B09
Structure Preserving Discretization of Gradient Flows

The project's objective is to investigate - analytically and numerically - the properties of adapted full discretizations for a class of evolution equations with gradient flow structure. Our motivation is two-fold. One goal is to define novel schemes for numerical solution of the evolution equations, that are stable and convergent, and in addition preserve various aspects of the variational structure and the associated qualitative properties of solutions on the discrete level. The second goal is to perform a qualitative analysis, e.g., to study the long-time behaviour, of the resulting dynamical system on a discrete space.

We are mainly interested in the discretization of fluid type equations for mass densities which arise as a steepest descent of an energy or entropy functional in the Wasserstein distance or a related mass transport metric. Specific examples that we intend to study are second order porous medium and fourth order lubrication equations, as well as, in later phases of the project, also more general equations of Cahn-Hilliard type and also genuine Euler-like hydrodynamic models. Some of the expected advantages of our structure preserving approach in comparison to a "generic" discretization are the following: the potential of the flow will be time-monotone; contraction estimates in the underlying metric will be preserved; and auxiliary Lyapunov functionals for the continuous flow will have discrete counterparts. These properties will also pave the way to proving stability and convergence. On top of that, the discretization will automatically guarantee such general features as non-negativity of solutions and the conservation of mass.

We plan to explore two different strategies to define adapted discretizations, which we label Lagrangian and Eulerian. The first uses a formulation of the underlying fluid dynamics in a co-moving frame. That is, we discretize the evolution of Lagrangian maps rather than the temporal change of the density directly. This approch is well-suited to gradient flows in the Wasserstein metric, where the Lagrangian map is a concatenation of infinitesimal plans for optimal mass transfer. Our discretization preserves this intuitive geometric interpretation. In the Eulerian approach, which is close to a classical finite element discretization, we will approximate the continuous densities by functions in a finite-dimensional ansatz space, e.g., by piecewise constant functions with respect to a fixed decomposition of the spatial domain. For a truely variational approach on these grounds, we will need to design of an efficient method for the calculation of the transportation distance between functions in the ansatz space.

The dynamics of the resulting discrete dynamical system will then be studied - analytically and numerically - in view of the following:
• energy dissipation, Lyapunov functionals, and derivation of discrete a priori estimates;
• qualitative properties of solutions like strict positivity and growth of the support;
• rate of equilibration in the long-time limit;
• existence and stability of special solutions, like quasi-self-similar profiles;
• convergence of the discrete solutions to a continuous one.