The heat equation as grid smoothing in a local optimization procedure

We consider a local optimization technique, where starting from a preliminary version of the grid under consideration, we try to improve the part of the grid that really needs this improvement. When this procedure is performed, the processed grids may result irregular, so a smoothing step must be taken into account. We propose a smoothing approach based on an iterative formula resembling the explicit difference schemes for the heat equation. This is a quite general approach, however to fix the ideas it is described in the context of quadrilateral grid generation and the variational approach is considered as the base method for the solution of planar grid generation. Some numerical experiments are presented to show the efficiency of the proposed method.