A Novel TGMFECN Dimension Reduction Method Based on POD for the Improved Nonlinear Fourth-Order Cahn-Hilliard Equations (Journal of Computational Mathematics, 2025)
An improved nonlinear fourth-order Cahn-Hilliard (INFOCH) equation is first developed to ensure that its numerical model is symmetric, positive definite, and solvable. Then, by introducing an auxiliary function, the INFOCH equation is decomposed into the nonlinear system of equations with second-order derivatives of spatial variables. Subsequently, by using the Crank-Nicolson (CN) technique to discretize the time derivative, a new time semi-discretized mixed CN (TSDMCN) scheme with second-order accuracy is constructed, and the existence, stability, and error estimates of TSDMCN solutions are analyzed. Thenceforth, a new two-grid mixed finite element (MFE) CN (TGMFECN) method is created by using two-grid MFE method to discretize the TSDMCN scheme, and the existence, stability, and error estimates of TGMFECN solutions are discussed. Next, it is most important that by using proper orthogonal decomposition to reduce the dimension of unknown coefficient vectors of TGMFECN solutions and keep the MFE basis functions unchanged, a new TGMFECN dimensionality reduction (TGMFECNDR) method with very few unknowns, unconditional stability, and second-order time precision is created, and the existence, stability, and error estimates of TGMFECNDR solutions are proved. Finally, the superiority of TGMFECNDR method and the correctness of the obtained theoretical results are showed by two sets of numerical experiments.
Keywords: The improved nonlinear fourth-order Cahn-Hilliard equation; Proper orthogonal decomposition; Time semi-discrete mixed Crank-Nicolson scheme; Two-grid mixed finite element Crank-Nicolson method; Two-grid mixed finite element Crank-Nicolson dimension reduction method; Existence; stability; and error estimate