On the positivity of discrete harmonic functions and the discrete Harnack inequality for piecewise linear finite elements

@article{Leykekhman2017OnTP,
  title={On the positivity of discrete harmonic functions and the discrete Harnack inequality for piecewise linear finite elements},
  author={Dmitriy Leykekhman and Michael Pruitt},
  journal={Math. Comput.},
  year={2017},
  volume={86},
  pages={1127-1145}
}
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… 

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