Gaussian spectral rules for the three-point second differences: I. A two-point positive definite problem in a semi-infinite domain

We suggest an approach to grid optimization for a second order finite-difference scheme for elliptic equations. A model problem corresponding to the three-point finite-difference semidiscretization of the Laplace equation on a semi-infinite strip is considered. We relate the approximate boundary Neumann-to-Dirichlet map to a rational function and calculate steps of our finite-difference grid using the Pade-Chebyshev approximation of the inverse square root. It increases the convergence order of the Neumann-to-Dirichlet map from second to exponential without increasing the stencil of the finite-difference scheme and losing stability.