Radial Basis Function Meshless Collocation Method for Partial Differential Equation

Abstract: Partial differential equations have been widely applied in physics, engineering and scientific computation. It is significant to research efficient numerical methods for solving partial differential equation in dealing with many physical equations in some fields, such as electromagnetics, acoustics and so on. The traditional numerical methods are based on mesh discretization mostly such as FEM and FDM. For the efficient algorithm of mesh discretization, the researchers carried out a lot of work. There is still not a general algorithm for the complex regional problems. Meshless methods have been proposed and achieved remarkable progress in numerical computations in last decade. It is based on the point approximation and needn't to pre-defined or generated grid, so that a new way is provided for the computational mechanics researchers to avoid the mesh discretization. Now more than ten kinds of meshless methods have been produced in different fields, each method is the combination of different approximation and discrete programs. RBF interpolation for partial differential equation is a very popular meshless method in recent years in the international aspects because of simple form, isotropic and so on. For the radial basis function approximation, discrete programs generally use the least squares method, Galerkin method, collocation method and so on. There are good convergence results of the least square and Galerkin approximation, but the collocation method has not much application and relational analysis although simple and practial in the calculation.First, we introduce a radial basis function and give a specific research for its naturein this paper. Then combining this radial basis function with collocation method, we constructed meshless methods for elliptic and parabolic equations by collocation and radial basis function. We also give theoratical proof of existence and uniqueness of sol-utions when using meshless method to solve partial differential equation by collocation with radial basis functions. Through the numerical examples,we give the empirical form-ula of the free parameter in the radial basis function in different values of dimensional space and their scope of application. We illustrate the effectiveness and practicality of the method through compared with the classic finite element method. We also illustrate the superiority of the method for partial differential equation through compared with the others…
Key words: radial basis function(RBF); meshless method; collocation method; partial differential equation(PDE)

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