Homogenization - Revision history

imported>Russell at 03:12, 13 March 2012

2012-03-13T03:12:27Z

@@ Line 1: / Line 1: @@
-Homogenization refers to the phenomenon (and the corresponding method of analysis) where solutions of a class of equations with highly oscillatory coefficients behave approximately like the solutions of a regular (say translation invariant) equation (''effective equation'') the approximation becoming more accurate as the oscillation of the coefficients is higher.
+Homogenization refers to the phenomenon-- and the corresponding methods of analysis-- where solutions of a class of equations with highly oscillatory coefficients behave approximately like the solutions of a regular, effective equation which is usually translation invariant.  The approximation becomes more accurate as the oscillation of the coefficients is higher.  At its core, the topic of homogenization should be viewed as a complexity reduction problem in the sense that the original equations typically contain too much detail to be efficiently solvable, whereas if one is willing to make the (hopefully small) error in passing to the translation invariant effective equation, then computing solutions and describing their properties is typically much easier.  Especially in many nonlinear problems there can be large gaps between the theorems of existence for an effective equation and the results which provide refined information about the effective equation which can be practically implemented.
-Typically, one needs some assumption in the way the oscillations organize across space, the most common in increasing order of generality include periodic coefficients, quasi-periodic coefficients, and stationary ergodic coefficients.
+Typically, one needs some assumption in the way the oscillations organize across space, the most common in increasing order of generality include periodic coefficients, quasi-periodic coefficients, and stationary ergodic coefficients.  An illustrative elementary example is given by the family of linear elliptic equations:
-An illustrative elementary example is given by the family of linear elliptic equations:
+\[(\epsilon)\;\;\;\; \left \{ \begin{array}{rl} a_{ij} \left (\frac{x}{\epsilon} \right ) u_{ij}^{(\epsilon)} (x)= f \left (\frac{x}{\epsilon} \right ) & x\in \Omega \\
+u^{(\epsilon)} = g(x) &  x \in \partial \Omega. \end{array}\right. \]
 Where the functions $a_{ij},f$ are assumed to be $\mathbb{Z}^n$-periodic, $\Omega \subset \mathbb{R}^n$ is a smooth bounded domain and $g: \partial \Omega \to \mathbb{R}$ is continuous. Of course the matrix $a_{ij}$ is assumed to be uniformly elliptic.
@@ Line 14: / Line 12: @@
 \[ \left \{ \begin{array}{rl} \bar a_{ij} \bar u_{ij} = \bar f & x\in \Omega \\
-\bar u = g(x) &  x \in \partial \Omega \end{array} \right. \]
+\bar u = g(x) &  x \in \partial \Omega. \end{array} \right. \]

imported>Russell at 01:55, 12 March 2012

2012-03-12T01:55:42Z

← Older revision		Revision as of 20:55, 11 March 2012
Line 8:		Line 8:
	u^{(\epsilon)} = g(x) & x \in \partial \Omega \end{array}\right. \]		u^{(\epsilon)} = g(x) & x \in \partial \Omega \end{array}\right. \]

	Where the functions $a_{ij},f$ are assumed to be $\mathbb{Z}^n$-periodic, $\Omega \subset \mathbb{R}^n$ is a smooth bounded domain and $g: \partial \Omega \to \mathbb{R}$ is continuous. Of course the matrix $a_{ij}$ is assumed to be uniformly ~~periodic~~.		Where the functions $a_{ij},f$ are assumed to be $\mathbb{Z}^n$-periodic, $\Omega \subset \mathbb{R}^n$ is a smooth bounded domain and $g: \partial \Omega \to \mathbb{R}$ is continuous. Of course the matrix $a_{ij}$ is assumed to be uniformly elliptic.

	In this setting, homogenization refers to the fact that as $\epsilon \to 0$, the unique solution $u^{(\epsilon)}$ of the problem above converges ''uniformly'' to a function $\bar u: \Omega\to\R$ which is the unique solution of		In this setting, homogenization refers to the fact that as $\epsilon \to 0$, the unique solution $u^{(\epsilon)}$ of the problem above converges ''uniformly'' to a function $\bar u: \Omega\to\R$ which is the unique solution of

imported>Nestor: Created page with "Homogenization refers to the phenomenon (and the corresponding method of analysis) where solutions of a class of equations with highly oscillatory coefficients behave approximate..."

2011-06-07T05:17:54Z

Created page with "Homogenization refers to the phenomenon (and the corresponding method of analysis) where solutions of a class of equations with highly oscillatory coefficients behave approximate..."

New page

Homogenization refers to the phenomenon (and the corresponding method of analysis) where solutions of a class of equations with highly oscillatory coefficients behave approximately like the solutions of a regular (say translation invariant) equation (''effective equation'') the approximation becoming more accurate as the oscillation of the coefficients is higher.

Typically, one needs some assumption in the way the oscillations organize across space, the most common in increasing order of generality include periodic coefficients, quasi-periodic coefficients, and stationary ergodic coefficients.

An illustrative elementary example is given by the family of linear elliptic equations:

\[(\epsilon)\;\;\;\; \left \{ \begin{array}{rl} a_{ij} \left (\frac{x}{\epsilon} \right ) u_{ij}^{(\epsilon)} = f \left (\frac{x}{\epsilon} \right ) & x\in \Omega \\
u^{(\epsilon)} = g(x) & x \in \partial \Omega \end{array}\right. \]

Where the functions $a_{ij},f$ are assumed to be $\mathbb{Z}^n$-periodic, $\Omega \subset \mathbb{R}^n$ is a smooth bounded domain and $g: \partial \Omega \to \mathbb{R}$ is continuous. Of course the matrix $a_{ij}$ is assumed to be uniformly periodic.

In this setting, homogenization refers to the fact that as $\epsilon \to 0$, the unique solution $u^{(\epsilon)}$ of the problem above converges ''uniformly'' to a function $\bar u: \Omega\to\R$ which is the unique solution of

\[ \left \{ \begin{array}{rl} \bar a_{ij} \bar u_{ij} = \bar f & x\in \Omega \\
\bar u = g(x) & x \in \partial \Omega \end{array} \right. \]