Uniform convergence, with respect to a small parameter, of a difference scheme for an elliptic problem in a band (Q1285459)

The periodic boundary problem for a singular perturbed elliptic equation in the strip \([0,1] \times ( - \infty, \infty)\) \[ L_{\varepsilon} \equiv -\varepsilon \biggl( \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} \biggr) - \frac{\partial u}{\partial x} + q(x,y)\cdot u = f(x,y),\quad x \in (0, 1), y \in (-\infty; +\infty), \] \[ u(0,y) = \mu_{1} (y),\;u(1,y) = \mu_{2} (y),\;u(x,y+1) = u(x,y), \quad x \in [0,1],\;y \in (-\infty, +\infty), \] where \(q (x,y) \geq 0\), the functions \(f(x,y)\), \(\mu_{1}(y)\), \(\mu_{2}(y)\) are periodic in \(y\) with the period \(T = 1\), \(\varepsilon \in ( 0, 1]\) is the small parameter, is considered. It is known that for singular perturbed problems the accuracy of classical numerical methods depends not only on a step size of the net but on a small parameter value as well. There is a necessity therefore in development of some special numerical methods with a uniform small parameter convergence. An approximate numerical method with a central difference derivative for the elliptic problem in the above mentioned strip is proposed. It is shown that on the piece-uniform Shishkin's net the scheme proposed uniformly diverges on the small parameter in a sense of the net norm \(L_{\infty}^{h}\) with the speed \(\mathcal{O} (( N^{-2} \ln^{2} N + \overline {N}^{-2})\sqrt{\ln N})\), where \(N\) and \(\overline{N}\) are the numbers of knots in \(x\) and \(y\) directions respectively.