An application of a multipoint differential-difference scheme to a boundary-value problem

For the boundary-value problem
\begin{gather} \Delta U(x,y)=f(x,y),\quad -a<x<a,\quad 0<y<b, \\ \begin{cases} U(-a,y)=\gamma_1(y), & U(x,0)=\gamma_3(x), \\ U(a,y)=\gamma_2(y), & U(x,b)=\gamma_4(x) \end{cases} \end{gather}
we construct a scheme of the method of lines with a central-difference approximation of the derivative $\dfrac{\partial^2U}{\partial y^2}$ for any odd pattern. In particular cases we investigate the behavior at the net refinement of the direct solution of the boundary-value problem for the determination of the difference between the approximate solution obtained by the method of lines and the exact solution of the problem (1), (2). We also consider some modifications of the method of lines: the number of the lines of the net is taken to be equal to that of the pattern. We give an estimate for the norm of the difference between the approximate solution obtained by this method and the exact solution of the problem (1), (2).