The Meshless Analog Equation Method:
A New Highly Accurate Truly Mesh-free Method
For Solving Partial Differential Equations
Author(s)
J. T. Katsikadelis
Abstract
A new purely meshless method to solve PDEs is presented. The method is based
on the concept of the analog equation of Katsikadelis, hence its name meshless
analog equation method (MAEM), which converts the original equation into a
simple solvable substitute one of the same order under a fictitious source. The
fictitious source is represented by MQ-RBFS. Integration of the analog equation
allows the approximation of the sought solution by new RBFs. Then inserting the
solution into the PDE and BCs and collocating at the mesh-free nodal points
yields a system of linear equations, which permit the evaluation of the expansion
coefficients. The method exhibits key advantages of over other RBF collocation
methods as it is highly accurate and the matrix of the resulting linear equations is
always invertible. The accuracy is increased using optimal values of the shape
parameters of the multiquadrics by minimizing the potential that produces the
PDE. Without restricting its generality, the method is illustrated by applying it to
the general second order elliptic PDE. The studied examples demonstrate the
efficiency and high accuracy of the developed method.
1 Introduction
The interest in mesh-free methods to solve PDEs has grown noticeably in the
past 20 years. This is mainly due to the fact that (i) Mesh generation over
complicated 2D and especially 3D domains is a very difficult problem and may
require long time, in some cases weeks or months, to create a well behaved mesh
(ii) The convergence rate of the traditional methods is of second order. The
mesh-free MQ-RBFs (multiquadric radial basis functions) method developed by
Keywords
Related Book
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