Mesh-free radial basis function network methods with domain decomposition for approximation of functions and numerical solution of Poisson’s equations.

*(English)*Zbl 0996.65131Summary: This paper presents the combination of new mesh-free radial basis function network (RBFN) methods and domain decomposition (DD) technique for approximating functions and solving Poisson’s equations. The RBFN method allows numerical approximation of functions and solution of partial differential equations (PDEs) without the need for a traditional ‘finite element’-type (FE) mesh while the combined RBFN-DD approach facilitates coarse-grained parallelization of large problems. Effect of RBFN parameters on the quality of approximation of function and its derivatives is investigated and compared with the case of single domain. In solving Poisson’s equations, an iterative procedure is employed to update unknown boundary conditions at interfaces. At each iteration, the interface boundary conditions are first estimated by using boundary integral equations (BIEs) and subdomain problems are then solved by using the RBFN method. Volume integrals in standard integral equation representation (IE), which usually require volume discretisation, are completely eliminated in order to preserve the mesh-free nature of RBFN methods. The numerical examples show that RBFN methods in conjunction with DD technique achieve not only a reduction of memory requirement but also a high accuracy of the solution.

##### MSC:

65N38 | Boundary element methods for boundary value problems involving PDEs |

65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

##### Keywords:

mesh-free radial basis function network methods; domain decomposition; boundary integral equations; function approximation; Poisson’s equation; numerical examples
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\textit{N. Mai-Duy} and \textit{T. Tran-Cong}, Eng. Anal. Bound. Elem. 26, No. 2, 133--156 (2002; Zbl 0996.65131)

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