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

This article extends the semidiscrete maximal $L^p$-regularity results in [27] 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 $e^{z{A}_h}$ and the resolvent operator $z(z-A_h{)}^{-1}$, respectively, associated to an elliptic finite element operator $A_h$. Maximal $L^p$-regularity, optimal ${\mathrm{\ell }}^p(L^q)$ error estimate, and ${\mathrm{\ell }}^p(W^{1,q})$ estimate are established for fully discrete finite element methods with multistep backward differentiation formula.