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

## TL;DR

This paper proves the analyticity and maximal regularity of semi-discrete finite element solutions for parabolic equations in nonconvex polygons and polyhedra, enabling stability analysis in maximum norm.

## Contribution

It establishes analyticity and maximal regularity results for finite element solutions in nonconvex domains, extending previous stability results to more general geometries.

## Key findings

- Proved analyticity of the finite element heat semigroup in L^q norms.
- Established maximal L^p-regularity for semi-discrete solutions.
- Reduced maximum-norm stability analysis to known Ritz projection stability.

## Abstract

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.