Inria Paris - France
In this talk we are going to present a mixed variational formulation for problems described by parametric PDEs. In particular, we focus on PDEs which can be seen as a Hamiltonian flow to which a dissipation mechanism is added, in the form of a gradient flow. Numerous PDEs can be written in this way (compressible gas-dynamics, dispersive waves, structure mechanics). In the first part of the talk we will show how this formulation, when the solution is part of a Hilbert scale, makes it possible to provide a geometrical interpretation of equilibria and conserved quantities. A mixed variational formulation is introduced, in order to semi-discretize the system in space. Under certain assumptions, we show that the resulting finite dimensional dynamical system is a discrete Hamiltonian system with a finite dimensional gradient flow dissipation, inheriting a number of conservations from the infinite dimensional system. In the second part of the talk we will show how the mixed formulation can be used in order to set up Reduced Order Models which are structure preserving. The results will be illustrated by some numerical examples. This is a joint work with Cecilia Pagliantini (Università di Pisa).