Maximum-norm stability and maximal L^p regularity of FEMs for parabolic equations with Lipschitz continuous coefficients

In this paper, we study the semi-discrete Galerkin finite element method for parabolic equations with Lipschitz continuous coefficients. We prove the maximum-norm stability of the semigroup generated by the corresponding elliptic finite element operator, and prove the space-time stability of the parabolic projection onto the finite element space in $L^{\mathrm{\infty }}({\mathrm{\Omega }}_T)$ and $L^p((0,T);L^q(\mathrm{\Omega }))$, $1<p,q<\mathrm{\infty }$. The maximal $L^p$ regularity of the parabolic finite element equation is also established.