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


Authors: D. Leykekhman and M. Pruitt
Journal: Math. Comp. 86 (2017), 1127-1145
MSC (2010): Primary 65N30, 65N15
DOI: https://doi.org/10.1090/mcom/3117
Published electronically: July 15, 2016
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Abstract: 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.


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Additional Information

D. Leykekhman
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
Email: dmitriy.leykekhman@uconn.edu

M. Pruitt
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
Email: michael.dennis.pruitt@gmail.com

DOI: https://doi.org/10.1090/mcom/3117
Keywords: Maximum norm, finite element method, pointwise error estimates, discrete Green's function, discrete Harnack inequality
Received by editor(s): July 21, 2014
Received by editor(s) in revised form: May 13, 2015, and October 5, 2015
Published electronically: July 15, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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