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