'/\partial x_i\) in it through the difference quotients \(u_{h x_i}^{\,a}\), \(u_{h x_1 x_2}^{\,a}\), and then transferring, by summation by parts, the difference with respect to \(x_i\) from \(u_{h x_1 x_2}^{\,a}\) to \(u_{x_j}^{\,a}\).
The estimate of
\[ \int_{\Omega}\sum_{i=1}^2 \left(\frac{\partial u^a}{\partial x_i}-\tilde u_{x_i}^{\,a}\right)^2\,dx \]
is carried out by splitting it into three parts:
\[ \int_{\Omega}\sum_{i=1}^2 \left(\frac{\partial u^a}{\partial x_i}-\tilde u_{x_i}^{\,a}\right)^2\,dx = \sum_{k=1}^2 \int_{\Omega_{hk}}\sum_{i=1}^2 \left(\frac{\partial u^a}{\partial x_i}-\tilde u_{x_i}^{\,a}\right)^2\,dx + \int_{\Omega_{he}}\sum_{i=1}^2 \left(\frac{\partial u^a}{\partial x_i}-\tilde u_{x_i}^{\,a}\right)^2\,dx. \]
\(\Omega_{h1}\) and \(\Omega_{h2}\) are the aggregates of grid squares respectively from the domains
\(\Omega_1\) and \(\Omega_2\), which are not intersected by the line \(S_1\). The function \(\widetilde u^a_{x_1}(x)\), for \(x=(x_1,x_2)\) varying in the square \(x_1^l \leq x_1 \leq x_1^l+h,\ x_2^m \leq x_2 \leq x_2^m+h\), is the average of the function \(\partial u^a/\partial x_1\) along the line \(x_2=x_2^m,\ x_1^l \leq x_1 \leq x_1^l+h\):
\[ \widetilde u^a_{x_1}(x)=\frac{1}{h}\int_{x_1^l}^{x_1^l+h} \frac{\partial u^a(\xi,x_2^m)}{\partial \xi}\,d\xi . \]
Similarly for \(\widetilde u^a_{x_2}\). Comparing \(\partial u^a/\partial x_i\) with its average and using the embedding theorems and estimates (4), one can obtain
\[ \sum_{k=1}^{2}\int_{\Omega_{hk}}\sum_{i=1}^{2} \left(\frac{\partial u^a}{\partial x_i}-\widetilde u^a_{x_i}\right)^2\,dx \leq C_8 h^2 \|f\|^2_{L_2(\Omega_1)}, \tag{13} \]
where \(C_8\) depends only on the form of the domains \(\Omega_1\) and \(\Omega_2\).
From the embedding theorems and the fact that the total area of the squares contained in \(\Omega_{he}\) does not exceed \(kh\), where \(k>0\) depends only on the form of \(S_1\), it follows that
\[ \int_{\Omega_{he}}\sum_{i=1}^{2} \left(\frac{\partial u^a}{\partial x_i}-\widetilde u^a_{x_i}\right)^2\,dx \leq C_9 h^{1-\delta}\|f\|^2_{L_2(\Omega_1)}, \tag{14} \]
where \(C_9\) depends only on the form of the domains \(\Omega_1,\Omega_2\), and \(0<\delta<1\).
From (11), (12), (13), (14) there follows estimate (8), which, together with estimate (5), makes it possible to solve problem (3) for an arbitrary domain.
Results similar to those obtained in the present work can be generalized to the three-dimensional case and to more complicated problems with discontinuous coefficients.
Recently a paper by V. K. Saul’ev \((^4)\) was published, in which similar questions are discussed.
The author expresses her sincere and deep gratitude to Prof. O. A. Ladyzhenskaya for supervising the work.
Leningrad State University
named after A. A. Zhdanov
Received
9 XI 1962
CITED LITERATURE
- O. A. Ladyzhenskaya, DAN, 96, No. 3, 433 (1954).
- O. A. Ladyzhenskaya, A Mixed Problem for a Hyperbolic Equation, 1953.
- O. A. Ladyzhenskaya, V. A. Solonnikov, Tr. Mat. Inst. im. V. A. Steklova, Academy of Sciences of the USSR, 59, 115 (1960).
- V. K. Saul’ev, DAN, 144, No. 3, 497 (1962).