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Finite element Galerkin methods for plate vibration problems: error analysis and efficient implementation
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DOI：10.1007/s11075-025-02264-w.png)
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Abstract
En 中文
The partial differential equation governing the vibration of a flat plate is reformulated as a Schr & ouml;dinger-type system subject to one of three types of boundary conditions, namely, hinged, mixed and clamped. To solve this system, a finite element Galerkin (FEG) method is used for the spatial discretization. For the case of hinged boundary conditions, an alternating direction implicit (ADI) Crank Nicolson (CN) FEG method is considered for the time-stepping. Optimal error estimates in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{L}<^>2$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{H}<^>1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{L}<^>{\infty }$$\end{document}-norms are derived. For each of the remaining boundary conditions, it is not possible to formulate an ADI CN method. As a consequence, for these two cases, using piecewise Hermite bicubics for the spatial discretization, the focus is on the development of efficient algorithms for determining a CN approximation based on a matrix decomposition method and a Schur complement approach, respectively. For these two cases, an optimal estimate in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{H}<^>1$$\end{document}-norm is derived. The results of numerical experiments demonstrate the accuracy of each method.
Keywords:
Vibration problem
Schr & ouml
dinger system
Finite element Galerkin method
Crank-Nicolson method
Alternating direction implicit method
Matrix decomposition
Schur complement
Error estimates
Numerical experiments