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A Preserving Precision Mixed Finite Element Dimensionality Reduction Method for Unsaturated Flow Problem

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  • Zhendong Luo

    (School of Digitalized Intelligence Engineering, Hunan Sany Polytechnic College, Changsha 410129, China)

  • Yuejie Li

    (Department of Mathematics and Computer Engineering, Ordos Institute of Technology, Ordos 017000, China)

Abstract

The unsaturated flow problem is of important applied background and its mixed finite element (MFE) method can be used to simultaneously calculate both water content and flux in soil, which is the most ideal calculation method. Nonetheless, it includes many unknowns. Thereby, herein we will employ the proper orthogonal decomposition (POD) to lower the dimension of unknown solution coefficient vectors in the MFE method for the unsaturated flow problem. Thus, we first examine the MFE method for the unsaturated flow problem and the existence and convergence of the classical MFE solutions. We then take advantage of the initial L MFE solution coefficient vectors to generate a set of POD basis vectors and utilize the most POD basis vectors to create the preserving precision MFE reduced-dimension (PPMFERD) format. Under the circumstances, the PPMFERD format has the same basis functions as the classical MFE format so that it can maintain the same accuracy as the classical MFE format, but it only includes a few unknowns, so it greatly lightens the calculating load, retards the accumulation of computing errors, saves CPU runtime, and improves the accuracy of the real-time calculation in the computational process. Next, we employ the analysis of matrices to demonstrate the existence and convergence of the PPMFERD solutions such that the theoretical analysis becomes very simple and elegant. Finally, we take advantage of some numerical simulations to check on the correctness of the PPMFERD method. It shows that the PPMFERD method is effective and feasible for simulating both water content and flux in unsaturated flow soil.

Suggested Citation

  • 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.
  • Handle: RePEc:gam:jmathe:v:10:y:2022:i:22:p:4391-:d:979718
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    References listed on IDEAS

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    1. Li, Baibing & Martin, Elaine B. & Morris, A. Julian, 2002. "On principal component analysis in L1," Computational Statistics & Data Analysis, Elsevier, vol. 40(3), pages 471-474, September.
    2. Zhendong Luo, 2022. "A Finite Element Reduced-Dimension Method for Viscoelastic Wave Equation," Mathematics, MDPI, vol. 10(17), pages 1-12, August.
    3. 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.
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    1. Xiaoli Hou & Fei Teng & Zhendong Luo & Hui Fu, 2024. "A New Reduced-Dimension Iteration Two-Grid Crank–Nicolson Finite-Element Method for Unsaturated Soil Water Flow Problem," Mathematics, MDPI, vol. 12(11), pages 1-26, June.

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