Meshfree explicit local radial basis function collocation method for diffusion problems.

*(English)*Zbl 1168.41003This paper formulates a simple explicit local version of the classical meshless radial basis function (RBF) collocation (Kansa) method [cf. E. J. Kansa, Comput. Math. Appl. 19, No. 8–9, 147–161 (1990; Zbl 0850.76048)]. The discussion is limited to solving the heat diffusion equation. The collocation is made locally over a set of overlapping domains of influence and the time-stepping is performed in an explicit way. Only small systems of linear equations with the dimension of the number of nodes included in the domain of influence have to be solved for each node. The boundary value problem (NAFEMS test) associated with the steady temperature field with simultaneous involvement of the Dirichlet, Neumann and Robin boundary conditions on a rectangle and the initial value problem associated with the Dirichlet jump problem on a square are discussed as numerical examples.

Reviewer: Johann Brauchart (Graz)

##### MSC:

41A63 | Multidimensional problems |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

##### Keywords:

heat diffusion problem; Kansa method; meshfree collocation method; multiquadrics radial basis functions
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\textit{B. Šarler} and \textit{R. Vertnik}, Comput. Math. Appl. 51, No. 8, 1269--1282 (2006; Zbl 1168.41003)

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##### References:

[1] | Atluri, S.N.; Shen, S., The meshless method, (2002), Tech Science Press Forsyth |

[2] | Liu, G.R., Mesh free methods, (2003), CRC Press Boca Raton, FL |

[3] | Atluri, S.N., The meshless method for domain and BIE discretisations, (2004), Tech Science Press Forsyth |

[4] | Šarler, B., Chapter 9: “meshless methods”, (), 225-247 |

[5] | Buhmann, M.D., Radial basis function: theory and implementations, (2003), Cambridge University Press Cambridge · Zbl 1038.41001 |

[6] | Kansa, E.J., Multiquadrics—A scattered data approximation scheme with applications to computational fluid dynamics-II. solutions to parabolic, hyperbolic and elliptic partial differential equations, Computers math. applic., 19, 147-161, (1990) · Zbl 0850.76048 |

[7] | Fasshauer, G.E., Solving partial differential equations by collocation with radial basis functions, (), 131-138 · Zbl 0938.65140 |

[8] | Power, H.; Barraco, W.A., Comparison analysis between unsymmetric and symmetric RBFCMS for the numerical solution of pdes, Computers math. applic., 43, 3-5, 551-583, (2002) · Zbl 0999.65135 |

[9] | Chen, W., New RBF collocation schemes and kernel RBFs with applications, Lecture notes in computational science and engineering, 26, 75-86, (2002) · Zbl 1016.65094 |

[10] | Mai-Duy, N.; Tran-Cong, T., Indirect RBFN method with thin plate splines for numerical solution of differential equations, Computer modeling in engineering & sciences, 4, 85-102, (2003) · Zbl 1148.76351 |

[11] | Sarler, B., A radial basis function collocation approach in computational fluid dynamics, Computer modeling in engineering & sciences, 7, 185-194, (2005) · Zbl 1189.76380 |

[12] | Mai-Duy, N.; Tran-Cong, T., Numerical solution of Navier-Stokes equations using multiquadric radial basis function networks, Neural networks, 14, 185-199, (2001) · Zbl 1047.76101 |

[13] | Sarler, B.; Perko, J.; Chen, C.S., Radial basis function collocation method solution of natural convection in porous media, Int. J. numer. methods heat & fluid flow, 14, 187-212, (2004) · Zbl 1103.76361 |

[14] | Kovačević, I.; Poredoš, A.; Sarler, B., Solving the Stefan problem by the RBFCM, Numer. heat transfer, part B: fundamentals, 44, 1-24, (2003) |

[15] | Chen, C.S.; Ganesh, M.; Golberg, M.A.; Cheng, A.H.-D., Multilevel compact radial basis functions based computational scheme for some elliptic problems, Computers math. applic., 43, 3-5, 359-378, (2002) · Zbl 0999.65143 |

[16] | Mai-Duy, N.; Tran-Cong, T., Mesh-free radial basis function network methods with domain decomposition for approximation of functions and numerical solution of Poisson’s equations, Engineering analysis with boundary elements, 26, 133-156, (2002) · Zbl 0996.65131 |

[17] | Lee, C.K.; Liu, X.; Fan, S.C., Local multiquadric approximation for solving boundary value problems, Computational mechanics, 30, 395-409, (2003) · Zbl 1035.65136 |

[18] | Tolstykh, A.I.; Shirobokov, D.A., On using radial basis functions in a “finite difference” mode with applications to elasticity problems, Computational mechanics, 33, 68-79, (2003) · Zbl 1063.74104 |

[19] | Shu, C.; Ding, H.; Yee, K.S., Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier-Stokes equations, Computer methods in applied mechanics and engineering, 192, 941-954, (2003) · Zbl 1025.76036 |

[20] | Cameron, A.D.; Casey, J.A.; Simpson, G.B., Benchmark tests for thermal analysis, (1986), NAFEMS National Agency for Finite Element Methods & Standards, Department of Trade and Industry, National Engineering Laboratory Glasgow |

[21] | Dalhuijsen, A.J.; Segal, A., Comparison of finite element techniques for solidification problems, International journal for numerical methods in engineering, 29, 1807-1829, (1986) · Zbl 0631.65127 |

[22] | Sarler, B.; Kuhn, G., Dual reciprocity boundary element method for convectioe-diffusive solid-liquid phase change problems, part 2: numerical examples, Engineering analysis with boundary elements, 21, 65-79, (1998) · Zbl 0956.76057 |

[23] | Carlsaw, H.S.; Jaeger, J.C., Conduction of heat in solids, (1995), Clarendon Press Oxford |

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