Adaptive strategies to improve the application of the generalized finite differences method in 2D and 3D

The generalized finite differences method allows the use of irregular clouds of nodes. The optimal values of the key parameters of the method vary depending on how the nodes in the cloud are distributed, and this can be complicated especially in 3D. Therefore, we establish 2 criteria to allow the automation of the selection process of the key parameters. These criteria depend on 2 discrete functions, one of them penalizes distances and the other one penalizes imbalances. In addition, we show how to generate irregular clouds of nodes more efficient than finer regular clouds of nodes. We propose an improved and more versatile h-adaptivemethod that allows adding, moving, and removing nodes. To decide which nodes to act on, we use an indicator of the error a posteriori. This h-adaptive method gives results more accurate than those ones presented for the generalized finite differences method so far and, in addition, with fewer nodes. In addition, thismethod can be used in time-dependant problems to increase the temporal step or to avoid instabilities. As an example, we apply it in problems of seismic waves propagation.