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:证明了带有Dirichlet边界条件的Poisson方程在边缘开口小于$\pi$的(可能是凹的)曲线多面体域内的等参有限元方法的弱最大值原理。该域包括光滑域和凸多面体的光滑变形。证明依赖于一个二重椭圆问题的分析,该问题具有由等参有限元引起的不连续系数矩阵。因此,文献中用于证明弱最大值原理的标准$H^2$椭圆正则性在该二重问题中不成立。为了克服这一难点,我们将解分解为光滑部分和非光滑部分,并分别用$H^2$和$W^{1,p}$估计这两部分。作为等参有限元方法在曲线多面体中的应用,证明了Poisson方程在曲线多面体中的等参有限元方法的最大范数最佳逼近性质。证明包含了对Schatz论据的非平凡修改,由于等参有限元的不一致性,我们需要构造一个全局光滑的流映射,将曲线多面体映射到一个扰动较大的区域上,从而建立Poisson方程的$W^{1,\infty}$正则性估计,且该估计与扰动无关。