## A Note on Multiple Scales: Homogenization Theory

As I have mentioned before, many problems in science can be profitably attacked from a large variety of different angles. Today I want to discuss one particular aspect of theoretical work in a technical way, in order to provide a bit more motivation to why a mathematical method of analysing a scientific problem provides insight that complements computational or experimental work. This is the technique of multiple-scales analysis, and specifically Homogenization. Most of the following was written as part of a broadening course I took in a Perturbation Methods lecture that is taught at Oxford for fourth-year undergraduates.

Here I want to summarize some of the ideas presented in [1], and briefly discuss them in the context of multiple scales analysis. Specifically, I want to emphasize the similarities and differences between one-dimensional homogenization for ordinary differential equations (multiple scales), and the approaches to partial differential equations that are discussed in the paper. Throughout, emphasis will be on the perturbative assumptions and analysis, although other considerations (e.g. numerical difficulties) will be mentioned.

1. Motivation

There are a variety of paradigms used to model the influence of complex media on macroscopic physics, such as the effects of a porous medium on transport of some physical quantity. An interesting approach is that of homogenization, where separation of scales is exploited to simplify the behaviour of the system. To give an example, consider a bounded domain ${\Omega \in {\mathbb R}^n}$ modelling some porous medium, so that to model the microscopic flow in ${\Omega}$ we must account for the complex geometry of the pores. This problem can be very difficult for many reasons, especially since even sophisticated numerical approaches are often not capable of resolving this fluid flow.

Homogenization is a mathematical approach where we consider a suitable average flow through this domain rather than the flow through the microstructure of the fluid domain itself. In many practical situations this is what we are actually interested in. Additionally, these approaches can sometimes be made rigorous, and at the very least give some motivation for phenomenological relationships discovered empirically, such as Darcy’s Law. This idea of rigorous justification is of course a spectrum, and research in areas like porous media fall throughout it. Nevertheless, it is useful to have a methodology that can be applied to a variety of microscopic geometries, and determine useful macroscopic values.

2. 1-D Homogenization

The case of homogenization for Ordinary Differential Equations was presented as an application of the Method of Multiple Scales. I will briefly outline how this can be applied to the equation,

$\displaystyle \frac{d}{dx}(D(x,\frac{x}{\epsilon})\frac{d u}{dx}) = f, \ \ \ \ \ (1)$

where ${D(x, y) > 0}$ and ${f(x, y)}$ are smooth and periodic in ${y}$ with period one. This might be a model, for example, of steady state transport in a one dimensional medium with a rapidly varying microstructure, modelled by the micro or fast variable ${y = \frac{x}{\epsilon}}$. The periodicity assumption warrants some discussion, but is a reasonable physical assumption for many problems of interest. See Technical Note 10 in [1] for more information. What we are interested in doing is averaging this equation in a suitable sense so that we don’t have to resolve the solution at the microscale, that is, when ${x}$ is order ${O(\epsilon)}$. What would be ideal is to recover an ODE in ${x}$ that has an average or effective diffusivity related to ${D}$.

We may proceed via multiple scales by formally treating ${x}$ and ${y}$ as different variables, and expanding ${u}$ as ${u(x,y) = u_0(x,y) + \epsilon u_1(x,y) + \epsilon^2 u_2(x,y) + ...}$. We may then further assumption that the ${u_i}$ are all periodic in ${y}$ with period one. Substituting this into (1), we find

$\displaystyle \begin{array}{rcl} &\partial_x(D(x,y)[\partial_x u_0 + \epsilon^{-1}\partial_y u_0 + \partial_y u_1])\\ + &\epsilon^{-1}\partial_y(D(x,y)[\partial_x u_0 + \epsilon \partial_x u_1 + \epsilon^{-1}\partial_y u_0 + \partial_y u_1 + \epsilon \partial_y u_2]) + O(\epsilon^2) = f. \end{array}$

Formally equating powers of ${\epsilon}$ we find at order ${\epsilon^{-2}}$,

$\displaystyle \begin{array}{rcl} \partial_y(D(x,y)\partial_y u_0) = 0. \end{array}$

If we multiply this equation by ${u_0}$, and integrate in ${y}$, we find

$\displaystyle \begin{array}{rcl} \int\partial_y(D(x,y)\partial_y u_0) u_0 dy = -\int D(x,y)|\partial_y u_0|^2dy = 0, \end{array}$

through integration by parts. Since ${D}$ is positive and ${u_0}$ is periodic in ${y}$, we have that ${u_0}$ does not depend on ${y}$. We now look at order ${\epsilon^{-1}}$,

$\displaystyle \begin{array}{rcl} \partial_y(D(x,y)\partial_y u_1) = -\partial_y (D(x,y) \partial_x u_0). \end{array}$

Integrating once over ${y}$ gives us,

$\displaystyle D(x,y)\partial_y u_1 = -D(x,y)\partial_x u_0 + k(x), \ \ \ \ \ (2)$

where ${k}$ is a constant of integration. Integrating this equation over ${y \in (0,1)}$, and using the periodicity of ${u_1}$, we can find ${k}$ as

$\displaystyle k = \frac{1}{\int_0^1 \frac{1}{D(x,y)}dy}\partial_x u_0 = \frac{1}{\langle 1/D\rangle}\partial_x u_0, \ \ \ \ \ (3)$

where we have used the brackets to denote averaging over the periodic cell. If we now look at the ${O(1)}$ problem we have,

$\displaystyle \begin{array}{rcl} \partial_x(D(x,y)[\partial_x u_0 + \partial_y u_1]) + \partial_y(D(x,y)[\partial_x u_1 + \partial_y u_2]) = f. \end{array}$

If we now average this equation over one period in ${y}$, we find by periodicity that

$\displaystyle \begin{array}{rcl} \partial_x(\langle D(x,y)\rangle\partial_x u_0 + \langle D(x,y)\partial_y u_1\rangle) = \langle f\rangle. \end{array}$

Now using the average of (2), we can substitute for the second term in this equation to find,

$\displaystyle \frac{d}{dx}(\frac{1}{\langle 1/D\rangle}\frac{d}{dx} u_0) = \langle f\rangle, \ \ \ \ \ (4)$

so that the harmonic mean of ${D}$ is the correct average, or effective diffusivity, that appears in our final ODE for ${u_0}$.

3. Homogenization for Partial Differential Equations

The above approach can fairly easily be extended to second order elliptic operators, but instead I want to compare it to the methods discussed in [1]. There, two techniques to homogenize PDE are compared with a general parabolic transport problem in mind. The first approach discussed is the use of volume averaging, which is a very physically-motivated technique to understand the relationship between the microstructure of a material, and its effective approximations. The second approach is by use of formal multiscale asymptotics, which is much closer to the 1-D method demonstrated above. For this reason I will focus the rest of the discussion on this second approach, and refer to the paper for details about the former. For the rest of this discussion we will use notation from the paper, recalling what seems appropriate. See section 4.1 of [1] for details if anything is unclear.

After a suitable nondimensionalization, many transport phenomena can be described by the following problem,

$\displaystyle \partial_t u = \nabla_{\boldsymbol{x}}\cdot(\boldsymbol{A}(\boldsymbol{x})\cdot\nabla_{\boldsymbol{x}} u) \quad \forall \boldsymbol{x} \in \Omega, \quad \forall t \in \mathbb{R}^+, \ \ \ \ \ (5)$

$\displaystyle u(\boldsymbol{x},t) = f(\boldsymbol{x},t) \quad \forall \boldsymbol{x} \in \partial \Omega, \ \ \ \ \ (6)$

$\displaystyle u(\boldsymbol{x},0) = 0 \quad \forall \boldsymbol{x} \in \Omega, \ \ \ \ \ (7)$

where ${\Omega \subseteq R^n}$ for ${n=1,2}$ or ${3}$. We also assume that ${\boldsymbol{A}}$ exhibits only high-frequency oscillations, that is, on length scales of order ${O(\epsilon)}$. As in the 1-D case, we are letting this diffusion tensor ${\boldsymbol{A}}$ model the microstructure of the medium. There is some discussion about when this is not applicable, such as when there are sharp phase boundaries, but it is a relatively good approximation for many physical problems. It should also be noted that ${\epsilon}$ is the ratio of the microscopic length scale, ${l}$, and the macroscopic length scale, ${L}$. That is, ${\epsilon = \frac{l}{L}}$.

To proceed, several assumptions are made about the problem that allow us to treat it in a similar fashion to the 1-D problem above. Here, however, both the physical and mathematical ideas become more technical. The assumptions made, and some simplified discussion, are as follows.

${\boldsymbol{A1}}$: We homogenize the problem by considering the limit of a sequence of fictitious problems ${(u_\epsilon)_{0<\epsilon<1}}$, rather than the original problem (5)(7), which implicitly contains a fixed finite, but nonzero, value of the length scale ratio ${\epsilon_0}$. This is in the spirit of zooming away from the microstructure in the limit, and so this limit should rightfully be considered an asymptotic approximation to the real geometry of the problem. Parameter fields for the problem use a similar subscript notation.

${\boldsymbol{A2}}$: We assume ${\boldsymbol{A_\epsilon}}$ scales as ${O(1)}$ component-wise, and that we can write ${\boldsymbol{A_\epsilon} = \boldsymbol{A_0}}$. The original nondimensionalization was with respect to the macroscopic length scale ${L}$, and so this assumption about scaling is focussing on the global, macroscopic behaviour. There are subtleties about the scalings chosen that are particular to each problem, and the paper has an important discussion of how these can lead to different macroscopic models. See Technical Note 12 in the paper for more details.

${\boldsymbol{A3}}$: We assume that we can use the formal two-scale expansion ${u(\boldsymbol{x}) = u^\dagger(\boldsymbol{x},\frac{\boldsymbol{x}}{\epsilon}) = \sum u_i(\boldsymbol{x},\frac{\boldsymbol{x}}{\epsilon})\epsilon^i}$. A crucial point here is that ${u_i = O(1)}$ for ${i=0,1,2,\dots}$.

${\boldsymbol{A4}}$: We assume that the microstructure of our problem is either periodic, or can be well-approximated by a conceptual periodic setting. The fundamental unit of periodicity is called the unit-cell, and would be considered the interval ${[0,1]}$ in the earlier example. Î½ Finally, we recall that averaging is with respect to the unit-cell being considered in the problem. If ${\mathcal{V}_\epsilon(\boldsymbol{x})}$ is the unit-cell with centroid ${\boldsymbol{x}}$ and corresponding volume ${V_\epsilon}$, we define the average of a function ${\psi(\boldsymbol{x},\boldsymbol{y},t)}$ as

$\displaystyle \begin{array}{rcl} \langle\psi\rangle_\epsilon(\boldsymbol{x},t) = \frac{1}{V_\epsilon}\int_{\mathcal{V}_\epsilon(\boldsymbol{x})}\psi(\boldsymbol{x},\boldsymbol{y},t)dV_{\boldsymbol{y}}. \end{array}$

Here, we are taking the unit-cells to be sequences of sets at each point ${\boldsymbol{x} \in \Omega}$, in order to define this average globally, and so the integral is only over ${\boldsymbol{y}}$. Think of this as a generalization of the earlier definition of average, ${\langle D \rangle = \int_0^1 D(x,y)dy}$.

I will briefly outline the method and results, and refer to the paper for the full details. Applying these assumptions sequentially, and following a process very similar to the 1-D example, we can arrive at a macroscale equation and a unit-cell problem. Doing so yields that ${u_0}$ is independent of ${\boldsymbol{y}}$, and the following equation for the unit-cell,

$\displaystyle \nabla_{\boldsymbol{y}}\cdot (\boldsymbol{A_0}\cdot \nabla_{\boldsymbol{y}} u_1)+\nabla_{\boldsymbol{y}}\cdot(\boldsymbol{A_0}\cdot \nabla_{\boldsymbol{x}} u_0)=0, \ \ \ \ \ (8)$

and the macroscale equation for ${u_0}$,

$\displaystyle \partial_t u_0 = \nabla_{\boldsymbol{x}} \cdot\langle\boldsymbol{A_0}\cdot(\nabla_{\boldsymbol{x}} u_0 + \nabla_{\boldsymbol{y}} u_1)\rangle. \ \ \ \ \ (9)$

Note that these two equations are exactly the macroscale and unit-cell problems for the 1-D problem if ${f=0}$ in the ODE, and if we added the time-derivative term.

Motivated by separation of variables, we write

$\displaystyle \begin{array}{rcl} u_1 = \boldsymbol{\chi}(\boldsymbol{x})\cdot\nabla_{\boldsymbol{x}} u_0, \end{array}$

and substitute it into the cell-problem, (8). We can now use the unit cell geometry and the definition of ${\boldsymbol{A_0}}$ to solve for ${\boldsymbol{\chi}}$, assuming it is periodic in ${\boldsymbol{y}}$. Substituting this separation of variables ansatz into (9) to get,

$\displaystyle \begin{array}{rcl} \partial_t u_0 = \nabla_{\boldsymbol{x}}\cdot(\boldsymbol{A_e}\cdot \nabla_{\boldsymbol{x}} u_0), \end{array}$

where

$\displaystyle \begin{array}{rcl} \boldsymbol{A_e} = \langle \boldsymbol{A_0}\cdot(\boldsymbol{I} + \nabla_{\boldsymbol{y}}. \boldsymbol{\chi})\rangle \end{array}$

Here, the average of the diffusion tensor directly takes the microscopic cell-problem into account. If we were to solve (2) by exploiting the linearity of the cell-problem, we could write the solution to that problem in an identical form. Instead, we went one step further since the geometry of the unit interval is so simple, and computed the effective diffusivity directly. The multidimensional problem has far more subtlety than the simple case, and even after averaging may be difficult to resolve numerically, but the similarities of the problems and their solutions is worth pointing out.

There are many important aspects that the paper covers that I did not discuss, as my intention was mostly to reflect on it in light of the Perturbation Methods lectures that I had attended. There are numerous other aspects worth discussing, such as efficient numerical coupling between the cell-problem and the macroscale equation, or formal convergence results of the asymptotic solutions. I would invite the interested reader to use [1] and the other links below as use starting points for further reading on the topic.

4. References and Useful Sources

[1] – Y. Davit, et. al., Homogenization via formal multiscale asymptotics and volume averaging: How do the two techniques compare? Advances in Water Resources, 62, Part B:178–206, December 2013.

[2] – Some good lecture notes on Multiple Scales methods.

[3] – Another paper comparing the physical meaning of homogenization with the mathematical theory.