Adaptive grids as parametrized scale-free networks

In this paper we present a possible model of adaptive grids for numerical resolution of differential problems, using physical or geometrical properties, as viscosity or velocity gradient of a moving fluid. The relation between the values of grid step and these entities is based on the mathematical scheme offered by the model of scale-free networks, due to Barábasi, so that the step can be connected to the other variables by a constitutive relation. Some examples and an application are discussed, showing that this approach can be further developed for treatment of more complex situations. 1 A scale-free network model for adaptive grids We consider a differential problem of advective viscous flow in a domain Ω, and let G a computational grid on Ω with an initial grid step h. We consider a sistem of cartesian coordinates for the representation of the nodes. In 1D the grid is a linear set of nodes, in 2D is a plane surface where nodes are vertices of squares with edge length equal to h, in 3D is a set of adjacent