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This article extends the semidiscrete maximal $L^p$-regularity results in[27] to multistep fully discrete finite element methods for parabolic equationswith more general diffusion coefficients in $W^{1,d+\beta}$, where $d$ is thedimension of space and $\beta>0$. The maximal angles of $R$-boundedness arecharacterized for the analytic semigroup $e^{zA_h}$ and the resolvent operator$z(z-A_h)^{-1}$, respectively, associated to an elliptic finite elementoperator $A_h$. Maximal $L^p$-regularity, optimal $\ell^p(L^q)$ error estimate,and $\ell^p(W^{1,q})$ estimate are established for fully discrete finiteelement methods with multistep backward differentiation formula.

Maximal regularity of multistep fully discrete finite element methods for parabolic equations