Gaussian spectral rules for the three-point second differences: I. A two-point positive definite problem in a semiinfinite domain Vladimir Druskin ∗ Leonid Knizhnerman † October 5, 1999 Abstract 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 semiinfinite 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 Pad´ e-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. Keywords: finite differences, exponential convergence, Neumann-to-Dirichlet map, Pad´ e-Chebyshev approximation, elliptic equations AMS subject classification: 65N06, 41A21, 15A15 Contents: 1. Introduction. 1.1. Model problem. 1.2. Connection with PDEs. 1.3. Outline. 2. Rational representation of the finite-difference Neumann-to-Dirichlet map and its error bound. 3. Algorithm and its properties. 3.1. Algorithm. 3.2. Step 1: Rational approximation of the square root. 3.3. Steps 2 through 4: Computation of the matrix and the grid. 4. Numerical experiments. 5. Concluding remarks. * Schlumberger-Doll Research, Old Quarry Road, Ridgefield, CT 06877-4108, U.S.A.; e-mail druskin@ridgefield.sdr.slb.com; phone (203) 431-5557; fax (203) 438-3819. † Central Geophysical Expedition, Narodnogo Opolcheniya St., House 40, Bldg. 3, Moscow 123298, Russia; e-mail mmd@cge.ru; phone 7 (205) 192-8156; fax 7 (205) 192-8088. 1