2020/06/26 by Ghislain Haine, Haine, Ghislain, Denis Matignon +3 · 1 citation
Engineering · Mathematics · #35L50 #65M60 #93C20 #Advanced Numerical Methods in Computational Mathematics #Dynamical Systems (math.DS) #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2006.15032
openalex publication_date 2020/06/26 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
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.