SIR Mathematics Investigation Abstract
ANALYSIS OF MULTIGRID METHODS
Presenter:
David Xia, Illinois Mathematics and Science Academy, 1500 West Sullivan Road, Aurora, IL 60506; dxia@imsa.edu
Mentor:
Dr. Paul Fischer, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439-4844
Abstract:
At the core of most physics problems there exists the to need to do extensive mathematical computations, and in the last 25 years there have been major advances in the application of multigrid methods to fulfill these tasks. Today, these advances have made multigrid one of the premier tools for rapidly solving partial differential equations with N unknowns. Multigrid algorithms are small iterative functions that can closely approximate solutions to PDE's, and the accuracy of these solutions depends on the number of iterations as well as the coarseness of the grid over which the function was solved. Even though multigrid is an approximation tool, there are ways to smooth the solutions after using iterative methods. To introduce the topic, I will discuss the close relationship between finite difference approximations to two-point boundary value problems and the iterative method to solving coarse grid problems. The focus of my research has been to analyze modern algorithms as well as the origin of the first multigrid algorithm that was derived from a finite difference discretization of Poisson's equation. In the presentation, I will present my analysis of the accuracy of these solutions, the mathematical reasoning behind the process, examples of multigrid method when used to approximate random functions, and the possible uses of multigrid in research and engineering.