The Reduced-Dimension Method for Crank–Nicolson Mixed Finite Element Solution Coefficient Vectors of the Extended Fisher–Kolmogorov Equation

Abstract

A reduced-dimension (RD) method based on the proper orthogonal decomposition (POD) technology and the linearized Crank–Nicolson mixed finite element (CNMFE) scheme for solving the 2D nonlinear extended Fisher–Kolmogorov (EFK) equation is proposed. The method reduces CPU runtime and error accumulation by reducing the dimension of the unknown CNMFE solution coefficient vectors. For this purpose, the CNMFE scheme of the above EFK equation is established, and the uniqueness, stability and convergence of the CNMFE solutions are discussed. Subsequently, the matrix-based RDCNMFE scheme is derived by applying the POD method. Furthermore, the uniqueness, stability and error estimates of the linearized RDCNMFE solution are proved. Finally, numerical experiments are carried out to validate the theoretical findings. In addition, we contrast the RDCNMFE method with the CNMFE method, highlighting the advantages of the dimensionality reduction method.

Keywords

• extended Fisher–Kolmogorov equation
• proper orthogonal decomposition
• mixed finite element
• uniqueness and stability
• error estimates
• numerical experiment