  Maximal regularity is a fundamental concept in the theory of partial
differential equations. In this paper, we establish a fully discrete version of
maximal regularity for a parabolic equation. We derive various stability
results in $L^p(0,T;L^q(\Omega))$ norm, $p,q\in (1,\infty)$ for the finite
element approximation with the mass-lumping to the linear heat equation. Our
method of analysis is an operator theoretical one using pure imaginary powers
of operators and might be a discrete version of G.~Dore and A.~Venni (On the
closedness of the sum of two closed operators. \emph{Math.\ Z.},
196(2):189--201, 1987). As an application, optimal order error estimates in
that norm are proved. Furthermore, we study the finite element approximation
for semilinear heat equations with locally Lipschitz continuous nonlinearity
and offer a new method for deriving optimal order error estimates. Some
interesting auxiliary results including discrete Gagliardo-Nirenberg and
Sobolev inequalities are also presented.
