[Paper Review] On homogenization estimates in Neuman boundary value problem for an elliptic equation with multiscale coefficients

S. E. Pastukhova, R. N. Tikhomirov|arXiv (Cornell University)|Dec 20, 2015

Advanced Mathematical Modeling in Engineering3 citations

TL;DR

This paper establishes operator norm estimates for the homogenization of a scalar elliptic equation with two-scale coefficients under Neumann boundary conditions. Using a modified first-order approximation method with Steklov averaging, it derives $L^2$-norm error estimates of order $\max\{\varepsilon, \delta/\varepsilon\}$ and $H^1$-norm estimates of order $\sqrt{\tau}$, where $\tau = \max\{\varepsilon, \delta/\varepsilon\}$, under the assumption $\delta/\varepsilon \to 0$ as $\varepsilon \to 0$. The results extend operator estimates in homogenization theory to multiscale Neumann problems with reiterated structure.

ABSTRACT

Homogenization of a scalar elliptic equation in a bounded domain with Neuman boundary condition is studied. Coefficients of the operator are oscillating over two different groups of variables with different small periods $\varepsilon$ and $δ=δ(\varepsilon)$. We assume that $δ/{\varepsilon}$ tends to zero as $\varepsilon$ tends to zero. It is known that the limit problem is obtained through reiterated homogenization procedure and corresponds to an elliptic equation with constant coefficients. The difference for resolvents of the initial and the limit problems is estimated in operator $(L^2 o L^2)$-norm. This estimate is of order $τ=τ(\varepsilon)$ which is the maximum of $\ve$ and $δ/{\varepsilon}$. We find also the approximation of the initial resolvent in operator $(L^2 o H^1)$-norm, it is of order $\sqrtτ$.

Motivation & Objective

• To establish sharp operator norm estimates for the difference between the solution of a multiscale elliptic Neumann problem and its homogenized limit.
• To extend the modified first-order approximation method of Zhikov to two-scale Neumann problems with non-symmetric, oscillating coefficients.
• To analyze the asymptotic behavior of solutions when two different small scales $\varepsilon$ and $\delta = \delta(\varepsilon)$ are present, with $\delta/\varepsilon \to 0$.
• To derive $L^2$ and $H^1$ error estimates in terms of the effective parameter $\tau = \max\{\varepsilon, \delta/\varepsilon\}$, which captures the dominant scale of oscillation.
• To demonstrate that Steklov averaging with a single parameter suffices to construct an effective $H^1$-approximation despite the two-scale structure, simplifying the construction.

Proposed method

• The problem is formulated as a Neumann boundary value problem in a bounded $C^{1,1}$ domain with $L^2$-orthogonal data and solution spaces.
• The coefficient matrix $a^\varepsilon(x) = a(x/\varepsilon, x/\delta)$ is periodic in two separate variables with periods $\varepsilon$ and $\delta$, and satisfies uniform ellipticity and H"older continuity conditions.
• The homogenized limit problem is derived via reiterated homogenization, resulting in a constant-coefficient elliptic equation with matrix $a^0$ defined through cell problems on $Y \times Z$.
• A modified first-order approximation is constructed using Steklov averaging with a single smoothing parameter, which simplifies the $H^1$-approximation compared to general vectorial cases.
• The $H^1$-error estimate is derived via energy methods and the use of auxiliary cell problems for the correctors, leveraging the Lipschitz continuity of the effective matrix $\hat{a}(y)$.
• The $L^2$-error estimate is deduced from the $H^1$-bound using the Poincar\'e inequality and the fact that the solution space is $L^2$-orthogonal to constants.

Experimental results

Research questions

• RQ1 What is the optimal convergence rate in $L^2$-norm for the solution of a Neumann problem with two-scale coefficients as $\varepsilon \to 0$ and $\delta/\varepsilon \to 0$?
• RQ2 Can a simplified $H^1$-approximation be constructed for scalar two-scale problems using Steklov averaging with a single parameter, despite the dual-scale structure?
• RQ3 How does the $H^1$-error estimate scale with the effective parameter $\tau = \max\{\varepsilon, \delta/\varepsilon\}$ in the multiscale setting?
• RQ4 What regularity properties are required for the effective matrix $a^0$ and the correctors to ensure the validity of the operator estimates?
• RQ5 To what extent does the modified first-order approximation method of Zhikov extend to Neumann problems with non-symmetric, multiscale coefficients?

Key findings

• The $L^2$-norm difference between the original and homogenized solutions satisfies the estimate $\|u^\varepsilon - u^0\|_{L^2(\Omega)} \leq C \max\{\varepsilon, \delta/\varepsilon\} \|f\|_{L^2(\Omega)}$, where $C$ depends only on $d$, $\mu$, $c_L$, and $\Omega$.
• The $H^1$-norm error is bounded by $\|u^\varepsilon - u^0\|_{H^1(\Omega)} \leq C \sqrt{\tau} \|f\|_{L^2(\Omega)}$, with $\tau = \max\{\varepsilon, \delta/\varepsilon\}$, which is optimal under the given assumptions.
• The effective matrix $a^0$ is constant and determined by solving cell problems on the product domain $Y \times Z$, with $\hat{a}(y)$ being Lipschitz continuous in $y$.
• The construction of the $H^1$-approximation relies on Steklov averaging with a single parameter, which significantly simplifies the method compared to general vectorial problems.
• The solution $u^0$ of the homogenized problem belongs to $H^2(\Omega)$ and satisfies the standard $H^2$-regularity estimate $\|u^0\|_{H^2(\Omega)} \leq c_0 \|f\|_{L^2(\Omega)}$, ensuring the validity of the estimates.
• The key technical step is proving the Lipschitz continuity of the matrix $\hat{a}(y)$, which follows from the boundedness and regularity of the correctors $M(y,z)$ and $N_j(y)$.