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

This research proposes a method for reducing 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 of the uniqueness, stability, and error estimates of the RDCNMFE solutions. The reduced-dimension method employs identical basis functions, unlike 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, numerical experiments are performed for both fourth-order and time-fractional fourth-order parabolic equations. The proposed method demonstrates its effectiveness not only for the fourth-order parabolic equations but also for time-fractional fourth-order parabolic equations, which further validate the universal applicability of the POD-based RDCNMFE method. Under a spatial discretization grid 40×40, the traditional CNMFE method requires 2×412 degrees of freedom at each time step, while the RDCNMFE method reduces the degrees of freedom to 2×6 through POD technology. The numerical results show that the RDCNMFE method is nearly 10 times faster than the traditional method. This clearly demonstrates the significant advantage of the RDCNMFE method in saving computational resources.