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The weak maximum principle of the isoparametric finite element method isproved for the Poisson equation under the Dirichlet boundary condition in a(possibly concave) curvilinear polyhedral domain with edge openings smallerthan $\pi$, which include smooth domains and smooth deformations of convexpolyhedra. The proof relies on the analysis of a dual elliptic problem with adiscontinuous coefficient matrix arising from the isoparametric finiteelements. Therefore, the standard $H^2$ elliptic regularity which is requiredin the proof of the weak maximum principle in the literature does not hold forthis dual problem. To overcome this difficulty, we have decomposed the solutioninto 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 amaximum-norm best approximation property of the isoparametric finite elementmethod for the Poisson equation in a curvilinear polyhedron. The proof containsnon-trivial modifications of Schatz's argument due to the non-conformity of theiso-parametric finite elements, which requires us to construct a globallysmooth flow map which maps the curvilinear polyhedron to a perturbed largerdomain on which we can establish the $W^{1,\infty}$ regularity estimate of thePoisson equation uniformly with respect to the perturbation.

Weak discrete maximum principle of isoparametric finite element methods in curvilinear polyhedra