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<p>This paper mainly studies the reduction dimensionality of two-grid mixed finite element (TGMFE) Crank-Nicolson (CN) (TGMEFCN) algorithm of the fourth-order nonlinear Rosenau (FONR) equation. For this objective, a novel TGMEFCN algorithm with time second-order precision and unconditional stableness is first developed by introducing an auxiliary function to decomposed the RONR equation as the system of nonlinear equations with second-order derivatives in spatial variables, using the CN scheme to discretize time derivative, and employing the TGMFE method to discretize spatial variables, which is composed of a system of nonlinear equations defined on a set of coarser meshes as well as a system of linear equations defined on a set of finer meshes with adequately high accuracy and can be easily solved. Then, most importantly, a novel TGMEFCN reduction dimensionality (TGMFECNRD) algorithm is established by employing proper orthogonal decomposition to reduce the dimensionality for the vectors of unknown coefficients of TGMFECN solutions. The largest contribution of this paper is theoretically to analyze the existence, stableness, and errors of the TGMFECNRD solutions, and in application, employ some numeric experiments to test the validity for the obtained theory conclusions and demonstrate the superiorities of the TGMFECNRD algorithm.</p>

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Reduced-Dimension Study of Two-Grid Mixed Finite Element Crank-Nicolson Algorithm for the Fourth-Order Nonlinear Rosenau Equation

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月27日

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  • Liang He,
  • Yihui Sun,
  • Zhenglong Chen,
  • Fei Teng,
  • Chao Shen,
  • Yue Jie Li,
  • Zhendong Luo

摘要

This paper mainly studies the reduction dimensionality of two-grid mixed finite element (TGMFE) Crank-Nicolson (CN) (TGMEFCN) algorithm of the fourth-order nonlinear Rosenau (FONR) equation. For this objective, a novel TGMEFCN algorithm with time second-order precision and unconditional stableness is first developed by introducing an auxiliary function to decomposed the RONR equation as the system of nonlinear equations with second-order derivatives in spatial variables, using the CN scheme to discretize time derivative, and employing the TGMFE method to discretize spatial variables, which is composed of a system of nonlinear equations defined on a set of coarser meshes as well as a system of linear equations defined on a set of finer meshes with adequately high accuracy and can be easily solved. Then, most importantly, a novel TGMEFCN reduction dimensionality (TGMFECNRD) algorithm is established by employing proper orthogonal decomposition to reduce the dimensionality for the vectors of unknown coefficients of TGMFECN solutions. The largest contribution of this paper is theoretically to analyze the existence, stableness, and errors of the TGMFECNRD solutions, and in application, employ some numeric experiments to test the validity for the obtained theory conclusions and demonstrate the superiorities of the TGMFECNRD algorithm.