Long-time behavior of a coupled heat-wave system using a structure-preserving finite element method

Abstract

This work is a numerical investigation of the coupling between the heat and wave equations, recast as an interconnection of open port-Hamiltonian systems (pHs). A structure-preserving discretization suited to open pHs, based on a mixed finite element approximation space that includes boundary inputs and outputs, is shown to yield a semi-discrete power balance analogous to the continuous one. In the frequency domain, the semi-discretization captures the finite accumulation point in the spectrum, associated with highly-oscillatory eigenfunctions localized at the interconnection interface. In the time domain, the polynomial and logarithmic energy decays proved by Zhang and Zuazua (Arch. Rational Mech.Anal. 184, 2007) are recovered using a Crank-Nicolson scheme.