Lakes and rivers in the landscape: A quasi-variational inequality approach

We consider an evolutionary quasi-variational inequality arising in a simplified model of a network of lakes and rivers forming upon a given relief of the Earth. We regularize this model and derive its finite element approximation, in which the water flow is confined to the mesh edges. The primal and mixed formulations of the discretized quasi-variational inequality are used in the numerical simulations. The corresponding steady state problems are also analyzed. Finally, we compare this approach to the lattice algorithms employed in geographic information systems for the automatic extraction of river networks from digital elevation data, and derive similar algorithms for our approximation.