Analyticity, maximal regularity and maximum-norm stability of semi-discrete finite element solutions of parabolic equations in nonconvex polyhedra
Extends theoretical foundations for finite element methods in nonconvex polyhedral domains, but is incremental as it builds on known Ritz projection stability.
Proved analyticity and maximal L^p-regularity of semi-discrete finite element solutions for parabolic equations in nonconvex polyhedra, reducing maximum-norm stability to that of the Ritz projection.
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.