A New Reduced-Dimension Iteration Two-Grid Crank–Nicolson Finite-Element Method for Unsaturated Soil Water Flow Problem

Xiaoli Hou, Fei Teng (), Zhendong Luo () and Hui Fu
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Xiaoli Hou: College of Environment and Ecology, Hunan Agricultural University, Changsha 410128, China
Fei Teng: College of Arts and Sciences, Shanghai Dianji University, Shanghai 201306, China
Zhendong Luo: School of Digitalized Intelligence Engineering, Hunan Sany Polytechnic College, Changsha 410129, China
Hui Fu: College of Environment and Ecology, Hunan Agricultural University, Changsha 410128, China

Mathematics, 2024, vol. 12, issue 11, 1-26

Abstract: The main objective of this paper is to reduce the dimensionality of unknown coefficient vectors of finite-element (FE) solutions in two-grid (CN) FE (TGCNFE) format for the nonlinear unsaturated soil water flow problem by using a proper orthogonal decomposition (POD) and to design a new reduced-dimension iteration TGCNFE (RDITGCNFE). For this objective, a new time semi-discrete CN (TSDCN) scheme for the nonlinear unsaturated soil water flow problem is first designed and the existence, stability, and error estimates of TSDCN solutions are demonstrated. Subsequently, a new TGCNFE format for the nonlinear unsaturated soil water flow problem is designed and the existence, unconditional stability, and error estimates of TGCNFE solutions are demonstrated. Next, a new RDITGCNFE format with the same FE basis functions as the TGCNFE format is built by the POD method and the existence, unconditional stability, and error estimates of RDITGCNFE solutions are discussed. Ultimately, the rightness of theory results and the superiority of the RDITGCNFE format are verified by two sets of numerical tests. It is worth noting that the RDITGCNFE format differs completely from all previous reduced-dimension methods, including the authors’ previous works. Therefore, the study of this paper can not only provide a new theoretical method for the dimensionality reduction of numerical models for nonlinear problems but also provide an algorithm implementation technology for the numerical simulation of practical engineering problems.

Keywords: nonlinear unsaturated soil water flow problem; two-grid Crank–Nicolson finite-element format; proper orthogonal decomposition; reduced-dimension iteration two-grid Crank–Nicolson finite-element format (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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