Stability, analyticity, and maximal regularity for parabolic finite element problems on smooth domains
Takahito Kashiwabara, Tomoya Kemmochi · Mathematics of Computation · 2019
In this paper, we consider the finite element semidiscretization for a parabolic problem on a smooth domain Ω ⊂ R N \Omega \subset \mathbb {R}^N with the Neumann boundary condition. We emphasize that the domain can be nonconvex in general. We discretize the this problem by the finite element method by constructing a family of polygonal or polyhedral domains { Ω h } h \{ \Omega _h \}_h that approximate the original domain Ω \Omega . The aim of this study is to derive the smoothing property for the discrete parabolic semigroup and the maximal regularity for the discrete elliptic operator. The main difficulty is the effect of the boundary-skin (symmetric difference) Ω △ Ω h \Omega \bigtriangleup \Omega _h . In order to address the effect of the boundary-skin, we introduce the tubular neighborhood of the original boundary ∂ Ω \partial \Omega .