Numerical Solutions to Poisson's Equation Over Non-Uniform Discretizations with Associated Fast Solvers
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Partial differential equations (PDE's) lay the foundation for the physical sciences and many engineering disciplines. Unfortunately, most PDE's can't be solved analytically. This limitation necessitates approximate solutions to these systems. This thesis focuses on a particular formulation for solving differential equations numerically known as the finite difference method (FDM). Traditional FDM calls for a uniform discretization of the domain over which the PDE is defined. In certain cases, the behavior of a PDE's solution is interesting in a particular region that we would like to better understand. Uniform discretization fails to increase resolution where desired. This manuscript investigates the approximation error of non-uniform discretizations and outlines attempts made at developing a fast-solver for efficiently handling the resultant non-symmetric system of linear equations.