Numerical Analysis of a Structure-Preserving Space-Discretization for an Anisotropic and Heterogeneous Boundary Controlled $N$-Dimensional Wave Equation As a 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. 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 are performed in 2D to illustrate the main theorems among several choices of classical finite element families. Several test cases are provided, including non-convex domain, anisotropic or heterogeneous cases and absorbing boundary conditions.

Rotational Shallow Water Equations with viscous damping and boundary control: structure-preserving spatial discretization

“…the triangle is inside the domain), the two dofs associated with this vertex are not correctly set to 0, while if the triangle share a whole edge, the four dofs on its ends are indeed equal to zero. This kind of difficulty needs to be further investigated in future works, as it seems to enlighten the compatibility requirements between finite element families, and especially with those at the boundary, as it has been proved in [31] for the wave equation.…”

“…This partitioned finite element method (PFEM) method has been applied to the discretization of various 2D and 3D pH models with non-autonomous boundary conditions [31]. Accurate convergence results in the sense of numerical analysis have been obtained [32] which suggest a heuristic for the optimal choice of finite element conforming spaces.…”