Dynamic node adaptive strategy for nearly singular problems on large domains

Many physical phenomena develop singular or nearly singular behavior in localized regions, e.g. boundary layers or blowup solutions. Using uniform grids for such problems becomes computationally prohibitive as the solution approaches singularity. For these problems, adaptive methods may be preferred over uniform grids methods. In large computational domains, because of the ill conditioning due to the large domain of the partial differential equation (PDE) problem, the existing node adaptive strategies perhaps encounter difficulty in detecting nearly singular regions. In this paper, we are interested in solving PDE problems on large domains, whose solution presents rapid variations or high gradients in some local regions of the domain. Our main purpose is to introduce a dynamic algorithm which finds regions with rapid variations and performs a local node adaptive strategy only in these nearly singular regions. In this algorithm, a step by step scheme is applied by using collocation points and thin plate spline radial basis functions. In spite of using local node adaptive strategy, the global solution exists in the whole computational domain. Another advantage of the new algorithm is its ability to keep the condition number and the required memory under control. The new algorithm is applied for two problems in two dimensions and the obtained results confirm the accuracy and efficiency of the proposed method.