Date of Award

3-2025

Document Type

Thesis

Degree Name

Master of Science in Applied Mathematics

Department

Department of Mathematics and Statistics

First Advisor

Jonah A. Reeger, PhD

Second Advisor

Benjamin F. Akers, PhD

Abstract

Recent progress has been made in the development of collocation-based iterative algorithms that approximate solutions to PDEs. These algorithms rely on the ability to identify regions within a domain where a finer discretization is required. Such iterative algorithms are beneficial particularly when solution functions have highly localized behavior. This thesis proposes an indicator for node refinement that is constructed by approximating the forward error. This proposed indicator also helps to establish confidence in the accuracy of a given solution estimate. The proposed error estimator is theoretically examined and compared with contemporary refinement indicators. It is shown that an iterative algorithm, when using the proposed estimator, approximates solutions to Poisson’s equation as efficiently or more efficiently than the same algorithm does when utilizing contemporary indicators. Moreover, this error estimator predicts the true forward error of an approximate solution far more accurately than other indicators. Analysis of this proposed estimator partially explains this superior performance.

AFIT Designator

AFIT-ENC-MS-25-M-241

Comments

An embargo was observed for this posting.

Approved for Public Release, Distribution Unlimited. PA case number 88ABW-2025-0201

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