On the estimation of the convergence rate of the difference scheme in the eigenvalue problems (Q2703345)

In the domain \(\Omega=\{x=(x_1,x_2): 0<x_{i}<l_{i}, i=1,2\}\) with boundary \(\Gamma\) the author considers the eigenvalue problem for the elliptic operator NEWLINE\[NEWLINE-\sum_{i,j=1}^{2}{\partial\over\partial x_{i}} \Biggl( K_{ij}(x){\partial u\over\partial x_{j}} \Biggr)= \lambda u, \quad x\in\Omega,\;u(x)=0,\;x\in \Gamma,NEWLINE\]NEWLINE where \(K_{ij}(x)\in W_{\infty}^{2}(\Omega)\), \(K_{ij}(x)=K_{ji}(x)\), and there are constants \(\theta_1\geq\theta_2>0\) such that for all \(x\in\Omega\) and \(\xi_1,\xi_2, \xi_1^2+\xi_2^2 \neq 0\) the following condition is fulfilled NEWLINE\[NEWLINE\theta_2(\xi_1^2+\xi_2^2) \leq\sum_{i,j=1}^{2}K_{ij}(x)\xi_{i}\xi_{j}\leq \theta_1(\xi_1^2+\xi_2^2).NEWLINE\]NEWLINE Using operators of the exact difference scheme a finite difference method of second order accuracy for eigenfunctions is constructed under the condition that eigenfunctions of the given eigenvalue problem belong to space \(W_{2}^3(\Omega)\).