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Volume 20, Issue 1
Numerical Analysis of a Structure-Preserving Space-Discretization for an Anisotropic and Heterogeneous Boundary Controlled $N$-Dimensional Wave Equation as a Port-Hamiltonian System

Ghislain Haine, Denis Matignon & Anass Serhani

DOI: 10.4208/ijnam2023-1005

Int. J. Numer. Anal. Mod., 20 (2023), pp. 92-133.

Published online: 2022-11

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  • 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.

  • Keywords

Port-Hamiltonian systems, $N$-dimensional wave equation, finite element method, structure-preserving discretization, numerical analysis.

  • AMS Subject Headings

65M60, 35L50, 93C20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-20-92, author = {Haine , GhislainMatignon , Denis and Serhani , Anass}, title = {Numerical Analysis of a Structure-Preserving Space-Discretization for an Anisotropic and Heterogeneous Boundary Controlled $N$-Dimensional Wave Equation as a Port-Hamiltonian System}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2022}, volume = {20}, number = {1}, pages = {92--133}, 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.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2023-1005}, url = {http://global-sci.org/intro/article_detail/ijnam/21206.html} }
TY - JOUR T1 - Numerical Analysis of a Structure-Preserving Space-Discretization for an Anisotropic and Heterogeneous Boundary Controlled $N$-Dimensional Wave Equation as a Port-Hamiltonian System AU - Haine , Ghislain AU - Matignon , Denis AU - Serhani , Anass JO - International Journal of Numerical Analysis and Modeling VL - 1 SP - 92 EP - 133 PY - 2022 DA - 2022/11 SN - 20 DO - http://doi.org/10.4208/ijnam2023-1005 UR - https://global-sci.org/intro/article_detail/ijnam/21206.html KW - Port-Hamiltonian systems, $N$-dimensional wave equation, finite element method, structure-preserving discretization, numerical analysis. AB -

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.

Haine , GhislainMatignon , Denis and Serhani , Anass. (2022). Numerical Analysis of a Structure-Preserving Space-Discretization for an Anisotropic and Heterogeneous Boundary Controlled $N$-Dimensional Wave Equation as a Port-Hamiltonian System. International Journal of Numerical Analysis and Modeling. 20 (1). 92-133. doi:10.4208/ijnam2023-1005
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