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

TL;DR

This paper develops and analyzes a structure-preserving finite element discretization for boundary-controlled anisotropic, heterogeneous wave equations modeled as port-Hamiltonian systems, ensuring optimal convergence and energy properties.

Contribution

It introduces a mixed Galerkin method that preserves the port-Hamiltonian structure and provides a comprehensive numerical analysis with optimal element choices and compatibility conditions.

Findings

Optimal finite element choices improve convergence

Compatibility conditions ensure Hamiltonian error is twice state error

Numerical simulations validate theoretical results across various scenarios

Abstract

The anisotropic and heterogeneous $N$-dimensional wave equation, controlled and observed at the boundary, is considered as a port-Hamiltonian system. A recent structure-preserving mixed Galerkin method is applied, leading directly to a finite-dimensional port-Hamiltonian system: its numerical analysis is carried out in a general framework. Optimal choices of mixed finite elements are then proved to reach the best trade-off between the convergence rate and the number of degrees of freedom for the state error. Exta compatibility conditions are identified for the Hamiltonian error to be twice that of the state error, and numerical evidence is provided that some combinations of finite element families meet these conditions. Numerical simulations in 2D are performed to illustrate the main theorems among several choices of classical finite element families. Several test cases are provided, including non-convex domain, anisotropic or hetergoneous cases and absorbing boundary conditions.