Local multiquadric approximation for solving boundary value problems
This paper presents a truly meshless approximation strategy for solving partial differential equations based on the local multiquadric (LMQ) and the local inverse multiquadric (LIMQ) approximations. It is different from the traditional global multiquadric (GMQ) approximation in such a way that it...
| Main Authors: | , , |
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| Other Authors: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
2014
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| Subjects: | |
| Online Access: | https://hdl.handle.net/10356/103125 http://hdl.handle.net/10220/19231 |
| Summary: | This paper presents a truly meshless approximation strategy for solving partial differential
equations based on the local multiquadric (LMQ) and the local inverse multiquadric (LIMQ)
approximations. It is different from the traditional global multiquadric (GMQ) approximation in
such a way that it is a pure local procedure. In constructing the approximation function, the only
geometrical data needed is the local configuration of nodes fallen within its influence domain.
Besides this distinct characteristic of localization, in the context of meshless-typed approximation
strategies, other major advantages of the present strategy include: (i) the existence of the shape
functions is guaranteed provided that all the nodal points within an influence domain are distinct;
(ii) the constructed shape functions strictly satisfy the Kronecker delta condition; (iii) the
approximation is stable and insensitive to the free parameter embedded in the formulation and;
(iv) the computational cost is modest and the matrix operations require only inversion of matrices
of small size which is equal to the number of nodes inside the influence domain. Based on the
present LMQ and LIMQ approximations, a collocation procedure is developed for solutions of
1D and 2D boundary value problems. Numerical results indicate that the present LMQ and LIMQ
approximations are more stable than their global counterparts. In addition, it demonstrates that
both approximation strategies are highly efficient and able to yield accurate solutions regardless
of the chosen value for the free parameter. |
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