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The Dimensionality Reduction of Crank–Nicolson Mixed Finite Element Solution Coefficient Vectors for the Unsteady Stokes Equation

Author

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  • Zhendong Luo

    (School of Intelligent Manufacturing, Hunan Sany Polytechnic College, Changsha 410129, China)

Abstract

By means of a proper orthogonal decomposition (POD) to cut down the dimensionality of unknown finite element (FE) solution coefficient vectors in the Crank–Nicolson (CN) mixed FE (CNMFE) method for two-dimensional (2D) unsteady Stokes equations in regard to vorticity stream functions, a reduced dimension recursive-CNMFE (RDR-CNMFE) method is constructed. In this case, the RDR-CNMFE method has the same FE basis functions and accuracy as the CNMFE method. The existence, stability, and errors of RDR-CNMFE solutions are analyzed by matrix analyzing, resulting in very simple theory analysis. Some numerical tries are used to check on the validity of the RDR-CNMFE method. The RDR-CNMFE method has second-order time accuracy and few unknowns so as to be able to shorten CPU runtime and retard the error cumulation in simulation calculating process, and improve real-time calculating accuracy.

Suggested Citation

  • Zhendong Luo, 2022. "The Dimensionality Reduction of Crank–Nicolson Mixed Finite Element Solution Coefficient Vectors for the Unsteady Stokes Equation," Mathematics, MDPI, vol. 10(13), pages 1-11, June.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:13:p:2273-:d:851187
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    Citations

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    Cited by:

    1. Zhendong Luo, 2022. "A Finite Element Reduced-Dimension Method for Viscoelastic Wave Equation," Mathematics, MDPI, vol. 10(17), pages 1-12, August.
    2. Yuejie Li & Zhendong Luo & Changan Liu, 2023. "The Mixed Finite Element Reduced-Dimension Technique with Unchanged Basis Functions for Hydrodynamic Equation," Mathematics, MDPI, vol. 11(4), pages 1-17, February.
    3. Xiaoyong Yang & Zhendong Luo, 2022. "An Unchanged Basis Function and Preserving Accuracy Crank–Nicolson Finite Element Reduced-Dimension Method for Symmetric Tempered Fractional Diffusion Equation," Mathematics, MDPI, vol. 10(19), pages 1-13, October.
    4. Yuejie Li & Zhendong Luo, 2023. "A Reduced-Dimension Extrapolating Method of Finite Element Solution Coefficient Vectors for Fractional Tricomi-Type Equation," Mathematics, MDPI, vol. 11(22), pages 1-13, November.

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