Proposing an Error Estimator for Collocated Discreet Least Square (CDLS) Meshless Method

The error estimation should be a main tool in every adaptivity process. This is the reason for the great importance of the estimation. It allows us to know the quality of the solution, and hence, if it is acceptable or not. Moreover, it provides some information about the changes that are necessary to be made in the mathematical model to reach, in an economic way, the desired solution. In this paper, a new error estimator for solving the hyperbolic problems to be used in conjunction with the Collocated Discreet Least Square Meshless (CDLS) method is presented. The error estimator is shown to be naturally related to the least-squares method,providing a suitable measure of the errors in the solution. The estimator is easily calculated by the use of already existing matrices of the least-squares computation, hence, it is very cheap. The proposed error estimator was implemented with CDLS method to solve three benchmark examples from the literature and the effect of collocation points on them was investigated. These examples are nonlinear burgers equation, dam break problem and the problem of shoaling a wave on sloping shallow waters. The results showed that the error estimator works very well in all numerical examples.