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**Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary.**
*(English)*
Zbl 0999.65137

Summary: The multiquadric radial basis function (MQ) method is a recent meshless collocation method with global basis functions. It was introduced for discretizing partial differential equations (PDEs) by E. J. Kansa [ibid. 19, No. 8/9, 127-145 and 147-161 (1990; Zbl 0692.76003 and Zbl 0850.76048)]. The MQ method was originally used for interpolation of scattered data, and it was shown to have exponential convergence for interpolation problems.

A. I. Fedoseyev, M. J. Friedman and E. J. Kansa [Continuation for nonlinear elliptic partial differential equations discretized by the multiquadric method, Preprint Math. NA/9812013 at E-PRINT, LANL (http://www.lanl.gov/ps/math.NA/9812013); Int. J. Bifur. & Chaos 10, No. 2 (2000)] have extended the Kansa-MQ method to numerical solution and detection of bifurcations in 1D and 2D parameterized nonlinear elliptic PDEs. We have found there that the modest size nonlinear systems resulting from the MQ discretization can be efficiently continued by a standard continuation software, such as AUTO. Wet have observed high accuracy with a small number of unknowns, as compared with most known results from the literature.

In this paper, we formulate an improved Kansa-MQ method with PDE collocation on the boundary (MQ PDECB): we add an additional set of nodes (which can lie inside or outside of the domain) adjacent to the boundary and, correspondingly, add an additional set of collocation equations obtained via collocation of the PDE on the boundary. Numerical results are given that show a considerable improvement in accuracy of the MQ PDECB method over the Kansa-MQ method, with both methods having exponential convergence with essentially the same rates.

A. I. Fedoseyev, M. J. Friedman and E. J. Kansa [Continuation for nonlinear elliptic partial differential equations discretized by the multiquadric method, Preprint Math. NA/9812013 at E-PRINT, LANL (http://www.lanl.gov/ps/math.NA/9812013); Int. J. Bifur. & Chaos 10, No. 2 (2000)] have extended the Kansa-MQ method to numerical solution and detection of bifurcations in 1D and 2D parameterized nonlinear elliptic PDEs. We have found there that the modest size nonlinear systems resulting from the MQ discretization can be efficiently continued by a standard continuation software, such as AUTO. Wet have observed high accuracy with a small number of unknowns, as compared with most known results from the literature.

In this paper, we formulate an improved Kansa-MQ method with PDE collocation on the boundary (MQ PDECB): we add an additional set of nodes (which can lie inside or outside of the domain) adjacent to the boundary and, correspondingly, add an additional set of collocation equations obtained via collocation of the PDE on the boundary. Numerical results are given that show a considerable improvement in accuracy of the MQ PDECB method over the Kansa-MQ method, with both methods having exponential convergence with essentially the same rates.

### MSC:

65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

35J65 | Nonlinear boundary value problems for linear elliptic equations |

35B32 | Bifurcations in context of PDEs |

### Keywords:

radial basis functions; multiquadratic method; numerical results; collocation method; bifurcations
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\textit{A. I. Fedoseyev} et al., Comput. Math. Appl. 43, No. 3--5, 439--455 (2002; Zbl 0999.65137)

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### References:

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