Numerical Analysis of a Second Order Crank-Nicolson Mixed Finite Element Method for the Swift-Hohenberg Equation

In this paper, we conduct an error analysis for a mixed finite element method combined with Crank-Nicolson time-stepping for the Swift-Hohenberg (SH) equation. The original fourth-order differential equation from the SH equation is reformulated as a coupled system comprising a nonlinear parabolic equation and an elliptic equation. This transformation allows us to propose a mixed finite element method that utilizes only continuous elements for the approximation of the system. We rigorously demonstrate that our proposed scheme is unconditionally uniquely solvable and maintains unconditional nonlinear energy stability. Additionally, we provide a thorough analysis to determine the convergence rate of the method. To substantiate our theoretical findings, we present numerical tests that confirm both the accuracy and energy stability of the scheme.