Document Type

Dissertation

Date of Award

5-31-2021

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

Department

Mathematical Sciences

First Advisor

Brittany Froese Hamfeldt

Second Advisor

David Shirokoff

Third Advisor

Michael Siegel

Fourth Advisor

Travis Askham

Fifth Advisor

Thomas L. Lewis

Abstract

Fully nonlinear elliptic partial differential equations (PDEs) arise in a number of applications. From mathematical finance to astrophysics, there is a great deal of interest in solving them. Eigenvalue problems for fully nonlinear PDEs with transport boundary conditions are of particular interest as alternative formulations of PDEs that require data to satisfy a solvability condition, which may not be known explicitly or may be polluted by noisy data. Nevertheless, these have not yet been well-explored in the literature. In this dissertation, a convergence framework for numerically solving eigenvalue problems for fully nonlinear PDEs is introduced. In addition, existing two-dimensional methods for nonlinear equations are extended to handle transport boundary conditions and eigenvalue problems. Finally, new techniques are designed to enable appropriate discretization of a large range of fully nonlinear three-dimensional equations.

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