Multigrid Acceleration Techniques and Applications to
the Numerical Solution of Partial Differential Equations

Jun Zhang
Department of Mathematics
The George Washington University
Washington, DC 20052, USA


Multigrid methods are extremely efficient for solving linear systems arising from discretized elliptic partial differential equations. For these problems, a few multigrid cycles are sufficient to obtain approximate solutions that are at least accurate to the level of the truncation error. However, difficulties associated with problems that are non-elliptic, or that have non-elliptic components, such as those described by the convection-diffusion equations and the incompressible Navier-Stokes equations with high Reynolds numbers, frequently cause a significant decrease in the efficiency of the standard multigrid methods. The convergence degradation gets worse when high-resolution discretization schemes are employed with the standard multigrid methods to obtain high accuracy numerical solutions. The purpose of this study is to develop efficient multigrid acceleration techniques to speed up the convergence of the multigrid iteration process and to apply these techniques to obtain high accuracy numerical solutions of the partial differential equations in computational fluid dynamics. It is shown by analysis and numerical computations that standard multigrid methods can be significantly accelerated and yield highly improved convergence at negligible extra cost. Some acceleration techniques developed in this research even reduce the cost of the standard multigrid methods, in addition to providing satisfactory convergence acceleration. Other techniques have been shown to be essential for some problems to converge. One important feature that distinguishes these acceleration techniques from existing ones is that they do not require that the coefficient matrix be symmetric and positive definite and thus have the potential to be applied to a wider range of practical problems. Another feature of these techniques is that they can be parallelized. Of particular importance to this work is the combination of these acceleration techniques with the high-order finite difference discretization schemes to construct stable multigrid solvers for obtaining fast and high accuracy numerical solutions of the convection-diffusion equations and of the incompressible Navier-Stokes equations with high Reynolds numbers.

Key words: Multigrid method, convergence acceleration, convection-diffusion equation, high-order compact discretization, minimal residual smoothing.

This dissertation was finished approximately in April 1996, but its official submission date was May 18, 1997. It has 147 pages.
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