In this dissertation we study multiscale numerical methods for nonlinear parabolic
equations, turbulent diffusion problems, and high contrast parabolic equations. We
focus on designing and analysis of multiscale methods which can capture the effects
of the small scale locally.
At first, we study numerical homogenization of nonlinear parabolic equations
in periodic cases. We examine the convergence of the numerical homogenization
procedure formulated within the framework of the multiscale finite element method.
The goal of the second problem is to develop efficient multiscale numerical techniques
for solving turbulent diffusion equations governed by celluar flows. The solution near
the separatrices can be approximated by the solution of a system of one dimensional
heat equations on the graph. We study numerical implementation for this asymptotic
approach, and spectral methods and finite difference scheme on exponential grids are
used in solving coupled heat equations. The third problem we study is linear parabolic
equations in strongly channelized media. We concentrate on showing that the solution
depends on the steady state solution smoothly.
As for the first problem, we obtain quantitive estimates for the convergence of
the correctors and some parts of truncation error. These explicit estimates show us
the sources of the resonance errors. We perform numerical implementations for the
asymptotic approach in the second problem. We find that finite difference scheme with exponential grids are easy to implement and give us more accurate solutions
while spectral methods have difficulties finding the constant states without major
reformulation. Under some assumption, we justify rigorously the formal asymptotic
expansion using a special coordinate system and asymptotic analysis with respect to
high contrast for the third problem.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/ETD-TAMU-2993 |
| Date | 15 May 2009 |
| Creators | Nam, Dukjin |
| Contributors | Efendiev, Yalchin |
| Source Sets | Texas A and M University |
| Language | en_US |
| Detected Language | English |
| Type | Book, Thesis, Electronic Dissertation, text |
| Format | electronic, application/pdf, born digital |
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