A fully discrete implicit scheme is proposed for the Swift-Hohenberg model, combining the third-order backward differentiation formula (BDF3) for the time discretization and the second-order finite difference scheme for the space discretization. Applying the Brouwer fixed-point theorem and the positive definiteness of the convolution coefficients of BDF3, the presented numerical algorithm is proved to be uniquely solvable and unconditionally energy stable, further, the numerical solution is shown to be bounded in the maximum norm. The proposed scheme is rigorously proved to be convergent in $L^2$ norm by the discrete orthogonal convolution (DOC) kernel, which transfer the four-level-solution form into the three-level-gradient form for the approximation of the temporal derivative. Consequently, the error estimate for the numerical solution is established by utilization of the discrete Gronwall inequality. Numerical examples in 2D and 3D cases are provided to support the theoretical results.

翻译：针对Swift-Hohenberg模型提出了一种全离散隐式格式，该格式结合了时间离散的三阶向后微分公式（BDF3）与空间离散的二阶有限差分格式。通过应用Brouwer不动点定理及BDF3卷积系数的正定性，本文证明了所提数值算法具有唯一可解性与无条件能量稳定性，进一步证明数值解在最大模下具有有界性。利用离散正交卷积（DOC）核——该核将四层解形式转化为三层梯度形式以逼近时间导数——严格证明了所提格式在L²范数下的收敛性。进而借助离散Gronwall不等式建立了数值解的误差估计。通过二维和三维算例验证了理论结果。