The anisotropic and heterogeneous \texorpdfstring{$N$}{N}-dimensional wave equation, controlled and observed at the boundary, is considered as a port-Hamiltonian system. The recent structure-preserving Partitioned Finite Element Method is applied, leading directly to a finite-dimensional port-Hamiltonian system, and its numerical analysis is done in a general framework, under usual assumptions for finite element. Compatibility conditions are then exhibited to reach the best trade off between the convergence rate and the number of degrees of freedom for both the state error and the Hamiltonian error. Numerical simulations in 2D are performed to illustrate the optimality of the main theorems among several choices of classical finite element families.
翻译:在边界上控制和观测的动向和多异\texorpdfstring{$N$N$N}维波方程式被视为港口-汉堡系统。最近采用了结构保存有限元素分部分法,直接导致一个有限维端-汉堡系统,其数字分析是在一个总的框架内,根据对有限元素的通常假设进行。然后展示兼容性条件,以便在国家误差和汉密尔顿误差的趋同率和自由度之间实现最佳平衡。在2D中进行数值模拟,以说明一些传统有限元素组选择的主要理论的最佳性。