  In general polygons and polyhedra, possibly nonconvex, the analyticity of the
finite element heat semigroup in the $L^q$ norm, $1\leq q\leq\infty$, and the
maximal $L^p$-regularity of semi-discrete finite element solutions of parabolic
equations are proved. By using these results, the problem of maximum-norm
stability of the finite element parabolic projection is reduced to the
maximum-norm stability of the Ritz projection, which currently is known to hold
for general polygonal domains and convex polyhedral domains.
