The Crank-Nicolson mixed finite element scheme and its reduced-order model for the Cahn-Hilliard equation

The Cahn-Hilliard equation is reduced to a second-order system by introducing an auxiliary variable 𝑣 = − 𝜀 2 ⁢ Δ ⁢ 𝑢 + 𝑓 ⁡ ( 𝑢 ) + 𝛽 ⁢ 𝑢 𝑡 . We formulate a Crank-Nicolson mixed finite element scheme. The stability of the scheme is proven, and error estimates for both u and v are derived. Furthermore, a reduced-order Crank-Nicolson mixed finite element model is constructed by applying proper orthogonal decomposition to the coefficient vectors of the first few Crank-Nicolson mixed finite element solutions. This reduced-order model is also proven stable, and we provide corresponding error estimates. Theoretical analysis demonstrates that the reduced-order method reduces the degrees of freedom from tens of thousands to just a few, significantly decreasing CPU time and storage costs. Numerical simulations confirm the effectiveness of both methods. The results demonstrate that both schemes achieve second-order spatial convergence for varying values of the viscosity coefficient β and interface width parameter ε. Under identical experimental conditions, the reduced-order Crank-Nicolson mixed finite element formulation required only about 1/100 of the CPU time of the Crank-Nicolson mixed finite element scheme. Moreover, the numerically observed energy-decay behavior is consistent with the theoretical derivation, as verified by multiple numerical examples.