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

Analyticity, maximal regularity and maximum-norm stability of semi-discrete finite element solutions of parabolic equations in nonconvex polyhedra