A numerical study of 2D integrated RBFNs incorporating Cartesian grids for solving 2D elliptic differential problems

Mai-Duy, Nam and Tran-Cong, Thanh (2010) A numerical study of 2D integrated RBFNs incorporating Cartesian grids for solving 2D elliptic differential problems. Numerical Methods for Partial Differential Equations, 26 (6). pp. 1443-1462. ISSN 0749-159X

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Abstract

This paper reports a numerical discretisation scheme, based on two-dimensional integrated radial-basis-function networks (2D-IRBFNs) and rectangular grids, for solving second-order elliptic partial differential equations
defined on 2D non-rectangular domains. Unlike finite-difference and 1D-IRBFN Cartesian-grid techniques, the present discretisation method is based on an approximation scheme that allows the field variable and its derivatives to be evaluated anywhere within the domain and on the boundaries, regardless of the shape of the problem domain. We discuss the following two particular strengths,
which the proposed Cartesian-grid-based procedure possesses,
namely (i) the implementation of Neumann boundary conditions on irregular boundaries and (ii) the use of high-order integration schemes to evaluate flux integrals arising from a control-volume discretisation on irregular domains. A new preconditioning scheme is suggested to improve the 2D-IRBFN matrix condition number. Good
accuracy and high-order convergence solutions are obtained.

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Item Type:	Article (Commonwealth Reporting Category C)
Refereed:	Yes
Item Status:	Live Archive
Additional Information:	Accepted version deposited in accordance with the copyright policy of the publisher.
Faculty/School / Institute/Centre:	Historic - Faculty of Engineering and Surveying - Department of Mechanical and Mechatronic Engineering (Up to 30 Jun 2013)
Faculty/School / Institute/Centre:	Historic - Faculty of Engineering and Surveying - Department of Mechanical and Mechatronic Engineering (Up to 30 Jun 2013)
Date Deposited:	18 Mar 2011 05:10
Last Modified:	03 Jul 2013 00:28
Uncontrolled Keywords:	integrated radial basis function network; Cartesian grid; irregular domain; Neumann boundary condition; control-volume discretisation; point-collocation discretisation
Fields of Research (2008):	01 Mathematical Sciences > 0103 Numerical and Computational Mathematics > 010302 Numerical Solution of Differential and Integral Equations 09 Engineering > 0913 Mechanical Engineering > 091307 Numerical Modelling and Mechanical Characterisation 01 Mathematical Sciences > 0101 Pure Mathematics > 010104 Combinatorics and Discrete Mathematics (excl. Physical Combinatorics)
Fields of Research (2020):	49 MATHEMATICAL SCIENCES > 4903 Numerical and computational mathematics > 490303 Numerical solution of differential and integral equations 40 ENGINEERING > 4017 Mechanical engineering > 401706 Numerical modelling and mechanical characterisation 49 MATHEMATICAL SCIENCES > 4904 Pure mathematics > 490404 Combinatorics and discrete mathematics (excl. physical combinatorics)
Socio-Economic Objectives (2008):	E Expanding Knowledge > 97 Expanding Knowledge > 970109 Expanding Knowledge in Engineering E Expanding Knowledge > 97 Expanding Knowledge > 970101 Expanding Knowledge in the Mathematical Sciences
Identification Number or DOI:	https://doi.org/10.1002/num.20502
URI:	http://eprints.usq.edu.au/id/eprint/18248

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