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.

翻译：本文针对具有小于$\pi$边角开口的（可能凹的）曲边多面体域（包含光滑域和凸多面体的光滑变形）中的泊松方程，在狄利克雷边界条件下证明了等参有限元方法的弱最大值原理。该证明依赖于对由等参有限元引起的具有不连续系数矩阵的对偶椭圆问题的分析。因此，文献中弱最大值原理证明所需的经典$H^2$椭圆正则性对此对偶问题不成立。为克服这一困难，我们将解分解为光滑部分和非光滑部分，并分别通过$H^2$估计和$W^{1,p}$估计对这两部分进行估算。作为弱最大值原理的应用，我们证明了曲边多面体中泊松方程的等参有限元方法具有最大范数最优逼近性质。该证明对Schatz论证进行了重要修正——由于等参有限元的非协调性，需要构造一个全局光滑流映射将曲边多面体映射到扰动后的较大域，从而能够建立关于扰动一致成立的泊松方程$W^{1,\infty}$正则性估计。