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

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