On the positivity of discrete harmonic functions and the discrete Harnack inequality for piecewise linear finite elements
Dmitriy Leykekhman, Michael Pruitt · Mathematics of Computation · 2015
The main aim of this paper is twofold. First, we investigate fine estimates of the discrete Green’s function and its positivity. We establish that in two dimensions on a smooth domain the discrete Green’s function with singularity in the interior of the domain must be strictly positive throughout the computational domain once the mesh is sufficiently refined. We also establish novel pointwise error estimates for the discrete Green’s function that are valid up to the boundary of the domain. Then, using these estimates we establish a discrete Harnack inequality for piecewise linear discrete harmonic functions. In contrast to the discrete maximum principle the result is valid for general quasi-uniform shape regular meshes except for a condition on the layer near the boundary. Such results may prove to be useful for the analysis of discrete solutions of fully nonlinear problems.