A comparison of numerical integration rules for the meshless local Petrov–Galerkin method

Abstract

The meshless local Petrov-Galerkin (MLPG) method is a meshfree procedure for solving partial differential equations. However, the benefit in avoiding the mesh construction and refinement is counterbalanced by the use of complicated non polynomial shape functions with subsequent difficulties, and a potentially large cost, when implementing numerical integration schemes. In this paper we describe and compare some numerical quadrature rules with the aim at preserving the MLPG solution accuracy and at the same time reducing its computational cost.

A new spectral meshless radial point interpolation (SMRPI) method: A well-behaved alternative to the meshless weak forms

“…(46) and explain how to implement spectral meshless radial point interpolation (SMRPI) method to obtain discrete equations. Eq.…”

“…For example, the nature of complicated non-polynomial shape functions may cause large cost when implementing numerical integration scheme [46]. On the other hand, in some of these methods like for example those methods which are based on moving least squares (MLS) and RBFs, they need to determine shape parameter which plays the important role in the accuracy of the methods, and also the resultant linear systems might be ill-conditioned and to omit this defect, some regularization methods are needed.…”

Product Gauss quadrature rules vs. cubature rules in the meshless local Petrov–Galerkin method

“…Usually, in two-dimensional problems the local integration domains are circles (for internal nodes) or the intersection of a circle with the global boundary (for nodes on or near to the global boundary). Therefore, much effort has been expended in improving the numerical integration in the MLPG [11,[14][15][16][17].…”

“…Our aim is to compare some integration rules taken from the existing literature on meshless methods [11,15,12,16] with some cubature formulas taken from the relevant literature on numerical integration [18][19][20][21] but not yet applied in meshless methods.…”

“…In [12] the MLPG is implemented by comparing two numerical quadrature schemes, the first one based on the Gauss-Legendre product rule with a change of variables from cartesian to polar coordinates, the same used in [10,11] and the second one constructed following the suggestion of [24] and [14,23], by applying a one-dimensional Gauss-Legendre formula for the ρ-integration and a midpoint rule for the θ-integral. In [16] the piecewise midpoint quadrature rule proposed in [15] is introduced for a better understanding of the behavior of the integration rules in the MLPG.…”