Reproducing Kernel Meshfree Method And Error Estimate

As a new numerical method, meshfree method is rapidly developed and one of the hottest research areas over the world. It can eliminate ,at least partly, the mesh and avoid the dependency of mesh such as finite element method.First, the principle of reproducing kernel method and the constructing method of approximate was introduced. A new reproducing kernel least square meshfree method was presented based on the principle of point least square method and reproducing kernel method. There are two numerical examples on partial differential equations,one is one dimension and the other is two. Compiling matlab code and solving nonlinear equation set with Newton method.It is a simple and high convergence rate method.Second, the error estimate of least square meshfree method for one order linear differential operator is got based on the error estimate of reproducing kernel approximation and least square method. Particularly, if the operator is coercive and elliptic, the result is shown in L2 space by Nitsche trick. About the nonlinear differential operator which preserves strongly monotone and Lipschitz continuity , the result is similar. Under some conditions , the error estimate of RKPM is presented for some nonlinear elliptic boundary problem.Third,because kernel function is also an important role in meshfree method. Splines are suited for their polynomial forms. Through the numerical example ,the result is not better as the order becomes higher. The comparison for different kernel functions is exhibited in a figure. If other conditions are the same, one is uniform convergence, the other is not stable.Last, the influence of penalty is not neglected for the method introduced in this paper. For the linear boundary problem, there is no much improvement on precision . The result becomes better as the penalty function is big enough for nonlinear boundary problem.