Corrector Theory in Random Homogenization of Partial Differential Equations

Title:

Corrector Theory in Random Homogenization of Partial Differential Equations

Author(s):

Jing, Wenjia

Thesis Advisor(s):

Bal, Guillaume

Date:

2011

Type:

Dissertations

Department:

Applied Physics and Applied Mathematics

Permanent URL:

http://hdl.handle.net/10022/AC:P:10336

Notes:

Ph.D., Columbia University.

Abstract:

We derive systematically a theory for the correctors in random homogenization of partial differential equations with highly oscillatory coefficients, which arise naturally in many areas of natural sciences and engineering. This corrector theory is of great practical importance in many applications when estimating the random fluctuations in the solution is as important as finding its homogenization limit. This thesis consists of three parts. In the first part, we study some properties of random fields that are useful to control corrector in homogenization of PDE. These random fields mostly have parameters in multidimensional Euclidean spaces. In the second part, we derive a corrector theory systematically that works in general for linear partial differential equations, with random coefficients appearing in their zeroorder, i.e., nondifferential, terms. The derivation is a combination of the studies of random fields and applications of PDE theory. In the third part of this thesis, we derive a framework of analyzing multiscale numerical algorithms that are widely used to approximate homogenization, to test if they succeed in capturing the limiting corrector predicted by the theory.

Subject(s):

Mathematics
Applied mathematics
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