Maximum principle and convergence analysis for the meshfree point collocation method

The discrete Laplacian operator is considered in the sense of the meshfree point collocation method which will be called the strong meshfree Laplacian operator. To define the strong meshfree Laplacian operator, we use the fast version of the generalized moving least square approximation, which can calculate the approximated derivatives of shape functions. Some types of the locally layered node distribution are defined in this paper, and two specific domains are constructed onto which we can distribute locally layered nodes. At such nodes, the discrete maximum principle can be shown to hold through the representation formula for the strong meshfree Laplacian operator. The discrete maximum principle, together with the reproducing property of the meshfree approximations, results in an a priori estimate for the strong meshfree Laplacian operator in the nodal solution space. Furthermore, the a priori estimate we have obtained guarantees the existence and the uniqueness of the numerical solution and plays a central role in achieving converged results for the Poisson problem with Dirichlet boundary conditions in the nodal solution space. The order of convergence of the nodal solutions can be raised up to O(h²) at the proposed type of nodes in specific domains. For generally shaped domains immersed in the previously mentioned domains, we can obtain the first order convergence result of O(h).