Finite difference operators from moving least squares interpolation

TU Braunschweig, Computational Mathematics, Pockelstrasse 14, 38106 Braunschweig, Germany. t.sonar@tu-bs.de

Received: 10 February 2004
Revised: 27 September 2006
Revised: 30 May 2007

Abstract

In a foregoing paper [Sonar, ESAIM: M2AN 39 (2005) 883–908] we analyzed the Interpolating Moving Least Squares (IMLS) method due to Lancaster and Šalkauskas with respect to its approximation powers and derived finite difference expressions for the derivatives. In this sequel we follow a completely different approach to the IMLS method given by Kunle [Dissertation (2001)]. As a typical problem with IMLS method we address the question of getting admissible results at the boundary by introducing “ghost points”. Most interesting in IMLS methods are the finite difference operators which can be computed from different choices of basis functions and weight functions. We present a way of deriving these discrete operators in the spatially one-dimensional case. Multidimensional operators can be constructed by simply extending our approach to higher dimensions. Numerical results ranging from 1-d interpolation to the solution of PDEs are given.