Collocated discrete least-squares (CDLS) meshless method: Error estimate and adaptive refinement

Meshless methods are new approaches for solving partial differential equations. The main characteristic of all these methods is that they do not require the traditional mesh to construct a numerical formulation. They require node generation instead of mesh generation. In other words, there is no pre-specified connectivity or relationships among the nodes. This characteristic make these methods powerful. For example, an adaptive process which requires high computational effort in mesh-dependent methods can be very economically solved with meshless methods. In this paper, a posteriori error estimate and adaptive refinement strategy is developed in conjunction with the collocated discrete least-squares (CDLS) meshless method. For this, an error estimate is first developed for a CDLS meshless method. The proposed error estimator is shown to be naturally related to the least-squares functional, providing a suitable posterior measure of the error in the solution. A mesh moving strategy is then used to displace the nodal points such that the errors are evenly distributed in the solution domain. Efficiency and effectiveness of the proposed error estimator and adaptive refinement process are tested against two hyperbolic benchmark problems, one with shocked and the other with low gradient smooth solutions. These experiments show that the proposed adaptive process is capable of producing stable and accurate results for the difficult problems considered. Copyright (C) 2007 John Wiley & Sons, Ltd.