The weak maximum principle of finite element methods for parabolic equations is proved for both semi-discretization in space and fully discrete methods with $k$-step backward differentiation formulae for $k = 1,... ,6$, on a two-dimensional general polygonal domain or a three-dimensional convex polyhedral domain. The semi-discrete result is established via a dyadic decomposition argument and local energy estimates in which the nonsmoothness of the domain can be handled. The fully discrete result for multistep backward differentiation formulae is proved by utilizing the solution representation via the discrete Laplace transform and the resolvent estimates, which are inspired by the analysis of convolutional quadrature for parabolic and fractional-order partial differential equations.

翻译：本文证明了二维一般多边形区域或三维凸多面体区域上抛物型方程的有限元方法弱最大值原理，包括空间半离散化以及采用$k$步后向差分公式（$k = 1,...,6$）的全离散方法。半离散结果通过二进分解论证和局部能量估计建立，其中可处理区域非光滑性。多步后向差分公式的全离散结果通过利用离散拉普拉斯变换的解表示及预解估计进行证明，其思想源于抛物型与分数阶偏微分方程卷积求积分析方法的启发。