Author ORCID Identifier
0000-0002-8771-7703
Document Type
Dissertation
Date of Award
5-31-2023
Degree Name
Doctor of Philosophy in Mathematical Sciences - (Ph.D.)
Department
Mathematical Sciences
First Advisor
Yassine Boubendir
Second Advisor
Brittany Froese Hamfeldt
Third Advisor
Michael Siegel
Fourth Advisor
David Shirokoff
Fifth Advisor
He Xiaoming
Abstract
The primary purpose of this dissertation is to expand upon the circle of domain decomposition methods (DDM) which are algorithms that reformulate a boundary value problem in terms of multiple localized problems on subdomains. The first project involves expanding upon DDMs in a relatively mature field: the Helmholtz equation for wave scattering applications. The proposed method is an adaptation of a continuous cross-point Finite Element Non-overlapping DDM algorithm. The usual unbounded computational domain is truncated and then the near-field wave pattern is solved with a parallelized finite element method. Several improvements over the standard transmission operator are discussed in this work. The resulting method accelerates the convergence and works well in conjunction with both the perfectly matched layer and the absorbing boundary condition.
In the second project, the efficacy of DDMs is explored for the Monge-Ampere (MA) equation, a fully nonlinear 2nd order degenerate elliptic equation. The recent years have seen development of novel discretization schemes of the MA equation. However, the methods to approximate the solutions to the resulting discrete nonlinear system has not been well studied. For this project, several DDMs are applied in conjunction with the theory of convergent monotone schemes for viscosity solutions. Numerical experiments are performed in one dimension (1D) and two dimensions (2D) to demonstrate convergence across a variety of parameters such as degeneracy level, number of subdomains, overlap percentage, and the discretization.
Recommended Citation
Takahashi, Tadanaga, "Domain decomposition methods for linear and non-linear elliptic problems" (2023). Dissertations. 1860.
https://digitalcommons.njit.edu/dissertations/1860
Included in
Numerical Analysis and Computation Commons, Numerical Analysis and Scientific Computing Commons
