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

Abstract

This thesis explores computational efficiency and accuracy of six node refinement methods for local adaptive kernel-based approximations of solutions to the two-dimensional Poisson equation. Using an adaptive kernel-based approximation algorithm, this research investigates performance of Delaunay triangulation-based methods (shifted barycenters and edge midpoints), refinement via approximate Fekete and discrete Leja points, and a meshless predefined shift refinement method across two domains with varying complexities. Computational experiments reveal that Delaunay triangulation-based methods achieve a practical balance between accuracy and efficiency, particularly in square domains. Refinement via approximate Fekete and discrete Leja points produce accurate results but incur greater computational costs, making them suitable for problems requiring precision in small, simple domains. Conversely, predefined shift refinement exhibits less flexibility and increased error, along with high computational expense, limiting its utility. The adaptive framework effectively handles localized features and mixed boundary conditions, with the error indicator from [39] guiding refinement. Results demonstrate that the error indicator approximates absolute error, ensuring targeted refinement and computational efficiency. This work advances understanding of adaptive refinement methods for kernel-based approximations, offering insights into their application for solving PDEs.

AFIT Designator

AFIT-ENC-MS-25-M-242

Comments

An embargo was observed for posting this work.

Distribution Statement A: Distribution Unlimited. Approved for public release. PA case number: 88ABW-2025-0193

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