Two-grid reduced-dimension extrapolated method of Crank-Nicolson finite element solution coefficient vectors for nonlinear parabolic equation

In this paper, we resort to proper orthogonal decomposition (POD) to reduce the dimension of unknown coefficient vectors of finite element (FE) solutions of two-grid Crank-Nicolson FE (TGCNFE) method for the nonlinear parabolic equation and develop a new two-grid reduced-dimension extrapolated CNFE (TGRDECNFE) method. To this end, we first develop a new TGCNFE method for the nonlinear parabolic equation and prove the existence, unconditional stability, and error estimates for the TGCNFE solutions. We then develop the TGRDECNFE method for the nonlinear parabolic equation by using the POD technique to lower the dimensionality of unknown coefficient vectors in the TGCNFE solutions and adopt the matrix analysis to analyze the existence, unconditional stability, and errors for the TGRDECNFE solutions. We finally employ two numerical examples to confirm our theoretical results and to explain the advantage of TGRDECNFE method. It has been verified on the whole network that both TGCNFE method and TGRDECNFE method herein are completely different from the existing works. Therefore, they are fire-new.