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<p>The partial differential equation governing the vibration of a flat plate is reformulated as a Schrödinger-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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textbf{L}^2\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\textbf{H}^1\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\textbf{L}^{\infty }\)</EquationSource> </InlineEquation>-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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\textbf{H}^1\)</EquationSource> </InlineEquation>-norm is derived. The results of numerical experiments demonstrate the accuracy of each method.</p>

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Finite element Galerkin methods for plate vibration problems: error analysis and efficient implementation

  • 期刊论文

  • 发表日期：

    2025年11月22日

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  • Morrakot Khebchareon,
  • Amiya K. Pani,
  • Graeme Fairweather

摘要

The partial differential equation governing the vibration of a flat plate is reformulated as a Schrödinger-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 \(\textbf{L}^2\) , \(\textbf{H}^1\) and \(\textbf{L}^{\infty }\) -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 \(\textbf{H}^1\) -norm is derived. The results of numerical experiments demonstrate the accuracy of each method.