On The Use Of Integrated RBFs In Galerkin
Approximation For Elliptic Problems
Author(s)
N. Mai-Duy & T. Tran-Cong
Abstract
This paper presents a new radial-basis-function (RBF) technique for solving elliptic
differential equations (DEs). Integrated RBF networks (IRBFN) are constructed
to satisfy (a) the boundary conditions in a local sense using the point-collocation
formulation, and (b) the DE in a global sense using the Galerkin formulation.
Several numerical examples are presented to demonstrate the attractiveness of the
present implementation.
Keywords: integrated RBFs, Galerkin formulation,Neumann boundary conditions,
multiple boundary conditions, domain decomposition.
1 Introduction
Many discretisation techniques can be viewed as variants of the method of
weighted residuals that can be stated in three well-known formulations, namely
the strong, weak and inverse statements [1]. By means of weighting functions in
a statement, the residuals for the DE and boundary conditions are made small in
some senses. Two popular ways used are (i) the point-collocation approach, where
the residuals are zero at certain points and (ii) the Galerkin-type approach, in which
the residuals are zero in the mean. Each approach has some advantages in certain
areas of application. The former is cost-effective as no integrations are required,
while the latter has a smoothing capability owing to its integral nature.
RBF collocation methods are considered as a powerful tool for the interpolation
of scattered data as well as for the solution of differential problems [2]. The methods
are capable of approximating arbitrarily-well continuous functions. A number
of RBFs such as multiquadric and Gaussian functions have spectral approximation
power. However, the condition number of the RBF interpolation matrix also grows
Keywords
integrated RBFs, Galerkin formulation,Neumann boundary conditions,
multiple boundary conditions, domain decomposition.
Related Book
Other papers in this volume
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