The aim of this paper is twofold. First, we prove $L^p$ estimates for a regularized Green's function in three dimensions. We then establish new estimates for the discrete Green's function and obtain some positivity results. In particular, we prove that the discrete Green's functions with singularity in the interior of the domain cannot be bounded uniformly with respect of the mesh parameter $h$. Actually, we show that at the singularity the discrete Green's function is of order $h^{-1}$, which is consistent with the behavior of the continuous Green's function. In addition, we also show that the discrete Green's function is positive and decays exponentially away from the singularity. We also provide numerically persistent negative values of the discrete Green's function on Delaunay meshes which then implies a discrete Harnack inequality cannot be established for unstructured finite element discretizations.

翻译：本文旨在实现两个目标。首先，我们证明了三维空间中正则化格林函数的$L^p$估计。随后，我们建立了离散格林函数的新估计，并获得了若干正性结果。特别地，我们证明了奇点位于区域内部的离散格林函数无法关于网格参数$h$一致有界。实际上，我们指出在奇点处离散格林函数的阶为$h^{-1}$，这与连续格林函数的性态一致。此外，我们还证明了离散格林函数是正的，且在远离奇点处呈指数衰减。同时，我们在Delaunay网格上观测到数值上持续存在的离散格林函数负值，这意味着对于非结构有限元离散无法建立离散Harnack不等式。