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The anisotropic and heterogeneous $N$-dimensional wave equation, controlledand observed at the boundary, is considered as a port-Hamiltonian system. Arecent structure-preserving mixed Galerkin method is applied, leading directlyto a finite-dimensional port-Hamiltonian system: its numerical analysis iscarried out in a general framework. Optimal choices of mixed finite elementsare then proved to reach the best trade-off between the convergence rate andthe number of degrees of freedom for the state error. Exta compatibilityconditions are identified for the Hamiltonian error to be twice that of thestate error, and numerical evidence is provided that some combinations offinite element families meet these conditions. Numerical simulations in 2D areperformed to illustrate the main theorems among several choices of classicalfinite element families. Several test cases are provided, including non-convexdomain, anisotropic or hetergoneous cases and absorbing boundary conditions.

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