Numerical analysis of a structure-preserving space-discretization for an anisotropic and heterogeneous boundary controlled N-dimensional wave equation as port-Hamiltonian system

Abstract

The anisotropic and heterogeneous N -dimensional wave equation, controlled and observed at the boundary, is considered as a port-Hamiltonian system. The recent structure-preserving Partitioned Finite Element Method is applied, leading directly to a finite-dimensional port-Hamiltonian system, and its numerical analysis is done in a general framework, under usual assumptions for finite element. Compatibility conditions are then exhibited to reach the best trade off between the convergence rate and the number of degrees of freedom for both the state error and the Hamiltonian error. Numerical simulations in 2D are performed to illustrate the optimality of the main theorems among several choices of classical finite element families.

Numerical approximation of port-Hamiltonian systems for hyperbolic or parabolic PDEs with boundary control

“…This method proves nice convergence properties, see e.g. [25] for a recent proof on the wave equation, that does not require the fulfilment of the usual inf-sup condition for mixed finite elements. This expresses a possible intrinsic locking free property of PFEM.…”

“…To perform the discretization in space, we must first specify the conforming finite element approximation spaces to be used (see [25]). Concerning the energy variables associated with the strain, we select the Raviart-Thomas finite element family known as RT k consisting of vector functions with a continuous normal component across the interfaces between the elements of a mesh.…”