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Mathematics 2017, 5(4), 54; doi:10.3390/math5040054

An Investigation of Radial Basis Function-Finite Difference (RBF-FD) Method for Numerical Solution of Elliptic Partial Differential Equations

1
Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand
2
Centre of Excellence in Mathematics, Commission on Higher Education, Ministry of Education, Si Ayutthaya Road, Bangkok 10400, Thailand
*
Author to whom correspondence should be addressed.
Received: 2 September 2017 / Revised: 13 October 2017 / Accepted: 17 October 2017 / Published: 23 October 2017
(This article belongs to the Special Issue Operators of Fractional Calculus and Their Applications)
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Abstract

The Radial Basis Function (RBF) method has been considered an important meshfree tool for numerical solutions of Partial Differential Equations (PDEs). For various situations, RBF with infinitely differentiable functions can provide accurate results and more flexibility in the geometry of computation domains than traditional methods such as finite difference and finite element methods. However, RBF does not suit large scale problems, and, therefore, a combination of RBF and the finite difference (RBF-FD) method was proposed because of its own strengths not only on feasibility and computational cost, but also on solution accuracy. In this study, we try the RBF-FD method on elliptic PDEs and study the effect of it on such equations with different shape parameters. Most importantly, we study the solution accuracy after additional ghost node strategy, preconditioning strategy, regularization strategy, and floating point arithmetic strategy. We have found more satisfactory accurate solutions in most situations than those from global RBF, except in the preconditioning and regularization strategies. View Full-Text
Keywords: numerical partial differential equations (PDEs); radial basis function-finite difference (RBF-FD) method; ghost node; preconditioning; regularization; floating point arithmetic numerical partial differential equations (PDEs); radial basis function-finite difference (RBF-FD) method; ghost node; preconditioning; regularization; floating point arithmetic
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Yensiri, S.; Skulkhu, R.J. An Investigation of Radial Basis Function-Finite Difference (RBF-FD) Method for Numerical Solution of Elliptic Partial Differential Equations. Mathematics 2017, 5, 54.

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