Nam, Dukjin (2008-08). Multiscale numerical methods for some types of parabolic equations. Doctoral Dissertation. Thesis uri icon

abstract

  • 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.
  • 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.

publication date

  • August 2008