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Homogeneous Difference Schemes on Irregular Meshes for a Differential Equation of the Fourth Order with Discontinuous Coefficients
A. K. Boyarchuk
The paper generalizes the results of [1] to the case of irregular meshes. Through-computation schemes are considered for a self-adjoint differential equation of the fourth order. The text in which the definition of a functional of rank \(\gamma\) is given coincides with the corresponding text of [1].
§ 1. Difference Boundary-Value Problem
1°. Boundary-value problem for the differential equation.
Consider the differential equation
\[ LLJ^{(k,p,q,f)}=\frac{d^2}{dx^2}\left[k(x)\frac{d^2U}{dx^2}\right] -\frac{d}{dx}\left[p(x)\frac{dU}{dx}\right]+q(x)U-f(x)=0 \tag{1.1} \]
with boundary conditions
\[ U(0)=U'(0)=U(1)=U'(1)=0. \tag{1.2} \]
We assume that the functions \(k,\ p,\ q\), and \(f\) on the interval \([0,1]\) satisfy the conditions:
\[ 0<c_1\le k(x)\le c_2,\quad 0\le p(x)\le c_3,\quad 0\le q(x)\le c_4,\quad |f(x)|\le c_5, \tag{1.3} \]
where \(c_j\ (j=1,2,3,4,5)\) are some positive constants. If the functions \(k,\ p,\ q\), and \(f\) have discontinuities of the first kind at some point \(\xi\in(0,1)\), then at this point the following conjugation conditions for the function \(U(x)\) must be satisfied:
\[ [U]=0,\quad [U']=0,\quad [kU'']=0,\quad [(kU'')'-pU']=0, \tag{1.4} \]
where \([\varphi]=\varphi_{\mathrm r}-\varphi_{\mathrm l}\), \(\varphi_{\mathrm r}=\varphi(\xi+0)\), \(\varphi_{\mathrm l}=\varphi(\xi-0)\). The boundary-value problem (1.1)—(1.4) will be called problem I.
2°. Notation adopted in the paper.
\[ 1)\quad \omega_{h_i}=\left\{ x_0=0,\ x_1=h_1,\ \ldots,\ x_i=\sum_{s=1}^{i}h_s,\ \ldots,\ x_N=\sum_{s=1}^{N}h_s=1 \right\}, \]
\[ h_i=x_i-x_{i-1},\quad x_{i+\frac12}=x_i+\frac{h_{i+1}}{2},\quad x_{i-\frac12}=x_i-\frac{h_i}{2},\quad h^*=\max h_i, \]
\[ h_*=\min h_i,\quad \mathbf h_i=\frac{h_i+h_{i+1}}{2},\quad sh_i^*=h_i+h_{i+1}+\cdots+h_{i+s-1}. \]
2) \(\|W_{h_i}\|_0=\max |W_i|\),
\[ \|\Psi_{h_i}\|_1=\sum_{i=2}^{N-1}|\nabla\eta_i|+\left|\frac{\nabla\eta_N}{h_N}\right|, \]
\[ \|\Psi_{h_i}\|_2=\sum_{i=1}^{N-1}|\varkappa_i|\,\mathbf h_i+|\varkappa_{N-1}|h_N,\qquad \nabla\eta_i=h_i\sum_{s=1}^{i-1}h_s\frac{\Psi_s}{a_s}, \]
\[ \varkappa_i=\sum_{s=2}^{i-1}h_s\Psi_s\sum_{k=s+1}^{i}h_k,\quad \Psi_{h_i}\text{ is a vector of a finite-dimensional space.} \]
3) \(\Delta y_i=y_{i+1}-y_i,\ \nabla y_i=y_i-y_{i-1}\),
\[ \Delta\left(\frac{\nabla y_i}{h_i}\right)=\frac{\Delta y_i}{h_{i+1}}-\frac{\nabla y_i}{h_i}. \]
4) \(C_{[0,1]}^{(m)}\) \(\bigl(Q_{[0,1]}^{(m)}\bigr)\) is the class of functions continuous (piecewise continuous) on \([0,1]\), together with their derivatives up to order \(m\) inclusive.
5) \(Q_{[0,1]}^{(m,1)}\) is the class of functions from \(Q_{[0,1]}^{(m)}\) whose \(m\)-th derivatives satisfy a Lipschitz condition on the intervals of their continuity; \(C_{[0,1]}^{(m,1)}\) is the class of functions defined analogously to \(Q_{[0,1]}^{(m,1)}\).
6) \(\rho(h_i)\) is a quantity tending to zero as \(h_i\to0\). We choose the mesh so that \(h^*=O(h_i)\). It follows from this that
\[ 0<\varkappa_0\le \frac{h_*}{h^*}\le \varkappa_1, \tag{1.5} \]
where \(\varkappa_0\) and \(\varkappa_1\) are constants independent of \(i\) and \(N\).
3°. Homogeneous difference schemes for equation (1.1). Consider difference schemes of the form
\[ L_{h_i}^{(k,p,q,f)}y_i=L_{h_i}^{(k,p,q)}y_i-\Phi_i=0, \tag{1.6} \]
where
\[ L_{h_i}^{(k,p,q)}y_i= \frac{1}{\mathbf h_i}\Delta\left\{ \frac{1}{h_i}\nabla\left[ \frac{a_i}{h_i}\Delta\left(\frac{\nabla y_i}{h_i}\right) \right]\right\} -\frac{1}{\mathbf h_i}\Delta\left(b_i\frac{\nabla y_i}{h_i}\right)+d_i y_i, \]
the coefficients of which \(a_i,\ b_i,\ d_i\), and \(\Phi_i\) are determined by means of template functionals given in the class \(Q^{(0)}\):
\[ A^{h_i}[\varphi(s)],\quad -1\le s\le 1, \]
\[ B^{h_i}[\varphi(s)],\quad -1\le s\le 0, \]
\[ D^{h_i}[\varphi(s)],\quad -\frac12\le s\le \frac12, \]
\[ F^{h_i}[\varphi(s)],\quad -\frac12\le s\le \frac12, \]
by the formulas:
\[ a_i=A^{(k,h_i)}=A^{h_i}[\bar k(s)],\quad \bar k(s)=k(x_i+sh_i), \]
\[ b_i=B^{(p,h_i)}=B^{h_i}[\bar p(s)],\quad \bar p(s)=p(x_i+sh_i), \]
\[ d_i = D^{(q,h_i)} = D^{h_i}[\bar q(s)],\quad \bar q(s)=q(x_i+sh_i), \]
\[ \Phi_i = F^{(f,h_i)} = F^{h_i}[\bar f(s)],\quad \bar f(s)=f(x_i+sh_i). \]
We shall assume that each coefficient of the scheme \(L_{h_i}^{(k,p,q,f)}\) depends only on one coefficient of the differential equation (1.1). To characterize the class of stencil functionals we shall use the concept of the rank of a functional introduced in [2].
A certain functional \(A^{h_i}[\varphi]\), depending on the parameter \(h_i\), will be called a functional of rank \(\gamma\) (\(\gamma \ge 0\)) if the following conditions are satisfied:
1) \(A^{h_i}[\varphi]\) has a differential of order \(\gamma\) with respect to \(h_i\):
\[ A^{h_i}[\varphi]=\sum_{\sigma=0}^{\gamma} h_i^\sigma A^{(\sigma)}[\varphi]+h_i^\gamma \rho(h_i,\varphi), \]
where \(|\rho(h_i,\varphi)| \le \rho(h_i)\), and each of the functionals \(A^{(\sigma)}[\varphi]\), \(\sigma=1,2,\ldots,\gamma\), has a differential of order \(\gamma-\sigma\) with respect to the argument \(\varphi\):
\[ A^{(\sigma)}[\varphi+\delta\Psi] = A^{(\sigma)}[\varphi] +\delta A_1^{(\sigma)}[\varphi,\Psi] +\cdots +\delta^{\gamma-\sigma} A_{\gamma-\sigma}^{(\sigma)}[\varphi,\Psi] +\delta^{\gamma-\sigma}\rho(\delta). \]
2) The functional \(A^{h_i}[\varphi]\) and, consequently, \(A^{(\sigma)}[\varphi]\) for all \(\sigma=1,2,\ldots,\gamma\) are homogeneous functionals of first degree,
\[ A^{h_i}[c\varphi]=cA^{h_i}[\varphi],\qquad A^{(\sigma)}[c\varphi]=cA^{(\sigma)}[\varphi], \]
where \(c\) is some positive constant.
3) The functionals \(A^{h_i}[\varphi]\) and \(A^{(\sigma)}[\varphi]\), \(\sigma=1,2,\ldots,\gamma\), are nondecreasing, i.e.
\[ A^{h_i}[\varphi_2]\ge A^{h_i}[\varphi_1],\quad \text{if } \varphi_2\ge \varphi_1, \]
and \(A^{h_i}[\varphi]\) is a normalized functional:
\[ A^{h_i}[1]=1 \quad \bigl(A^{(0)}[1]=1,\ A^{(\sigma)}[1]=0 \text{ for } \sigma\ge 1\bigr). \]
If \(A^{h_i}[\varphi]\) is a linear functional, then all \(A^{(\sigma)}[\varphi]\) are also linear. Therefore condition 2) and the requirement of differentiability with respect to the argument \(\varphi\) are satisfied automatically. We shall call the scheme \(L_{h_i}^{(k,p,q,f)}\) a scheme of \(n\)-th rank if all stencil functionals \(A^{h_i}[\varphi]\), \(B^{h_i}[\varphi]\), \(D^{h_i}[\varphi]\), and \(F^{h_i}[\varphi]\) have rank \(n\). We assume that \(D^{h_i}[\varphi]\) and \(F^{h_i}[\varphi]\) are linear functionals, while \(A^{h_i}[\varphi]\) and \(B^{h_i}[\varphi]\), generally speaking, are nonlinear functionals. Let us note the obvious inequalities
\[ 0<c_1\le a_i\le c_2,\quad 0\le b_i\le c_3,\quad 0\le d_i\le c_4,\quad |\Phi_i|\le c_5 . \tag{1.7} \]
4°. Approximation error of the scheme \(L_{h_i}^{(k,p,q,f)}\). The difference scheme \(L_{h_i}^{(k,p,q,f)}\) can be characterized by the approximation error on the solution of the differential equation (1.1)
\[ \Psi_i=(L^{(k,p,q,f)}U)_{x=x_i}-L_{h_i}^{(k,p,q,f)}U_i . \tag{1.8} \]
Theorem 1. The approximation error of any scheme \(L_{h_i}^{(k,p,q,f)}\) for \(k,p,q,f\in C_{[0,1]}^{(2,1)}\) can be represented in the form
\[ \Psi_i=\frac{1}{h_i}\Delta\left(\frac{1}{h_i}\nabla\Psi_i^{(a)}\right)+\frac{1}{h_i}\Delta\Psi_i^{(b)}+\Psi_i^*, \tag{1.9} \]
where
\[ \Psi_i^{(a)}=(kU'')_i-\frac{1}{h_i}a_i\Delta\left(\frac{\nabla U_i}{h_i}\right),\qquad \Psi_i^{(b)}=\frac{b_i}{h_i}\nabla U_i-(pU')_{i-\frac12}, \]
\[ \Psi_i^*=[(qU)_i-d_iU_i]+(\Phi_i-f_i)+ \frac{h_{i+1}-h_i}{3}\bigl[(pU')'-qU+f\bigr]_{x=x_i}+ \]
\[ +\frac{h_{i+1}-h_i}{4}(pU')_i''+O(h_i^2), \]
and \(U=U(x)\) is the solution of the equation \(L^{(k,p,q,f)}U=0\).
We omit the proof because of its elementary nature. Let \(L_{h_i}^{(k,p,q,f)}\) be a second-order scheme. Expanding \(a_i,\ b_i,\ d_i\), and \(\Phi_i\) in series, we obtain
\[ a_i=k_i+h_ik_i'A_1^{(0)}[s]+O(h_i^2), \]
\[ b_i=p_{i-\frac12}+h_ip'_{i-\frac12}\left(\frac12+B_1^{(0)}[s]\right)+O(h_i^2), \]
\[ d_i=q_i+h_iD_1^{(0)}[s]q_i'+O(h_i^2),\qquad \Phi_i=f_i+h_if_i'F_1^{(0)}[s]+O(h_i^2), \]
\[ \Delta\left(\frac{\nabla U_i}{h_i}\right) =h_i\left[U_i''-\frac{h_{i+1}-h_i}{3}U_i'''+O(h_i^2)\right], \qquad \frac{\nabla U_i}{h_i}=U'_{i-\frac12}+O(h_i^2), \]
\[ \Psi_i^{(a)}=\frac{h_{i+1}-h_i}{3}k_iU_i'''-h_ik_i'U_i''A_1^{(0)}[s]+O(h_i^2), \]
\[ \Psi_i^{(b)}=h_ip'_{i-\frac12}U'_{i-\frac12}\left(\frac12+B_1^{(0)}[s]\right)+O(h_i^2). \]
We shall say that the difference scheme \(L_{h_i}^{(k,p,q,f)}\) has order of approximation \(h_{i+1}-h_i\) if the following conditions are satisfied:
\[ \Psi_i^{(a)}=2(h_{i+1}-h_i)\varphi_i+O(h_i^2),\qquad \Psi_i^{(b)}=O(h_i^2),\qquad \Psi_i^*=2(h_{i+1}-h_i)\omega_i+O(h_i^2), \]
where \(\varphi(x)\) and \(\omega(x)\) are functions differentiable on the interval \((0,1)\).
Theorem 2. The necessary conditions for order of approximation \(h_{i+1}-h_i\) for the difference scheme \(L_{h_i}^{(k,p,q,f)}\), of second order in the class \(k,p,q,f\in C_{[0,1]}^{(2,1)}\), have the form
\[ A_1^{(0)}[s]=0,\qquad B_1^{(0)}[s]+\frac12=0,\qquad D_1^{(0)}[s]=0,\qquad F_1^{(0)}[s]=0. \tag{1.10} \]
5°. Difference boundary-value problem. To Problem I we associate the difference boundary-value problem
\[ L_{h_i}^{(k,p,q)}y_i=\Phi_i\quad (i=2,3,\ldots,N-2), \tag{1.11} \]
\[ y_1=0,\qquad y_{N-1}=0, \tag{1.12} \]
\[ \frac{\nabla y_1}{h_i}-\frac{h_1}{2h_1}\Delta\left(\frac{\nabla y_1}{h_1}\right)=0,\qquad \frac{\nabla y_N}{h_N}+\frac{h_N}{2h_{N-1}}\nabla\left(\frac{\nabla y_N}{h_N}\right)=0. \tag{1.13} \]
It is easy to see that the difference boundary conditions (1.12), (1.13) approximate the boundary conditions (1.2) with accuracy up to \(O(h_1^2)\) and \(O(h_{N-1}^2)\). We shall call problem (1.11)—(1.13) problem II.
6°. Difference boundary-value problem for the error of the solution. Let \(U(x)\) and \(y_{h_i}\) denote, respectively, the solutions of problems I and II. For the error of the solution \(z_i=y_i-U_i\) we obtain the following difference boundary-value problem:
\[ L_{h_i}^{(k,p,q)} z_i=\Psi_i \quad (i=2,3,\ldots,N-2), \tag{1.14} \]
\[ z_1=\nu_1,\qquad z_{N-1}=\mu_1, \tag{1.15} \]
\[ \frac{\nabla z_1}{h_1}-\frac{h_1}{2h_1}\Delta\left(\frac{\nabla z_1}{h_1}\right)=\nu_2,\qquad \frac{\nabla z_N}{h_N}+\frac{h_N}{2h_{N-1}}\nabla\left(\frac{\nabla z_N}{h_N}\right)=\mu_2, \tag{1.16} \]
where \(\Psi_i\) is the approximation error of the scheme \(L_{h_i}^{(k,p,q,f)}\) on the solution of equation (1.1), \(\nu_1=-U_1,\ \mu_1=-U_{N-1}\),
\[ \nu_2=-\left(\frac{\nabla U_1}{h_1}-\frac{h_1}{2h_1}\Delta\left(\frac{\nabla U_1}{h_1}\right)\right),\qquad \mu_2=-\left(\frac{\nabla U_N}{h_N}+\frac{h_N}{2h_{N-1}}\nabla\left(\frac{\nabla U_N}{h_N}\right)\right). \]
7°. Difference Green formulas. With the aid of the summation-by-parts formula
\[ \sum_{i=2}^{N-2}\frac{1}{h_i}\Delta\left(\frac{1}{h_i}\nabla W_i\right)V_i h_i = \sum_{i=1}^{N-1} W_i\Delta\left(\frac{\nabla V_i}{h_i}\right)+ \tag{1.17} \]
\[ +\left( V_{i-1}\frac{\nabla W_{i-1}}{h_{i-1}}-W_{i-1}\frac{\nabla V_i}{h_i}\right)_{i=N} -\left( V_{i+1}\frac{\Delta W_{i+1}}{h_{i+2}}-W_{i+1}\frac{\Delta V_i}{h_{i+1}}\right)_{i=0} \]
we obtain the first and second difference Green formulas, which are the principal tool in estimating the error of the solution:
\[ \sum_{i=2}^{N-2} V_i L_{h_i}^{(k,p)} W_i h_i = \sum_{i=1}^{N-1}\frac{a_i}{h_i}\Delta\left(\frac{\nabla W_i}{h_i}\right)\Delta\left(\frac{\nabla V_i}{h_i}\right) + \sum_{i=2}^{N-2} b_i\frac{\nabla W_i}{h_i}\nabla V_i+ \]
\[ +\left\{ \frac{V_{i-1}}{h_{i-1}}\nabla\left[ \frac{a_{i-1}}{h_{i-1}}\nabla\left(\frac{\nabla W_i}{h_i}\right) \right] - \frac{a_{i-1}}{h_{i-1}}\nabla\left(\frac{\nabla W_i}{h_i}\right)\frac{\nabla V_i}{h_i} \right\}_{i=N} \tag{1.18} \]
\[ -\left\{ \frac{V_{i+1}}{h_{i+2}}\Delta\left[ \frac{a_{i+1}}{h_{i+1}}\Delta\left(\frac{\Delta W_i}{h_{i+1}}\right) \right] - \frac{a_{i+1}}{h_{i+1}}\Delta\left(\frac{\Delta W_i}{h_{i+1}}\right)\frac{\Delta V_i}{h_{i+1}} \right\}_{i=0} - \]
\[ -\frac{b_{N-1}}{h_{N-1}}\nabla W_{N-1}V_{N-1} + \frac{b_2}{h_2}V_1\nabla W_2; \]
\[ \sum_{i=2}^{N-2}\left(V_iL_{h_i}^{(k,p)}W_i-W_iL_{h_i}^{(k,p)}V_i\right)h_i = \]
\[
=\left\{\left[\frac{V_{i-1}}{h_{i-1}}\nabla\left(\frac{a_{i-1}}{h_{i-1}}\nabla\left(\frac{\nabla W_i}{h_i}\right)\right)
-\frac{a_{i-1}}{h_{i-1}}\nabla\left(\frac{\nabla W_i}{h_i}\right)\frac{\nabla V_i}{h_i}\right]\right.
\]
\[
\left.
-\left[\frac{W_{i-1}}{h_{i-1}}\nabla\left(\frac{a_{i-1}}{h_{i-1}}\nabla\left(\frac{\nabla V_i}{h_i}\right)\right)
-\frac{a_{i-1}}{h_{i-1}}\nabla\left(\frac{\nabla V_i}{h_i}\right)\frac{\nabla W_i}{h_i}\right]\right\}_{i=N}
\tag{1.19}
\]
\[
-\left\{\left[\frac{V_{i+1}}{h_{i+2}}\Delta\left(\frac{a_{i+1}}{h_{i+1}}\Delta\left(\frac{\Delta W_i}{h_{i+1}}\right)\right)
-\frac{a_{i+1}}{h_{i+1}}\Delta\left(\frac{\Delta W_i}{h_{i+1}}\right)\frac{\Delta V_i}{h_{i+1}}\right]\right.
\]
\[
\left.
-\left[\frac{W_{i+1}}{h_{i+2}}\Delta\left(\frac{a_{i+1}}{h_{i+1}}\Delta\left(\frac{\Delta V_i}{h_{i+1}}\right)\right)
-\frac{a_{i+1}}{h_{i+1}}\Delta\left(\frac{\Delta V_i}{h_{i+1}}\right)\frac{\Delta W_i}{h_{i+1}}\right]\right\}_{i=0}
\]
\[
-\frac{1}{h_{N-1}}\,b_{N-1}\left(V_{N-1}\nabla W_{N-1}-W_{N-1}\nabla V_{N-1}\right)
+\frac{b_2}{h_2}\left(V_1\nabla W_2-W_1\nabla V_2\right).
\]
8°. Difference Green’s function of problem II. We define the difference Green’s function \(G_{ij}\) of problem II by the conditions:
1) the function \(G_{ij}\), as a function of the variable \(i\) for fixed \(j\), satisfies the equation
\[ L_{h_i}^{(k,p,q)}G_{ij}=\frac{\delta_{ij}}{h_i},\qquad 1<i<N-1,\qquad \delta_{ij}= \begin{cases} 1,& i=j,\\ 0,& i\ne j; \end{cases} \tag{1.20} \]
2) the function \(G_{ij}\) satisfies the difference boundary conditions
\[ G_{1j}=0,\qquad G_{N-1,j}=0; \tag{1.21} \]
\[ \frac{\nabla G_{1j}}{h_1}-\frac{h_1}{2h_1}\Delta\left(\frac{\nabla G_{1j}}{h_1}\right)=0,\qquad \frac{\nabla G_{Nj}}{h_N}+\frac{h_N}{2h_{N-1}}\nabla\left(\frac{\nabla G_{Nj}}{h_N}\right)=0. \tag{1.22} \]
We seek the Green’s function in the form
\[ G_{ij}= \begin{cases} \overline{G}_{ij}=\alpha_j V_i^{(1)}+\beta_j V_i^{(2)},& i<j,\\[2mm] \overline{\overline{G}}_{ij}=\gamma_j V_i^{(3)}+\kappa_j V_i^{(4)},& i>j, \end{cases} \tag{1.23} \]
where \(V_i^{(m)}\) \((m=1,2,3,4)\) are difference functions satisfying the equation
\(L_{h_i}^{(k,p,q)}V_i^{(m)}=0\), as well as the conditions:
\[ V_1^{(1)}=0,\qquad V_1^{(2)}=0, \]
\[ \frac{\nabla V_1^{(1)}}{h_1}-\frac{h_1}{2h_1}\Delta\left(\frac{\nabla V_1^{(1)}}{h_1}\right)=0,\qquad \frac{\nabla V_1^{(2)}}{h_1}-\frac{h_1}{2h_1}\Delta\left(\frac{\nabla V_1^{(2)}}{h_1}\right)=0, \]
\[ \frac{a_1}{h_1}\Delta\left(\frac{\nabla V_1^{(1)}}{h_1}\right)=1,\qquad \frac{a_1}{h_1}\Delta\left(\frac{\nabla V_1^{(2)}}{h_1}\right)=0, \]
\[ \frac{1}{h_2}\Delta\left[\frac{a_1}{h_1}\Delta\left(\frac{\nabla V_1^{(1)}}{h_1}\right)\right]=0,\qquad \frac{1}{h_2}\Delta\left[\frac{a_1}{h_1}\Delta\left(\frac{\nabla V_1^{(2)}}{h_1}\right)\right]=1, \tag{1.24} \]
\[ V_{N-1}^{(3)}=0,\qquad V_{N-1}^{(4)}=0, \]
\[ \frac{\nabla V_N^{(3)}}{h_N}+\frac{h_N}{2h_{N-1}}\nabla\left(\frac{\nabla V_N^{(3)}}{h_N}\right)=0,\qquad \frac{\nabla V_N^{(4)}}{h_N}+\frac{h_N}{2h_{N-1}}\nabla\left(\frac{\nabla V_N^{(4)}}{h_N}\right)=0, \]
\[ \frac{a_{N-1}}{\hbar_{N-1}}\,\Delta\left(\frac{\nabla V_{N-1}^{(3)}}{h_{N-1}}\right)=1,\qquad \frac{a_{N-1}}{\hbar_{N-1}}\,\Delta\left(\frac{\nabla V_{N-1}^{(4)}}{h_{N-1}}\right)=0, \]
\[ \frac{1}{h_{N-1}}\nabla\left[\frac{a_{N-1}}{\hbar_{N-1}}\,\Delta\left(\frac{\nabla V_{N-1}^{(3)}}{h_{N-1}}\right)\right]=0,\qquad \frac{1}{h_{N-1}}\nabla\left[\frac{a_{N-1}}{\hbar_{N-1}}\,\Delta\left(\frac{\nabla V_{N-1}^{(4)}}{h_{N-1}}\right)\right]=-1. \]
We find the coefficients \(\alpha_j,\ \beta_j,\ \gamma_j\), and \(\varkappa_j\) from the conditions
\[ \overline{G}_{ij}=\overline{\overline{G}}_{ij}\ (i=j),\qquad \frac{\nabla \overline{G}_{ij}}{h_i}=\frac{\nabla \overline{\overline{G}}_{ij}}{h_i}\ (i=j), \]
\[ \frac{a_i}{\hbar_i}\Delta\left(\frac{\nabla \overline{G}_{ij}}{h_i}\right) = \frac{a_i}{\hbar_i}\Delta\left(\frac{\nabla \overline{\overline{G}}_{ij}}{h_i}\right)\ (i=j), \tag{1.25} \]
\[ \frac{1}{h_i}\nabla\left[\frac{a_i}{\hbar_i}\Delta\left(\frac{\nabla \overline{G}_{ij}}{h_i}\right)\right] - \frac{1}{h_i}\nabla\left[\frac{a_i}{\hbar_i}\Delta\left(\frac{\nabla \overline{\overline{G}}_{ij}}{h_i}\right)\right] =-1\ (i=j). \]
The system (1.25) was studied in I for the case when \(h_i=h=\mathrm{const}\). The function \(G_{ij}\) on a nonuniform mesh has the same properties as on a uniform mesh: for the function \(G_{ij}\) and its difference quotients the inequalities
\[ |G_{ij}|\leq M,\qquad \left|\frac{\nabla G_{ij}}{h_j}\right|\leq M,\qquad \left|\frac{1}{\hbar_j}\Delta\left(\frac{\nabla G_{ij}}{h_j}\right)\right|\leq M, \]
\[ \left|\frac{1}{h_j}\nabla\left[\frac{a_j}{\hbar_j}\Delta\left(\frac{\nabla G_{ij}}{h_j}\right)\right]\right|\leq M, \tag{1.26} \]
hold, where \(M>0\) is a constant independent of \(h_i\). With the aid of the second difference Green’s formula we immediately establish the symmetry property of the Green’s function: \(G_{ij}=G_{ji}\). The proof of the above-mentioned properties of the function \(G_{ij}\) is no different from the proof given in I [1] for a uniform mesh.
§ 2. ACCURACY IN THE CLASS OF SMOOTH COEFFICIENTS
1°. A priori estimates.
Lemma 1. If \(\Psi_i\) has the form (1.9), and \(z_{h_i}\) is the solution of problem (1.14)—(1.16), then for \(z_i\) the estimate
\[ |z_i|\leq M'\left\{\|\Psi_i^{(a)}\|_1+\|\Psi_i^*\|_2+\sum_{j=2}^{N-1}|\Psi_j^{(b)}|h_j+\right. \]
\[ \left. +|\Psi_{N-1}^{(a)}|h_N+|\Psi_1^{(a)}|h_1+|\mu_1|+|\mu_2|+|\nu_1|+|\nu_2|\right\}, \tag{2.1} \]
where
\[ \|\Psi_i^{(a)}\|_1=\sum_{j=2}^{N-1}|\nabla\eta_j|+\left|\frac{\nabla\eta_N}{h_N}\right|,\qquad \|\Psi_i^*\|_2=\sum_{j=1}^{N-1}|\varkappa_j|h_j+|\varkappa_{N-1}|h_N, \]
\[ \nabla\eta_i=h_i\sum_{s=1}^{i-1}\hbar_s\,\frac{\Psi_s^{(a)}}{a_s},\qquad \varkappa_i=\sum_{s=2}^{i-1}\hbar_s\Psi_s^*\sum_{k=s+1}^{i}h_k, \]
\(M'>0\) is a constant independent of \(h_i\).
Proof. Using the function \(G_{ij}\), we represent the solution of problem (1.14)—(1.16) in the form
\[ z_i=\sum_{j=2}^{N-2} G_{ij}\Psi_j h_j+\omega, \tag{2.2} \]
where
\[ \omega= \frac{\mu_1}{h_{N-1}}\nabla\left[ \frac{a_{N-1}}{h_{N-1}}\nabla\left(\frac{\nabla G_{iN}}{h_N}\right) \right] - \frac{\mu_2}{h_{N-1}}\left[ a_{N-1}\nabla\left(\frac{\nabla G_{iN}}{h_N}\right) \right] - \]
\[ - \frac{\nu_1}{h_2}\Delta\left[ \frac{a_1}{h_1}\Delta\left(\frac{\nabla G_{i1}}{h_1}\right) \right] + \frac{\nu_2}{h_1}\left[ a_1\Delta\left(\frac{\nabla G_{i1}}{h_1}\right) \right] - \]
\[ - \frac{\mu_1}{h_{N-1}} b_{N-1}\nabla G_{i,N-1} + \frac{\nu_1}{h_2} b_2\nabla G_{i2}. \]
Taking (1.9) into account and applying the summation-by-parts formula, we obtain
\[ z_i= \sum_{j=1}^{N-1}\frac{1}{h_j} \Delta\left(\frac{\nabla G_{ij}}{h_j}\right)\Psi_j^{(a)}h_j + \frac{\nabla G_{i1}}{h_1}\Psi_1^{(a)} - \frac{\nabla G_{iN}}{h_N}\Psi_{N-1}^{(a)} - \]
\[ - \sum_{j=2}^{N-1}\frac{\nabla G_{ij}}{h_j}\Psi_j^{(b)}h_j + \sum_{j=2}^{N-2}G_{ij}\Psi_j^*h_j+\omega . \tag{2.3} \]
Let us introduce the difference vector-functions \(\eta_{h_i}\) and \(\chi_{h_i}\), defined by the conditions:
\[ \frac{a_i}{h_i}\Delta\left(\frac{\nabla\eta_i}{h_i}\right) = \Psi_i^{(a)} \quad (i=1,2,\ldots,N-1),\qquad \eta_0=\eta_1=0, \tag{2.4} \]
\[ \frac{1}{h_i}\Delta\left(\frac{\nabla\chi_i}{h_i}\right) = \Psi_i^* \quad (i=2,3,\ldots,N-2),\qquad \chi_1=\chi_2=0. \tag{2.5} \]
Solving problems (2.4) and (2.5), we find
\[ \nabla\eta_i = h_i\sum_{s=1}^{i-1} h_s\frac{\Psi_s^{(a)}}{a_s} \quad (i=2,3,\ldots,N), \tag{2.6} \]
\[ \chi_i= \sum_{s=2}^{i-1} h_s\Psi_s^*\sum_{k=s+1}^{i}h_k \quad (i=3,4,\ldots,N-1). \tag{2.7} \]
Equality (2.3) takes the form
\[ z_i= -\sum_{j=2}^{N-1}\frac{1}{h_j} \nabla\left[ \frac{a_j}{h_j}\Delta\left(\frac{\nabla G_{ij}}{h_j}\right) \right]\nabla\eta_j + \frac{a_{N-1}}{h_{N-1}} \Delta\left(\frac{\nabla G_{i,N-1}}{h_{N-1}}\right) \frac{\nabla\eta_N}{h_N} + \]
\[ + \frac{\nabla G_{i1}}{h_1}\Psi_1^{(a)} - \frac{\nabla G_{iN}}{h_N}\Psi_{N-1}^{(a)} - \sum_{j=2}^{N-1}\frac{\nabla G_{ij}}{h_j}\Psi_j^{(b)}h_j + \]
\[ + \sum_{j=1}^{N-1} \frac{1}{h_j}\Delta\left(\frac{\nabla G_{ij}}{h_j}\right)\varkappa_j h_j -\frac{\Delta G_{iN}}{h_N}\varkappa_{N-1}+\omega . \tag{2.8} \]
Taking into account inequality (1.26) and the difference boundary conditions (1.21)—(1.22), we obtain the estimate
\[
|z_i|\leq M'\left\{
\sum_{j=2}^{N-i}|\nabla\eta_j|
+\left|\frac{\nabla\eta_N}{h_N}\right|
+|\Psi_1^{(a)}|h_1
+|\Psi_{N-1}^{(a)}|h_N
\right.
\]
\[
\left.
+\sum_{j=2}^{N-1}|\Psi_j^{(b)}|h_j
+\sum_{j=1}^{N-1}|\varkappa_j|h_j
+|\varkappa_{N-1}|h_N
+|\mu_1|+|\mu_2|+|\nu_1|+|\nu_2|
\right\},
\tag{2.9}
\]
where \(M'>0\) is a constant independent of \(h_i\). The lemma is proved.
Lemma 2. Let \(z_{h_i}\) be the solution of problem (1.14)—(1.16), and let \(\Psi_i\) have the form (1.9) at all mesh points, with the exception of the points \(x_{n-1}, x_n, x_{n+1}, x_{n+2}\). Then for \(z_i\) the inequalities
\[
|z_i|\leq M''\left\{
\sum_{j=1}^{n-1}|\Psi_j^{(a)}|h_j
+\sum_{j=n+2}^{N-1}|\Psi_j^{(a)}|h_j
+\sum_{j=2}^{n-2}|\Psi_j^{(b)}|h_j
\right.
\]
\[
\left.
+\sum_{j=n+4}^{N-1}|\Psi_j^{(b)}|h_j
+\sum_{j=2}^{n-2}|\Psi_j^*|h_j
+\sum_{j=n+3}^{N-2}|\Psi_j^*|h_j
+|\Psi_{n-1}^{(a)}|
\right.
\]
\[
\left.
+\left|\frac{\nabla\Psi_{n-1}^{(a)}}{h_{n-1}}\right|
+|\Psi_{n+2}^{(a)}|
+\left|\frac{\nabla\Psi_{n+3}^{(a)}}{h_{n+3}}\right|
+|\Psi_{n-1}^{(b)}|+|\Psi_{n+3}^{(b)}|
\right.
\]
\[
\left.
+|\mu_1|+|\mu_2|+|\nu_1|+|\nu_2|+\|\Psi_j\|_3
\right\},
\tag{2.10}
\]
where
\[
\|\Psi_j\|_3=
|\Psi'_{\,n-1}|h_{n-1}h_nh_n
+|2\Psi_{n-1}h_{n-1}h_n+\Psi_n h_nh_{n+1}|h_{n+1}
\]
\[
+|3\Psi_{n-1}h_{n-1}h_n^*+2\Psi_nh_nh_{n+1}+\Psi_{n+1}h_{n+1}h_{n+2}^*|
\]
\[
+|\Psi_{n-1}h_{n-1}+\Psi_nh_n+\Psi_{n+1}h_{n+1}+\Psi_{n+2}h_{n+2}|,
\]
\[ s h_i^* = h_i+h_{i+1}+\cdots+h_{i+s-1}. \]
Proof. Represent \(z_i\) in the form of a sum
\[ z_i=\bar z_i+\overline{\bar z}_i, \tag{2.11} \]
where
\[ \bar z_i= \sum_{j=2}^{n-2}G_{ij}\Psi_jh_j +\sum_{j=n+3}^{N-2}G_{ij}\Psi_jh_j+\omega, \qquad \overline{\bar z}_i= \sum_{j=n-1}^{n+2}G_{ij}\Psi_jh_j . \]
Summing by parts, we obtain
\[ \sum_{j=2}^{n-2}G_{ij}\Psi_jh_j = \sum_{j=1}^{n-1}\frac{1}{h_j} \Delta\left(\frac{\nabla G_{ij}}{h_j}\right)\Psi_j^{(a)}h_j -\frac{\Delta G_{i,n-1}}{h_n}\Psi_{n-1}^{(a)}+ \]
\[ +G_{i,n-2}\Psi_{n-1}^{(b)}+G_{i,n-1}\frac{\nabla\Psi_{n-1}^{(a)}}{h_{n-1}} +\sum_{j=2}^{n-2}G_{ij}\Psi_j^*h_j -\sum_{j=2}^{n-2}\frac{1}{h_j}\nabla G_{ij}\Psi_j^{(b)}h_j; \tag{2.12} \]
\[ \sum_{j=n+3}^{N-2}G_{ij}\Psi_jh_j = \sum_{j=n+2}^{N-1}\frac{1}{h_j}\Delta\left(\frac{\nabla G_{ij}}{h_j}\right)\Psi_j^{(a)}h_j +\frac{\nabla G_{i,n+2}}{h_{n+2}}\Psi_{n+2}^{(a)} - \]
\[ -G_{i,n+3}\Psi_{n+3}^{(b)} -G_{i,n+2}\frac{\nabla\Psi_{n+3}^{(a)}}{h_{n+3}} +\sum_{j=n+3}^{N-2}G_{ij}\Psi_j^*h_j -\sum_{j=n+4}^{N-1}\frac{1}{h_j}\nabla G_{ij}\Psi_j^{(b)}h_j. \tag{2.13} \]
Define the function \(\varphi_i\) from the conditions
\[ \frac{1}{h_i}\Delta\left(\frac{\nabla\varphi_i}{h_i}\right)=\Psi_i \quad (i=n-1,\ n,\ n+1,\ n+2), \]
\[ \varphi_{n-2}=\varphi_{n-1}=0. \tag{2.14} \]
Solving problem (2.14), we find
\[ \varphi_n=\Psi_{n-1}h_{n-1}h_n,\qquad \varphi_{n+1}=2\Psi_{n-1}h_{n-1}h_n+\Psi_n h_nh_{n+1}, \]
\[ \varphi_{n+2}=3\Psi_{n-1}h_{n-1}h_n^*+2\Psi_n h_nh_{n+1} +\Psi_{n+1}h_{n+1}h_{n+2}, \]
\[ \varphi_{n+3}=4\Psi_{n-1}h_{n-1}h_n^* +3\Psi_n h_nh_{n+1}^* +2\Psi_{n+1}h_{n+1}h_{n+2} +\Psi_{n+2}h_{n+2}h_{n+3}. \]
Starting from the equality
\[ \bar z_i= \sum_{j=n-1}^{n+2}G_{ij}\Psi_jh_j +\frac{1}{h_n}\Delta\left(\frac{\nabla G_{in}}{h_n}\right)\varphi_nh_n+ \]
\[ +\frac{1}{h_{n+1}}\Delta\left(\frac{\nabla G_{i,n+1}}{h_{n+1}}\right)\varphi_{n+1}h_{n+1} -\frac{\nabla G_{i,n+2}}{h_{n+2}}\varphi_{n+2} +G_{i,n+2}\frac{\nabla\varphi_{n+3}}{h_{n+3}}, \tag{2.15} \]
and inequality (1.26), we obtain the estimate
\[ |\bar z_i|\le M\{|\Psi_{n-1}|h_{n-1}h_nh_n +|2\Psi_{n-1}h_{n-1}h_n+\Psi_nh_nh_{n+1}|h_{n+1}+ \]
\[ +|3\Psi_{n-1}h_{n-1}h_n^*+2\Psi_nh_nh_{n+1} +\Psi_{n+1}h_{n+1}h_{n+2}|+ \]
\[ +|\Psi_{n-1}h_{n-1}+\Psi_nh_n+\Psi_{n+1}h_{n+1} +\Psi_{n+2}h_{n+2}|\}, \tag{2.16} \]
where \(M>0\) is a constant independent of \(h_i\). Estimating \(\bar z_i\) and joining inequality (2.16), we obtain estimate (2.10).
2°. Estimate of the error of the solution in the class of discontinuous functions.
Theorem 3. If \(L_{h_i}^{(k,p,q,f)}\) is a scheme of second order, for which conditions (1.10) are satisfied, then for the error of the solution in the class \(k,p,q,\ f\in C_{[0,1]}^{(2,1)}\) the estimate
\[ \|z_{h_i}\|_0\le M'''(h^*)^2, \tag{2.17} \]
is valid, where \(M'''>0\) is a constant independent of \(h_i\).
The proof of the theorem follows from Lemma 1.
§ 3. ACCURACY IN THE CLASS OF DISCONTINUOUS FUNCTIONS
1°. Approximation error in a neighborhood of a point of discontinuity of the functions \(k, p, q, f\). If \(\xi \in (x_n, x_{n+1})\) is a point of discontinuity of the functions \(k, p, q, f\), then for
\(i=n-1, n, n+1, n+2\), \(\Psi_i\) is a quantity of order \(\dfrac{1}{h_i^2}\); at all other nodes of the grid \(\Psi_i\) has the form (1.9). To determine the necessary convergence conditions in the class of discontinuous functions we shall need estimates of the expressions
\[ \varphi_1= \left|3\Psi_{n-1}h_{n-1}h_n^*+ 2\Psi_n h_n h_{n+1}+ \Psi_{n+1}h_{n+1}h_{n+2}\right|, \tag{3.1} \]
\[ \varphi_2= \left|\Psi_{n-1}h_{n-1}+ \Psi_n h_n+ \Psi_{n+1}h_{n+1}+ \Psi_{n+2}h_{n+2}\right|. \tag{3.2} \]
2°. Necessary convergence conditions. In estimating (3.1) and (3.2), the principal role is played by the expressions:
\[ \begin{aligned} S_1={}& \frac{a_{n+2}}{\mathbf{h}_{n+2}} \Delta\left(\frac{\nabla U_{n+2}}{h_{n+2}}\right) - \frac{a_{n-1}}{\mathbf{h}_{n-1}} \Delta\left(\frac{\nabla U_{n-1}}{h_{n-1}}\right) \cdot \frac{h_{n-1}+h_n+h_{n+1}+h_{n+2}}{h_{n-1}} \\ &\quad + \frac{a_{n-2}}{\mathbf{h}_{n-2}} \cdot \frac{h_n+h_{n+1}+h_{n+2}}{h_{n-1}} \Delta\left(\frac{\nabla U_{n-2}}{h_{n-2}}\right), \end{aligned} \tag{3.3} \]
\[ \begin{aligned} S_2={}& \frac{a_{n+3}}{\mathbf{h}_{n+3}h_{n+3}} \Delta\left(\frac{\nabla U_{n+3}}{h_{n+3}}\right) - \frac{a_{n+2}}{\mathbf{h}_{n+2}h_{n+3}} \Delta\left(\frac{\nabla U_{n+2}}{h_{n+2}}\right) \\ &\quad - \frac{a_{n-1}}{\mathbf{h}_{n-1}h_{n-1}} \Delta\left(\frac{\nabla U_{n-1}}{h_{n-1}}\right) + \frac{a_{n-2}}{\mathbf{h}_{n-2}h_{n-1}} \Delta\left(\frac{\nabla U_{n-2}}{h_{n-2}}\right) \\ &\quad - \left( b_{n+3}\frac{\Delta U_{n+2}}{h_{n+3}} - b_{n-1}\frac{\nabla U_{n-1}}{h_{n-1}} \right). \end{aligned} \tag{3.4} \]
If \(L_{h_i}^{(k,p,q,f)}\) is a second-rank scheme for which conditions (1.10) are fulfilled, then \(S_1\) can be written in the following form:
\[ S_1=\omega\left( \frac{a_{n+2}}{k_{\mathrm{r}}} - \frac{h_{n-1}+h_n+h_{n+1}+h_{n+2}}{h_{n-1}} \frac{a_{n-1}}{k_{\mathrm{l}}} + \frac{h_n+h_{n+1}+h_{n+2}}{h_{n-1}} \frac{a_{n-2}}{k_{\mathrm{l}}} \right)+O(\tilde h), \tag{3.5} \]
where \(\omega=(kU'')_{\mathrm{r}}=(kU'')_{\mathrm{l}}\),
\(\tilde h=\max(h_{n-2},h_{n-1},\ldots,h_{n+3})\). Taking into account the properties of template functionals, we obtain
\[ \varphi_1=O(\tilde h), \tag{3.6} \]
since \(S_1=O(\tilde h)\).
Let us proceed to the estimate of \(\varphi_2\). For this we estimate \(S_2\):
\[ S_2= \left( k'_{\mathrm{r}}U''_{\mathrm{r}} + \frac{h_{n+2}+h_{n+3}+h_{n+4}}{3h_{n+3}} k_{\mathrm{r}}U'''_{\mathrm{r}} - p_{\mathrm{r}}U'(\xi) \right) - \]
\[ -\left(k_l' U_l''+\frac{h_{n-2}+h_{n-1}+h_n}{3h_{n-1}}\,k_l U_l'''-p_l U'(\xi)\right)+O(\bar h), \tag{3.7} \]
where \(\bar h=\max(\widetilde h,h_{n+4})\). If
\[ h_{n-2}=h_{n-1}=h_n, \tag{3.8} \]
\[ h_{n+2}=h_{n+3}=h_{n+4}, \tag{3.9} \]
then \(S_2\) and, consequently, \(\varphi_2\) are quantities of order \(O(\bar h)\). We have proved
Theorem 4. The necessary conditions for convergence in the class of discontinuous functions have the form (3.8) and (3.9).
\(3^\circ\). On accuracy in the class of discontinuous functions.
Theorem 5. If \(L_{h_i}^{(k,p,q,f)}\) is a second-rank scheme for which conditions (1.10) and the necessary convergence conditions (3.8) and (3.9) are satisfied, then the solution of difference boundary-value problem II in the class \(k,p,q,f\in Q_{[0,1]}^{(2,1)}\) has first-order accuracy
\[ \|z_{h_i}\|_0 \le \bar M h^*, \tag{3.10} \]
where \(\bar M>0\) is a constant independent of \(h_i\).
The proof follows from Lemma 2. The difference scheme with coefficients
\[ a_i=\left(\int_0^1 \frac{(1-s)\,ds}{k(x_i+sh_i)} +\int_0^1 \frac{(1-s)\,ds}{k(x_i-sh_i)}\right)^{-1}, \qquad b_i=\int_{-1}^0 p(x_i+sh_i)\,ds, \]
\[ d_i=\int_{-\frac12}^{\frac12} q(x_i+sh_i)\,ds, \qquad \Phi_i=\int_{-\frac12}^{\frac12} f(x_i+sh_i)\,ds \]
ensures, in the class \(Q_{[0,1]}^{(2,1)}\), second-order accuracy when the necessary convergence conditions are satisfied. This follows from [1]. A. A. Samarskii, together with Hao Zhou, investigated homogeneous difference schemes on nonuniform grids for an equation of the fourth order, as he kindly informed us. The results of these investigations, unknown to us, have not yet appeared in print.
§ 4. STABILITY OF THE DIFFERENCE SCHEMES \(L_{h_i}^{(k,p,q,f)}\)
\(1^\circ\). Some definitions.
We write problems I and II, respectively, in the form of operator equations
\[ LU=f, \tag{4.1} \]
\[ L_{h_i}y_{h_i}=\Phi_{h_i}, \tag{4.2} \]
where \(\Phi_{h_i}=(0,0,\Phi_2,\ldots,\Phi_{N-2},0,0)\) is an \(N+1\)-dimensional vector. \(L_{h_i}\) will be called the difference operator corresponding to the differential operator \(L\). The space of \(N+1\)-dimensional vectors \(W_{h_i}=(W_0,W_1,\ldots,\)
\(W_N\)) with scalar product and norm defined, respectively, by the equalities
\[ (W_{h_i}, V_{h_i})=\sum_{i=0}^{N} W_i V_i h_i,\qquad \|W_{h_i}\|_0=\max_{\omega_{h_i}} |W_i|, \]
will be called the space \(E_{[0,1]h_i}\). We shall say that a certain difference operator \(T_{h_i}\) is symmetric if, for arbitrary two vectors \(W_{h_i}\) and \(V_{h_i}\) from its domain of definition, the equality
\[ (T_{h_i}W_{h_i}, V_{h_i})=(W_{h_i}, T_{h_i}V_{h_i}) \]
holds. A symmetric difference operator \(T_{h_i}\) is called positive definite if, for an arbitrary vector \(W_{h_i}\) from its domain of definition, the inequality
\[ (T_{h_i}W_{h_i}, W_{h_i})\ge \chi^2 \|W_{h_i}\|_0^2 \]
holds.
Definition. We shall say that the difference scheme \(M_{h_i}^{(k,p,q,f)}\) is stable if the solution of the operator equation
\[ M_{h_i}W_{h_i}=F_{h_i} \tag{4.3} \]
exists for arbitrary vectors \(F_{h_i}\) of type \(\Phi_{h_i}\) and depends continuously on them, this continuous dependence being uniform in \(h_i\): for arbitrary \(\varepsilon>0\) there exists a \(\delta>0\), independent of \(h_i\), such that for the given vector \(W_{h_i}\) satisfying equation (4.3), and for an arbitrary vector \(V_{h_i}\) satisfying the equation \(M_{h_i}V_{h_i}=\widetilde F_{h_i}\), where \(\|F_{h_i}-\widetilde F_{h_i}\|_0<\delta\), the inequality
\[ \|W_{h_i}-V_{h_i}\|_0<\varepsilon \tag{4.4} \]
holds.
The definition of a stable difference scheme was given by P. S. Bondarenko in [3].
2°. On the stability of the schemes \(L_{h_i}^{(k,p,q,f)}\).
Theorem 6. The operator \(L_{h_i}\) is positive definite.
Proof. Applying Green’s formula (1.18), we obtain
\[ \begin{aligned} (L_{h_i}W_{h_i}, V_{h_i}) &=\sum_{i=1}^{N-1}\frac{a_i}{h_i}\, \Delta\left(\frac{\nabla W_i}{h_i}\right) \Delta\left(\frac{\nabla V_i}{h_i}\right) +\sum_{i=2}^{N-2} b_i\,\frac{\nabla W_i}{h_i}\nabla V_i \\ &\quad+\sum_{i=2}^{N-2} d_i W_i V_i h_i +\frac{2a_1}{h_1^3}\,\nabla V_1\nabla W_1 +\frac{2a_{N-1}}{h_N^3}\,\nabla W_N\nabla V_N . \end{aligned} \tag{4.5} \]
The symmetry of the operator \(L_{h_i}\) is proved by formula (4.5). Putting \(V_{h_i}=W_{h_i}\), we obtain
\[ \begin{aligned} (L_{h_i}W_{h_i}, W_{h_i}) &=\sum_{i=1}^{N-1}\frac{a_i}{h_i} \left[\Delta\left(\frac{\nabla W_i}{h_i}\right)\right]^2 +\sum_{i=2}^{N-2} b_i\,\frac{(\nabla W_i)^2}{h_i} \\ &\quad+\sum_{i=2}^{N-2} d_i W_i^2 h_i +\frac{2a_1}{h_1^3}(\nabla W_1)^2 +\frac{2a_{N-1}}{h_N^3}(\nabla W_N)^2 . \end{aligned} \tag{4.6} \]
Neglecting the term
\[ \sum_{i=2}^{N-2}\left[ b_i\frac{(\nabla W_i)^2}{h_i}+d_i W_i^2 h_i\right] +\frac{2a_{N-1}}{h_N^3}(\nabla W_N)^2\geqslant 0, \]
we obtain the inequality
\[ \left(L_{h_i}W_{h_i},\,W_{h_i}\right)\geqslant \frac{c_1}{h^*}\left\{\sum_{i=1}^{N-1} \left[\Delta\left(\frac{\nabla W_i}{h_i}\right)\right]^2 +\frac{(\nabla W_1)^2}{h_1^2}\right\}. \tag{4.7} \]
Since \(W_1=0\), we have \(W_s=\sum_{i=1}^{s-1}\Delta W_i\). Applying the Cauchy inequality
\[ \left(\sum_k a_k b_k\right)^2\leqslant \sum_k a_k^2 \sum_k b_k^2, \]
we obtain the inequality
\[ \begin{aligned} W_s^2 &=\left(\sum_{i=1}^{s-1}\frac{\Delta W_i}{h_i}\cdot h_i\right)^2 \leqslant \sum_{i=1}^{s-1}\frac{(\Delta W_i)^2}{h_i^3}\cdot h_i \sum_{i=1}^{s-1} h_i^2 \leqslant \\ &\leqslant \sum_{i=1}^{N-1}\frac{(\Delta W_i)^2}{h_i^3}\,h_i \sum_{i=1}^{N-1}h_i^2 \leqslant \frac{h^*}{h_*^3}\sum_{i=1}^{N-1}(\Delta W_i)^2 h_i, \end{aligned} \tag{4.8} \]
which also holds for the nodal point at which \(W_s^2\) attains its greatest value; therefore the inequality
\[ \sum_{i=1}^{N-1}(\Delta W_i)^2 h_i \geqslant \frac{h_*^3}{h^*}\,\|W_{h_i}\|_0^2 \tag{4.9} \]
is valid.
It is obvious that
\[ \frac{\Delta W_s}{h_{s+1}} = \sum_{i=1}^{s}\Delta\left(\frac{\nabla W_i}{h_i}\right) +\frac{\nabla W_1}{h_1}. \]
Applying the inequality \((\xi+\eta)^2\leqslant 2(\xi^2+\eta^2)\), as well as the Cauchy inequality, we obtain
\[ \begin{aligned} \left(\frac{\Delta W_s}{h_{s+1}}\right)^2 &= \left(\sum_{i=1}^{s}\Delta\left(\frac{\nabla W_i}{h_i}\right) +\frac{\nabla W_1}{h_1}\right)^2 \\ &\leqslant 2\left[ \left(\sum_{i=1}^{s}\Delta\left(\frac{\nabla W_i}{h_i}\right)\right)^2 +\frac{(\nabla W_1)^2}{h_1^2} \right] = \\ &= 2\left[ \left(\sum_{i=1}^{s}\Delta\left(\frac{\nabla W_i}{h_i}\right)h_i\frac1{h_i}\right)^2 +\frac{(\nabla W_1)^2}{h_1^2} \right] \leqslant \tag{4.10}\\ &\leqslant 2\left[ \sum_{i=1}^{s} \left[\Delta\left(\frac{\nabla W_i}{h_i}\right)\right]^2 h_i^2 \sum_{i=1}^{s}\frac{h_i}{h_i^3} +\frac{(\nabla W_1)^2}{h_1^2} \right]\leqslant \\ &\leqslant 2\left[ \frac{(h^*)^2}{h_*^3} \sum_{i=1}^{N-1} \left[\Delta\left(\frac{\nabla W_i}{h_i}\right)\right]^2 +\frac{(\nabla W_1)^2}{h_1^2} \right], \end{aligned} \]
from which we obtain the inequality
\[ (\Delta W_s)^2\leqslant 2\left[ \frac{(h^*)^4}{h_*^3} \sum_{i=1}^{N-1} \left[\Delta\left(\frac{\nabla W_i}{h_i}\right)\right]^2 +\frac{(h^*)^2}{h_1^2}(\nabla W_1)^2 \right]\leqslant \]
\[ \leqslant 2 \left[\frac{(h_i^*)^4}{h_*^3} \left(\sum_{i=1}^{N-1}\left[\Delta\left(\frac{\nabla W_i}{h_i}\right)\right]^2 +\frac{(\nabla W_1)^2}{h_1^2}\right)\right]. \tag{4.11} \]
Multiplying the left- and right-hand sides of inequality (4.11) by \(h_s\) and summing over the grid, we obtain the inequality
\[ \sum_{s=1}^{N-1}(\Delta W_s)^2 h_s \leqslant 2\,\frac{(h_i^*)^4}{h_*^3} \left(\sum_{i=1}^{N-1}\left[\Delta\left(\frac{\nabla W_i}{h_i}\right)\right]^2 +\frac{(\nabla W_1)^2}{h_1^2}\right). \tag{4.12} \]
Comparing inequalities (4.7), (4.9), (4.12) and taking (1.5) into account, we obtain the inequality
\[ \left(L_{h_i}W_{h_i},\,W_{h_i}\right) \geqslant \frac{c_1}{2}\, \frac{h_*^6}{(h_i^*)^6} \|W_{h_i}\|_0^2 \geqslant \varkappa^2\|W_{h_i}\|_0^2, \tag{4.13} \]
where \(\varkappa\) is a constant independent of \(h_i\). The theorem is proved. Let us formulate a well-known theorem (see [4]).
Theorem 7. In order that a linear operator \(B\) have an inverse operator, it is necessary and sufficient that the homogeneous equation \(BU=0\) have the unique solution \(U=0\).
Theorem 8. On the set of vectors of the type \(\Phi_{h_i}\) there exists an operator \(L_{h_i}^{-1}\), inverse to the operator \(L_{h_i}\), uniformly bounded on this set (with respect to \(h_i\)).
Proof. From the linearity of the operator \(L_{h_i}\) it follows that the equation \(L_{h_i}V_{h_i}=0\) has the solution \(V_{h_i}=0\). From the positive definiteness of the operator \(L_{h_i}\) it follows that this solution is unique. By Theorem 7 there exists an operator \(L_{h_i}^{-1}\), inverse to the operator \(L_{h_i}\). We shall show its uniform boundedness with respect to \(h_i\). With the aid of the operator \(L_{h_i}^{-1}\), we write the solution of the equation
\[ L_{h_i}V_{h_i}=\Phi_{h_i}^{*} \tag{4.14} \]
in the form
\[ V_{h_i}=L_{h_i}^{-1}\Phi_{h_i}^{*}. \tag{4.15} \]
From inequality (4.13) we obtain the inequality
\[ \left\|L_{h_i}^{-1}\Phi_{h_i}^{*}\right\|_0^2 \leqslant \frac{1}{\varkappa^2} \left(\Phi_{h_i}^{*},\,L_{h_i}^{-1}\Phi_{h_i}^{*}\right). \tag{4.16} \]
With the aid of the inequality \((W_{h_i}, W_{h_i})\leqslant \|W_{h_i}\|_0^2\), we obtain
\[ \frac{1}{\varkappa^2} \left(\Phi_{h_i}^{*},\,L_{h_i}^{-1}\Phi_{h_i}^{*}\right) \leqslant \frac{1}{\varkappa^2} \left\{\left(\Phi_{h_i}^{*},\Phi_{h_i}^{*}\right) \left(L_{h_i}^{-1}\Phi_{h_i}^{*},L_{h_i}^{-1}\Phi_{h_i}^{*}\right)\right\}^{\frac12} \leqslant \]
\[ \leqslant \frac{1}{\varkappa^2} \|\Phi_{h_i}^{*}\|_0 \left\|L_{h_i}^{-1}\Phi_{h_i}^{*}\right\|_0, \tag{4.17} \]
whence
\[ \left\|L_{h_i}^{-1}\Phi_{h_i}^{*}\right\|_0 \leqslant \frac{1}{\varkappa^2}\|\Phi_{h_i}^{*}\|_0, \tag{4.18} \]
since \(\Phi_{h_i}^{*}\ne 0\). We have obtained that
\[ \|L_{h_i}^{-1}\|_{0}\leq \frac{1}{\varkappa^{2}}, \]
which proves the assertion of the theorem.
Theorem 9. The difference scheme \(L_{h_i}^{(k,p,q,f)}\) is stable.
Proof. Inequality (4.18) can be written in the form
\[ \|W_{h_i}\|_{0}\leq \frac{1}{\varkappa^{2}}\|\Phi_{h_i}^{*}\|_{0}, \tag{4.19} \]
which was required to be proved.
References
- Hao Shou. Zhurnal vychisl. matem. i matem. fiz., 3, No. 5, 1963, pp. 841–860.
- Tikhonov A. N., Samarskii A. A. Zhurnal vychisl. matem. i matem. fiz., 1, No. 1, 1961, pp. 5–63.
- Bondarenko P. S. Visnyk KDU, series astronomy, mathematics and mechanics, issue 2, No. 2, 1959, pp. 26–44.
- Mikhlin S. G. Variational Methods in Mathematical Physics. Gostekhizdat, Moscow, 1957.
Received by the editors
November 20, 1964
Kyiv State University
named after T. G. Shevchenko