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Error estimates for moving least square approximations

Summary: Error estimates for moving least square
Mar a G. Armentano and Ricardo G. Dur an
In this paper we obtain error estimates for moving least square approximations in the
one dimensional case. For the application of this method to the numerical solution of
di erential equations it is fundamental to have error estimates for the approximations of
derivatives. We prove that, under appropriate hypothesis on the weight function and the
distribution of points, the method produces optimalorder approximationsof the function
and its rst and second derivatives. As a consequence, we obtain optimal order error
estimates for Galerkin approximations of coercive problems. Finally, as an application of
the moving least square method we consider a convection-di usion equation and propose
a way of introducing up-wind by means of a non-symmetric weight function. We present
several numerical results showing the good behavior of the method.
Key words. error estimates, moving least square, Galerkin approximations, convection-di usion.
AMS Subject Classi cation. 65L70, 65L10, 65D10.
1 Introduction
The moving least square (MLS) as approximation method has been introduced by Shep-
ard 12] in the lowest order case and generalized to higher degree by Lancaster and Salkauskas
6]. The object of those works was to provide an alternative to classic interpolation useful to


Source: Armentano, María Gabriela - Departamento de Matemática, Universidad de Buenos Aires


Collections: Mathematics