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# Comprehensive Exam Presentation - Damyn Chipman

## April 8 @ 2:00 pm - 3:00 pm MDT

Title of this talk: Recent Progress Towards and Motivation for Fast, Direct Methods for Elliptic Partial Differential Equations

Presenter: Damyn Chipman

## Abstract

We present our motivation and the current progress in the literature for developing fast, direct methods for solving elliptic partial differential equations (PDEs). Ellipitic PDEs can be difficult to solve due to their global nature and sometimes ill-conditioned operators. The common approach to solve elliptic PDEs is through iterative methods, which we give an overview of. In addition, to motivate the practicality of direct methods, we also overview the advantages and disadvantages of iterative and direct methods. Recent work in the area of direct methods has led to the theory and development of the Hierarchical Poincaré-Steklov (HPS) method, a direct method that builds a global solution operator through a domain decomposition into a binary tree of rectangular patches. The HPS method can be accelerated to achieve order $\mathcal{O}(N)$ asymptotical performance through hierarchically structured matricies. We also present our motivation for following the recent literature towards fast, direct methods, namely in solving a dispersive term correction to the shallow water equations. This results in a system called the Serre-Green-Nagdhi model and require an elliptic-like solve every timestep.

Committee Members: Donna Calhoun, Michal Kopera, Cathie Olschanowsky