The Study of Domain Decomposition Algorithm Based on Radial Basis Meshless Method for Solving PDEs

Abstract: Partial differential equations have been widely applied in physics, engineering and scientific computation. It is significant to research efficient numerical methods for solving PDEs in dealing with many physical equations in some fields, such as electromagnetics, acoustics and so on. The traditional numerical methods are based on mesh approximation mostly such as FEM and FDM, so it is hard to handle the discontinuity and great distortion disaccord with the original grid lines. Meshless methods have been proposed and achieved remarkable progress in numerical computations in last decade, the main advantage of which is that it is based on spot approximation and mesh can be eliminated wholly or partly instead of mesh discretization or generation initially. This feature reduces difficulty in mesh generation of the structure, as FEM in which mesh generation can be a very time-consuming and expensive task.In this paper, three schemes of domain decomposition method(DDM) based on radial basis meshless method are constructed to solve PDEs by combining radial basis meshless collocation method with DDM. RBF meshess collocation method has many advantages in solving PDEs, for example mesh can be eliminated wholly; it has no relation with space dimension and has convergence rate of O(h~(d+1)), here h is collocation points density and d is dimension. However, RBF is defined in the global domain and it is found that the resulting coefficient matrix is full, and highly ill-conditioned, which hinders the applicability of RBFs method to solve large scale problems. The main idea of DDM is that the original large domain is divided into two or more subdomains, which provides an alternative way to avoid the ill-conditioning problem resulted from using the RBFs by solving many small subdomain problems instead of one global large domain problem. So this algorithm keeps the both advantages of RBF messless collocation method and the DDM.Algorithm construction, error estimate, convergence analysis and numerical examples are included in the paper. Convergence analysis and the numerical examples show that this scheme not only reduces condition number of coefficient matrix but also improves computational accuracy and efficiency than RBF meshless collocation method. It is an efficient method for solving PDEs…
Key words: radial basis function(RBF); RBF collocation method; Schwarz altenative method; domain decomposition method(DDM); conditional number

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