Stability, analyticity and maximal regularity of semidiscrete isoparametric finite element solutions of parabolic equations in curvilinear polyhedra

In this paper, we investigate the isoparametric finite element semidiscretization for a parabolic problem on a curvilinear polyhedral domain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega subset-of-or-equal-to double-struck upper R Superscript upper N"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mo> ⊆ </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\Omega \subseteq \mathbb {R}^N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with homogeneous Dirichlet boundary condition. The domain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"> <mml:semantics> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:annotation encoding="application/x-tex">\Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> may include nonconvex corners, i.e., with edge openings possibly greater than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi> π </mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We establish the analyticity and maximal regularity of the discrete semigroup by employing a transformation method to address the domain perturbation effect <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega not-equals normal upper Omega Subscript h"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mo> ≠ </mml:mo> <mml:msub> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\Omega \neq \Omega _h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . As an application of the logarithmically quasi-maximal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript normal infinity"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">L^\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -regularity, we derive a quasi-optimal maximum-norm error estimate for the semidiscrete isoparametric finite element methods on curvilinear polyhedral domains with edge openings smaller than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi> π </mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which includes a term of quasi-optimal order due to domain perturbation.