A Reduced-Order Extrapolation (ROE) Method for Solution Coefficient Vectors in the Mixed Finite Element (MFE) Method for the Two-Dimensional (2D) Fourth-Order Hyperbolic Equation

Abstract

This study focuses on a reduced-order extrapolation method for the coefficient vectors of the mixed finite element solution for two-dimensional fourth-order hyperbolic equation. We first establish the mixed finite element scheme for the equation and give the matrix model of the mixed finite element scheme and the existence, stability and error estimates of its solutions. Then, we derive a reduced-order extrapolation mixed finite element matrix model with a small number of unknowns, where the proper orthogonal decomposition method is used to save central processing unit time, and prove the existence, stability and error estimates of the reduced-order extrapolation mixed finite element solutions with the help of matrix knowledge. More importantly, the reduced-order extrapolation mixed finite element matrix model have the same basis functions and error accuracy as the mixed finite element matrix model. Finally, some numerical experiments confirm the effectiveness of the reduced-order extrapolation mixed finite element matrix model, where the central processing unit time is greatly reduced and the accuracy is maintained.INDEX TERMS Fourth-order hyperbolic equation, mixed finite element method, reduced-order extrapolation, proper orthogonal decomposition, existence and stability as well as error analysis.