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Web
page http://rbf-pde.uah.edu is supported
by Alex Fedoseyev, Center
for Microgravity and Materials Research ,
University of Alabama in Huntsville, Huntsville,
Alabama, U.S.A.
Please send your comments and suggestions to Ed
Kansa and Alex Fedoseyev.
Radial Basis Function for the solution of PDE
The RBF-PDE is the Radial Basis Function for the solution of PDE. This is a meshless collocation method with global basis functions. It is known to have exponentional convergence for interpolation problems. One can descretize nonlinear elliptic PDEs by RBF method. This results in modest size systems of nonlinear algebraic equations which can be efficiently solved by standard software such as LINPACK, LAPACK etc. Examples are published for 1D and 2D PDEs. These examples show high accuracy with small number of unknowns, as compared with known results from the literature.
OVERVIEW (by Ed KANSA)
1 Introduction.
The numerical solution of partial differential equations (PDEs) has been dominated by either finite difference methods (FDM), finite element methods (FEM), and finite volume methods (FVM). These methods can be derived from the assumptions of the local interpolation schemes. These methods require a mesh to support the localized approximations; the construction of a mesh in three or more dimensions is a non-trivial problem. Typically with these methods only the function is continuous across meshes, but not its partial derivatives.
In practice, only low order approximations are used because of the notorious
polynomial snaking problem. While higher order schemes are necessary for
more accurate approximations of the spatial derivatives, they are not sufficient
without monotonicity constraints. Because of the low order schemes typically
employed, the spatial truncation errors can only be controlled by using
progressively smaller meshes. The mesh spacing, h, must be sufficiently
fine to capture the functions partial derivative behavior and to avoid
unnecessarily large amounts of numerical artifacts contaminating the solution.
Spectral methods while offering very high order spatial schemes typically
depend upon tensor product grids in higher dimensions....
(Complete overview is here : HTML
version, DVI file, PDF
file, Postscript file)
List of people with E-mail addresses, Web-links is being appended. If you want your name to be added to the list, please send E-mail to Ed Kansa.
Links - Web
resources
Boundary
element method Web page
CALL FOR PAPERS (Expired on 9/1/00, over 20 papers will be published)
Special Issue of Computers and Mathematics with Applications on RBF-PDE
Publisher: Pergamon Press
We are soliciting applied and/or theoretical papers that apply radial basis function to the solution of partial differential equations, integral equations, or integro-partial differential equations. There are no specific page limitations imposed, but we request that it be kept under 30 pages, single spaced. All papers are to be written in English.
Please write the papers and figures in LaTeX to accelerate publication; however, commonly used word processors are acceptable.
All papers will be required to be accepted by at least two peer-reviewers.
The deadline for submission of papers is 1 September, 2000. The special issue is expected to appear in early 2001 if the papers can be converted to TEX.
Please send manuscripts electronically as attachements or 3 hard copies to either:
Dr. Edward J. Kansa, E-mail: kansa1@llnl.gov
Mail Stop L-200
Lawrence Livermore National Laboratory
Livermore, CA 94551-0808 USA
or
Prof. Yiu-Chung Hon, E-mail: maychon@cityu.edu.hk
Mathematics Department
City University of Hong Kong
Tat Chee Avenue
Kowloon Tong Hong Kong
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