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Reduced Order Models (ROM) provide a low-dimensional alternative form of a system of differential equations. Such a form permits faster computation of solutions. In this paper, Poisson's Equation in two dimensions, the Heat Equation in one dimension, and a Nonlinear Reaction-Diffusion equation in one dimension are solved using the Galerkin formulation of the Finite Element Method (FEM) in conjunction with Newton's Method. Reduced Order Modeling (ROM) by Proper Orthogonal Decomposition (POD) is then used to accelerate the solution of successive linear systems required by Newton's Method. This is done to show the viability of the method on a simple problem. The Navier-Stokes (NS) Equations are introduced and solved by FEM. A ROM using both POD and clustering by Centroidal Voronoi Tesselation (CVT) are then used to solve the NS equations, and the results are compared with the FEM solution. The specific NS problem we consider has inhomogeneous Dirichlet boundary conditions and the treatment of the boundary conditions is explained. The resulting decrease in computation time required for solving the various equations are compared with ROM methods.
Finite Element Methods, Navier-Stokes Equations, Nonlinear PDEs, Reduced Order Modeling
Date of Defense
October 5, 2012.
A Thesis submitted to the Department of Scientiﬁc Computing in partial fulfillment of the requirements for the degree of Master of Science.
Includes bibliographical references.
Janet Peterson, Professor Directing Thesis; Tomasz Plewa, Committee Member; Sachin Shanbhag, Committee Member.
Florida State University
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