A new adaptive-node refinement algorithm based on multiquadric method for the nearly singular problems

Abstract

In this paper, a new adaptive-node refinement algorithm for multiquadric (MQ) method applied to nearly singular PDEs, has been presented. Besides, for the first time, the concept of the gradient has been employed in the refinement index. Regions with high gradients are identified based on the average value of the proposed function solution. The solution is approximated using a first-order derivative of interpolation with MQ. In the framework of the adaptive algorithm, the average of the proposed function was used as an indicator that determines where the point distribution can be refined and nodes can be added or removed based on this indicator. Different applications of the proposed adaptive algorithm are investigated through numerical examples. It has been revealed that the proposed algorithm is able to identify the singularities both in the domain and near boundaries and the numerical results of the MQ method confirm the accuracy and efficiency of the algorithm. The main advantage of this algorithm is that in the first steps, the regions with high gradients are identified correctly and the convergence speed of the algorithm increases continuously.