Mathematics (Nov 2022)
A Preserving Precision Mixed Finite Element Dimensionality Reduction Method for Unsaturated Flow Problem
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.
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