A Meshless Method Based On Taylor Series
Author(s)
Y. Tampango, M. Potier-Ferry & Y. Koutsawa
Abstract
We study a meshless method based on Taylor series approximation. This method
solves quasi-exactly the Partial Differential Equation (PDE) in the domain. The
boundary conditions are applied by using a least square method as proposed
by Zhang et al. for stabilizing collocation method. Sometimes, the convergence
can only be obtained by splitting the domain in to several subdomains: a new
bridging technique is proposed to match several polynomial approximations.
Finally, numerical techniques like the Domb Sykes plot permits us to estimate
the radius of convergence of the series.
Keywords: meshless method, junction method, convergence analysis, Taylor series
expansion.
1 Introduction
Meshless methods are considered as promising alternatives to overcome the
difficulties related to mesh generation. Each class of meshless method can
be characterized by the form of the used shape functions, the most common
ones being based on moving least square techniques [1], radial functions [2]
or fundamental solutions [3]. In this paper we discuss another recent method
proposed in Z´ez´e et al. [4], where the shape functions are high degree polynoms
that are computed by solving locally the Partial Differential Equation. For instance
in the case of the bi-dimensional Laplace equation, this analytical solution leads
to a family of polynomial shape functions Re(x + iy)n and Im(x + iy)n. In this
respect the presented method is boundary only, because the PDE is solved quasiexactly
inside the domain. This method has several advantages. First the number
of shape functions is much smaller than with the p-version of finite elements [5, 6].
Keywords
meshless method, junction method, convergence analysis, Taylor series
expansion.
Related Book
Other papers in this volume
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