The fluid dynamic approach to equidistribution methods for grid adaptation

The equidistribution methods based on LpLp Monge–Kantorovich optimization and on the deformation method are analyzed primarily in the context of grid adaptation. The first class of methods can be obtained from a variational principle leading to a fluid dynamic formulation based on time-dependent equations for the mass density and the momentum density. In this context, deformation methods arise from a similar fluid formulation by making a specific assumption on the time evolution of the density (but with some degree of freedom for the momentum density). In general, deformation methods do not arise from a variational principle. However, it is possible to prescribe an optimal deformation method, related to L1L1 Monge–Kantorovich optimization, by making a further assumption on the momentum density. Thus, the fluid dynamic formulation provides a unified description of equidistribution methods. Some numerical examples using the LpLp fluid dynamic formulation are also explored.