A Lagrangian Approach Based on the Natural Neighbor Interpolation
Abstract
A Lagrangian formulation is constructed using the so-called Moving-Particle Semi-Implicit Method. The approximation scheme is then modified to incorporate the natural neighbor interpolation in the definition of the shape functions. In this way, a multi-scale method results. This method is presented as an alternative for solving partial-differential equations (PDE). Second-order convergence is achieved when uniform rectangular (finite-difference-type) grids are used. Laplace and Poisson equations are solved in somewhat more general geometries as examples; the method demonstrates the same second-order accuracy as classical finite-elements (FEM), but allows a different treatment of geometry potentially powerful for local mesh adaptation in particular.