Date of Completion

7-22-2020

Embargo Period

7-22-2020

Keywords

Finite Elements, Green's Functions, Harnack, Harmonic Functions

Major Advisor

Dmitriy Leykekhman

Associate Advisor

Vasileios Chousionis

Associate Advisor

Jeffrey Connors

Field of Study

Mathematics

Degree

Doctor of Philosophy

Open Access

Open Access

Abstract

The aim of this thesis is twofold. First, we will establish new estimates for the discrete Green's function and obtain some positivity results. In particular, we establish 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 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 establish numerically persistent negative values of the discrete Green's function on Delaunay meshes which then implies a discrete Harnack inequality cannot be establish for unstructured Finite Element discretizations. Of independent interest we also prove Lp estimates for a regularized Green's function in three dimensions which may have implications in establishing best approximation results in optimal control.

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