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Exponential convergence and H‐c multiquadric collocation method for partial differential equations
Author(s) -
Cheng A. H.D.,
Golberg M. A.,
Kansa E. J.,
Zammito G.
Publication year - 2003
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.10062
Subject(s) - mathematics , radial basis function , collocation (remote sensing) , partial differential equation , exponential function , collocation method , partial derivative , regularized meshless method , convergence (economics) , orthogonal collocation , finite element method , function (biology) , residual , gaussian , mathematical optimization , mathematical analysis , differential equation , algorithm , singular boundary method , boundary element method , ordinary differential equation , computer science , economic growth , biology , quantum mechanics , machine learning , evolutionary biology , artificial neural network , thermodynamics , physics , economics
The radial basis function (RBF) collocation method uses global shape functions to interpolate and collocate the approximate solution of PDEs. It is a truly meshless method as compared to some of the so‐called meshless or element‐free finite element methods. For the multiquadric and Gaussian RBFs, there are two ways to make the solution converge—either by refining the mesh size h , or by increasing the shape parameter c . While the h ‐scheme requires the increase of computational cost, the c ‐scheme is performed without extra effort. In this paper we establish by numerical experiment the exponential error estimate ϵ ∼ O (λ √ c̄h ) where 0 < λ < 1. We also propose the use of residual error as an error indicator to optimize the selection of c . © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 571–594, 2003