  The weak maximum principle of the isoparametric finite element method is
proved for the Poisson equation under the Dirichlet boundary condition in a
(possibly concave) curvilinear polyhedral domain with edge openings smaller
than $\pi$, which include smooth domains and smooth deformations of convex
polyhedra. The proof relies on the analysis of a dual elliptic problem with a
discontinuous coefficient matrix arising from the isoparametric finite
elements. Therefore, the standard $H^2$ elliptic regularity which is required
in the proof of the weak maximum principle in the literature does not hold for
this dual problem. To overcome this difficulty, we have decomposed the solution
into a smooth part and a nonsmooth part, and estimated the two parts by $H^2$
and $W^{1,p}$ estimates, respectively.
  As an application of the weak maximum principle, we have proved a
maximum-norm best approximation property of the isoparametric finite element
method for the Poisson equation in a curvilinear polyhedron. The proof contains
non-trivial modifications of Schatz's argument due to the non-conformity of the
iso-parametric finite elements, which requires us to construct a globally
smooth flow map which maps the curvilinear polyhedron to a perturbed larger
domain on which we can establish the $W^{1,\infty}$ regularity estimate of the
Poisson equation uniformly with respect to the perturbation.

Numerical Analysis;Numerical Analysis