On the consistency and convergence of particle-based meshfree discretization schemes for the Laplace operator

- The accuracy of the particle-based discretization schemes for the Laplacian is verified.
- Numerical test shows that some schemes are convergent and others are divergent.
- New scheme for the Laplacian is proposed to enhance accuracy and computational efficiency.
- An application of new discretization scheme for the Poisson equation is demonstrated.

The Laplace operator appears in the governing equations of continua describes dissipative dynamics, and it also emerges in some second order partial differential equations such as the Poisson equation. In this paper, accuracy and its convergence rates of some meshfree discretization schemes for the Laplace operator are studied as a verification. Moreover, a novel meshfree discretization scheme for the second order differential operator which enables us to use smaller dilation parameter of the compact support of the weight function is proposed, and its application for the meshfree discretization of the Poisson equation demonstrates an improvement of the solution accuracy.