Ghislain Haine

Ghislain Haine, Karim Ramdani

A new iterative algorithm for solving initial data inverse problems from partial observations has been recently proposed in Ramdani, Tucsnak and Weiss [15]. Based on the concept of observers (also called Luenberger observers), this algorithm covers a large class of abstract evolution PDE's. In this paper, we are concerned with the convergence analysis of this algorithm. More precisely, we provide a complete numerical analysis for semi-discrete (in space) and fully discrete approximations derived using finite elements in space and finite differences in time. The analysis is carried out for abstract Schrödinger and wave conservative systems with bounded observation (locally distributed).

Antoine Bendimerad-Hohl, Ghislain Haine, Laurent Lefèvre et al.

We study the structure-preserving space discretization of port-Hamiltonian (pH) systems defined with differential constitutive relations. Using the concept of Stokes-Lagrange structure to describe these relations, these are reduced to a finite-dimensional Lagrange subspace of a pH system thanks to a structure-preserving Finite Element Method. To illustrate our results, the 1D nanorod case and the shear beam model are considered, which are given by differential and implicit constitutive relations for which a Stokes-Lagrange structure along with boundary energy ports naturally occur. Then, these results are extended to the nonlinear 2D incompressible Navier-Stokes equations written in a vorticity-stream function formulation. It is first recast as a pH system defined with a Stokes-Lagrange structure along with a modulated Stokes-Dirac structure. A careful structure-preserving space discretization is then performed, leading to a finite-dimensional pH system. Theoretical and numerical results show that both enstrophy and kinetic energy evolutions are preserved both at the semi-discrete and fully-discrete levels.

Lorenzo Agostini, Michel Fournié, Ghislain Haine

The numerical simulation of the inviscid Burgers' equation is often hindered by spurious oscillations near discontinuities. To mitigate this issue, a viscous term can be introduced, leading to the viscous Burgers' equation. In this work, port-Hamiltonian formulations for both the inviscid and the viscous Burgers' equations are proposed, enabling a representation that incorporates both convective and dissipative effects. Boundary control and observation are naturally handled within this framework. Applying a dedicated finite element method, a finite-dimensional port-Hamiltonian system is derived. The relationship between time step, spatial resolution, and viscosity required to achieve numerical stability is analyzed. Numerical experiments validate the effectiveness of the approach.