This article extends the semidiscrete maximal $L^p$-regularity results in Li (2019, Analyticity, maximal regularity and maximum-norm stability of semi-discrete finite element solutions of parabolic equations in nonconvex polyhedra. Math. Comp., 88, 1--44) to multistep fully discrete finite element methods for parabolic equations with more general diffusion coefficients in $W^{1,d+\beta }$, where $d$ is the dimension of space and $\beta>0$. The maximal angles of $R$-boundedness are characterized for the analytic semigroup $e^{zA_h}$ and the resolvent operator $z(z-A_h)^{-1}$, respectively, associated to an elliptic finite element operator $A_h$. Maximal $L^p$-regularity, an optimal $\ell ^p(L^q)$ error estimate and an $\ell ^p(W^{1,q})$ estimate are established for fully discrete finite element methods with multistep backward differentiation formulae.

中文翻译：

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抛物方程多步完全离散有限元法的最大正则性

本文扩展了 Li 中的半离散最大 $L^p$-regularity 结果（2019, Analyticity, maximal regularity and maximum-norm stability of semi-discretefinite element solutions of parabolic equations in nonconvex polyhedra. Math. Comp., 88, 1 --44) 用于抛物线方程的多步完全离散有限元方法，在 $W^{1,d+\beta }$ 中具有更一般的扩散系数，其中 $d$ 是空间维数，$\beta>0$。$R$-有界的最大角分别为解析半群 $e^{zA_h}$ 和分解算子 $z(z-A_h)^{-1}$ 表征，与椭圆有限元算子 $ 相关联A_h$。最大 $L^p$-正则性，最优 $\ell ^p(L^q)$ 误差估计和 $\ell ^p(W^{1,