Anisotropic heterogeneous n-D heat equation with boundary control and observation: II. Structure-preserving discretization

Abstract

The heat equation with boundary control and observation can be described by means of three different Hamiltonians, the internal energy, the entropy, or a classical Lyapunov functional, as shown in the companion paper (Serhani et al. (2019a)). The aim of this work is to apply the partitioned finite element method (PFEM) proposed in Cardoso-Ribeiro et al. (2018) to the three associated port-Hamiltonian systems. Differential Algebraic Equations are obtained. The strategy proves very efficient to mimic the continuous Stokes-Dirac structure at the discrete level, and especially preserving the associated power balance. Anisotropic and heterogeneous 2D simulations are finally performed on the Lyapunov formulation to provide numerical evidence that this strategy proves very efficient for the accurate simulation of a boundary controlled and observed infinite-dimensional system.