Discrete maximal parabolic regularity for Galerkin finite element methods for non-autonomous parabolic problem s

The main goal of the paper is to establish time semidiscrete and space-time fully discrete maximal parabolic regularity for the lowest order time discontinuous Galerkin solution of linear parabolic equations with time-dependent coefficients. Such estimates have many applications. As one of the applications we establish best approximations type results with respect to the $L^p(0,T;L^2(\mathrm{\Omega }))$ norm for $1\le p\le \mathrm{\infty }$.