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The Reduced-Order Extrapolating Method about the Crank-Nicolson Finite Element Solution Coefficient Vectors for Parabolic Type Equation

Author

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  • Zhendong Luo

    (School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China)

Abstract

This study is mainly concerned with the reduced-order extrapolating technique about the unknown solution coefficient vectors in the Crank-Nicolson finite element (CNFE) method for the parabolic type partial differential equation (PDE). For this purpose, the CNFE method and the existence, stability, and error estimates about the CNFE solutions for the parabolic type PDE are first derived. Next, a reduced-order extrapolating CNFE (ROECNFE) model in matrix-form is established with a proper orthogonal decomposition (POD) method, and the existence, stability, and error estimates of the ROECNFE solutions are proved by matrix theory, resulting in an graceful theoretical development. Specially, our study exposes that the ROECNFE method has the same basis functions and the same accuracy as the CNFE method. Lastly, some numeric tests are shown to computationally verify the validity and correctness about the ROECNFE method.

Suggested Citation

  • Zhendong Luo, 2020. "The Reduced-Order Extrapolating Method about the Crank-Nicolson Finite Element Solution Coefficient Vectors for Parabolic Type Equation," Mathematics, MDPI, vol. 8(8), pages 1-11, August.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:8:p:1261-:d:393217
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    References listed on IDEAS

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    1. Luo, Zhendong & Teng, Fei & Chen, Jing, 2018. "A POD-based reduced-order Crank–Nicolson finite volume element extrapolating algorithm for 2D Sobolev equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 146(C), pages 118-133.
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    Cited by:

    1. Zhendong Luo & Yuejie Li, 2022. "A Preserving Precision Mixed Finite Element Dimensionality Reduction Method for Unsaturated Flow Problem," Mathematics, MDPI, vol. 10(22), pages 1-17, November.
    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.

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