Two-grid dimension reduction method of Crank-Nicolson mixed finite element solution coefficient vectors for the fourth-order extended Fisher-Kolmogorov equation

Abstract

In this article, we mainly resort to a proper orthogonal decomposition (POD) method to reduce the dimensionality of unknown mixed finite element (MFE) solution coefficient vectors in the two-grid Crank-Nicolson MFE (TGCNMFE) method for the fourth-order extended Fisher-Kolmogorov (FOEFK) equation and to build a new two-grid reduced-dimension extrapolated Crank-Nicolson MFE (TGRDECNMFE) method. For this purpose, we first propose a new TGCNMFE method for the FOEFK equation and demonstrate the existence, unconditional stability, and error estimates of the TGCNMFE solutions. Thereafter, we use the POD method to reduce the dimensionality of the unknown TGCNMFE solution coefficient vectors of the FOEFK equation to develop the new TGRDECNMFE method, and use matrix analysis to analyze the existence, unconditional stability, and errors of TGRDECNMFE solutions. Finally, we perform numerical experiments to confirm our theoretical results and show the advantages of the TGRDECNMFE method.