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

Authors:
Vladimir Druskin and Shari Moskow

Journal:
Math. Comp. **71** (2002), 995-1019

MSC (2000):
Primary 65N06, 65N35

DOI:
https://doi.org/10.1090/S0025-5718-01-01349-7

Published electronically:
November 19, 2001

MathSciNet review:
1898743

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Abstract: A method for calculating special grid placement for three-point schemes which yields exponential superconvergence of the Neumann to Dirichlet 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.

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Additional Information

**Vladimir Druskin**

Affiliation:
Schlumberger-Doll Research, Old Quarry Rd, Ridgefield, Connecticut 06877

Email:
druskin@ridgefield.sdr.slb.com

**Shari Moskow**

Affiliation:
Department of Mathematics, University of Florida, 358 Little Hall, P.O. Box 118105, Gainesville, Florida 32611-8105

Email:
moskow@math.ufl.edu

DOI:
https://doi.org/10.1090/S0025-5718-01-01349-7

Keywords:
Second order scheme,
exponential superconvergence,
pseudospectral,
Galerkin-Petrov,
rational approximations

Received by editor(s):
December 2, 1999

Received by editor(s) in revised form:
July 12, 2000, and September 26, 2000

Published electronically:
November 19, 2001

Article copyright:
© Copyright 2001
American Mathematical Society