Work

A POD-Based Reduced-Dimension Method for Solution Coefficient Vectors in the Crank–Nicolson Mixed Finite Element Method for the Fourth-Order Parabolic Equation

Year: 2025

Type: article

Abstract: The research proposes a method to reduce the dimension of the coefficient vector for Crank-Nicolson mixed finite element (CNMFE) solutions to solve the fourth-order variable coefficient parabolic equation. Initially, the CNMFE schemes and corresponding matrix schemes for the equation are established, followed by a thorough discussion of the uniqueness, stability, and error estimates for the CNMFE solutions. Next, a matrix-form reduced-dimension CNMFE (RDCNMFE) method is developed utilizing Proper Orthogonal Decomposition (POD) technology, with an in-depth discussion on the uniqueness, stability, and error estimates of the RDCNMFE solutions. The reduced-dimension method employ identical basis functions compared to standard CNMFE methods. It significantly reduces the number of unknowns in the computations, thereby effectively decreasing computational time, while there is no loss of accuracy. Finally, the viability and proficiency are substantiated by examining two numerical experiments.

Source: Fractal and Fractional

DOI: 10.3390/fractalfract9030137

Authors: Xiaohui Chang, Hong Li

Institution: Inner Mongolia University

Language: English

Topic: Model Reduction and Neural Networks

Subfield: Statistical and Nonlinear Physics

Field: Physics and Astronomy

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,258

Funder: National Natural Science Foundation of China