DISSERTATION OF JUN ZHANG
Multigrid Acceleration Techniques and Applications to
the Numerical Solution of Partial Differential Equations
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.
Return to Jun Zhang's home page.