Date of Completion
Finite Elements, Green's Functions, Harnack, Harmonic Functions
Field of Study
Doctor of Philosophy
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.
Miller, Andrew, "On the Positivity of the Discrete Green's Function for Unstructured Finite Elements Discretizations in Three Dimensions" (2020). Doctoral Dissertations. 2582.