Three-point finite-difference schemes, Padé and the spectral Galerkin method. I. One-sided impedance approximation

A method for calculating special grid placement for three-point schemes which yields exponential superconvergence of the Neumann to Dirich-let map has been suggested earlier. Here we show that such a grid placement can yield impedance which is equivalent to that of a spectral Galerkin method, or more generally to that of a spectral Galerkin-Petrov method. In fact we show that. for every stable Galerkin-Petrov method there is a three-point scheme which yields the same solution at the boundary. We discuss the application of this result to partial differential equations and give numerical examples. We also show equivalence at one corner of a two-dimensional optimal grid with a spectral Galerkin method.

[1]  Sergey Pavlovich Suetin,et al.  On the Rate of Convergence of Padé Approximants of Orthogonal Expansions , 1992 .

[2]  J. Boyd.  Chebyshev & Fourier Spectral Methods , 1989 .

[3]  J. Cooper,et al.  Theory of Approximation , 1960, Mathematical Gazette.

[4]  P. Petrushev,et al.  Rational Approximation of Real Functions , 1988 .

[5]  Siam J. Sci.  SPECTRAL ELEMENTS ON INFINITE DOMAINS , 1998 .

[6]  I. S. Kats.  SPECTRAL FUNCTIONS OF A STRING , 1983 .

[7]  Amir Averbuch,et al.  A Fast Spectral Solver for a 3D Helmholtz Equation , 1999, SIAM J. Sci. Comput..

[8]  A. Gončar,et al.  ON MARKOV'S THEOREM FOR MULTIPOINT PADÉ APPROXIMANTS , 1978 .

[9]  S. Orszag.  Spectral methods for problems in complex geometries , 1980 .

[10]  Radakovič.  The theory of approximation , 1932 .

[11]  Vladimir Druskin,et al.  Gaussian spectral rules for second order finite-difference schemes , 2000, Numerical Algorithms.

[12]  Bengt Fornberg,et al.  A practical guide to pseudospectral methods: Introduction , 1996 .

[13]  Vladimir Druskin,et al.  Gaussian Spectral Rules for the Three-Point Second Differences: I. A Two-Point Positive Definite Problem in a Semi-Infinite Domain , 1999, SIAM J. Numer. Anal..

[14]  W. J. Thron,et al.  Encyclopedia of Mathematics and its Applications. , 1982 .

[15]  Vladimir Druskin,et al.  Optimal finite difference grids and rational approximations of the square root I. Elliptic problems , 2000 .

[16]  Vladimir Druskin,et al.  Application of the Difference Gaussian Rules to Solution of Hyperbolic Problems , 2000 .

[17]  I. Babuska,et al.  Finite Element Analysis , 2021 .

[18]  Kelly Black.  Spectral Elements on Infinite Domain , 1998, SIAM J. Sci. Comput..

[19]  B. Parlett.  The Symmetric Eigenvalue Problem , 1981 .