[Paper Review] Homogenization of the Neumann problem for higher-order elliptic equations with periodic coefficients

T. A. Suslina|arXiv (Cornell University)|May 21, 2017

Advanced Mathematical Modeling in Engineering16 references4 citations

TL;DR

This paper establishes sharp operator error estimates for the homogenization of higher-order elliptic Neumann problems with periodic coefficients. It derives approximations for the resolvent $(A_{N, ho}-\zeta I)^{-1}$ in $L_2$ and $H^p$ norms, showing error bounds of order $\varepsilon^{1/2}\rho_\flat(\zeta)^{1/2} + \varepsilon|\zeta+1|\rho_\flat(\zeta)$, with a corrector term $\varepsilon^p K_N^0(\zeta;\varepsilon)$, under optimal regularity and rank conditions on the differential operator and coefficient matrix.

ABSTRACT

Let $\mathcal{O}\subset\mathbb{R}^d$ be a bounded domain of class $C^{2p}$. In $L_2(\mathcal{O};\mathbb{C}^n)$, we study a selfadjoint strongly elliptic operator $A_{N,\varepsilon}$ of order $2p$ given by the expression $b({\mathbf D})^* g({\mathbf x}/\varepsilon) b({\mathbf D})$, $\varepsilon >0$, with the Neumann boundary conditions. Here $g({\mathbf x})$ is a bounded and positive definite $(m imes m)$-matrix-valued function in ${\mathbb R}^d$, periodic with respect to some lattice; $b({\mathbf D})=\sum_{|α|=p} b_α{\mathbf D}^α$ is a differential operator of order $p$ with constant coefficients; $b_α$ are constant $(m imes n)$-matrices. It is assumed that $m\geqslant n$ and that the symbol $b({\boldsymbol ξ})$ has maximal rank for any $0 e {\boldsymbol ξ}\in {\mathbb C}^d$. We find approximations for the resolvent $\left(A_{N,\varepsilon}-ζI ight)^{-1}$ in the $L_2(\mathcal{O};\mathbb{C}^n)$-operator norm and in the norm of operators acting from $L_2(\mathcal{O};\mathbb{C}^n)$ to the Sobolev space $H^p(\mathcal{O};\mathbb{C}^n)$, with error estimates depending on $\varepsilon$ and $ζ$.

Motivation & Objective

• To analyze the homogenization of self-adjoint, strongly elliptic, higher-order differential operators of order $2p$ with periodic coefficients in a bounded $C^{2p}$ domain.
• To derive operator norm error estimates for the resolvent $ (A_{N, ho} - \zeta I)^{-1} $ in $ L_2 $ and $ H^p $ spaces as $ \varepsilon \to 0 $, where $ \varepsilon $ is the scale of periodicity.
• To construct a corrector term $ \varepsilon^p K_N^0(\zeta;\varepsilon) $ that improves the approximation of the resolvent in the $ H^p $ norm.
• To establish error bounds that depend explicitly on the spectral parameter $ \zeta \in \mathbb{C} \setminus [c_\flat, \infty) $, capturing both low- and high-frequency behavior.

Proposed method

• The analysis uses an operator-theoretic approach based on scaling transformations, Floquet-Bloch theory, and analytic perturbation theory.
• The differential operator is factorized as $ A_{N,\varepsilon} = b(\mathbf{D})^* g(\mathbf{x}/\varepsilon) b(\mathbf{D}) $, where $ b(\mathbf{D}) $ is a $ p $-th order constant-coefficient operator with maximal rank symbol.
• The effective operator $ A_N^0 $ is constructed via the homogenized matrix $ g^0 $, derived from the periodic structure of $ g(\mathbf{x}) $.
• Error estimates are derived via spectral analysis in the Fourier domain, using the resolvent identity and estimates on the spectral projectors and corrector operators.
• The corrector $ K_N^0(\zeta;\varepsilon) $ is explicitly defined and shown to capture the oscillatory behavior of the solution at scale $ \varepsilon $, with $ \|K_N^0(\zeta;\varepsilon)\|_{L_2 \to H^p} = O(\varepsilon^{-p}) $.
• The estimates are proven in two operator norms: $ L_2 \to L_2 $ and $ L_2 \to H^p $, with bounds depending on $ \varepsilon $, $ \zeta $, and the spectral function $ \rho_\flat(\zeta) $.

Experimental results

Research questions

• RQ1 What is the rate of convergence of the resolvent $ (A_{N,\varepsilon} - \zeta I)^{-1} $ to the homogenized resolvent $ (A_N^0 - \zeta I)^{-1} $ in the $ L_2 $-operator norm as $ \varepsilon \to 0 $?
• RQ2 How can the convergence be improved beyond the leading-order approximation using a corrector term in the $ H^p $-norm?
• RQ3 What is the dependence of the error on the spectral parameter $ \zeta \in \mathbb{C} \setminus [c_\flat, \infty) $?
• RQ4 Under what conditions does the corrector term vanish or simplify, and how does this affect the error estimate?
• RQ5 Can sharp operator error estimates be derived for higher-order elliptic Neumann problems with periodic coefficients, extending previous results for second-order operators?

Key findings

• The resolvent $ (A_{N,\varepsilon} - \zeta I)^{-1} $ converges to $ (A_N^0 - \zeta I)^{-1} $ in the $ L_2(\mathcal{O};\mathbb{C}^n) \to L_2(\mathcal{O};\mathbb{C}^n) $ norm with error $ \leq \mathfrak{C}_1 \left( \varepsilon^{1/2} \rho_\flat(\zeta)^{1/2} + \varepsilon |\zeta+1| \rho_\flat(\zeta) \right) $ for $ 0 < \varepsilon \leq \varepsilon_1 $.
• In the $ L_2(\mathcal{O};\mathbb{C}^n) \to H^p(\mathcal{O};\mathbb{C}^n) $ norm, the error is bounded by $ \widetilde{\mathfrak{C}}_{10} \left( \varepsilon^{1/2} \rho_\flat(\zeta)^{1/2} + \varepsilon |\zeta+1| \rho_\flat(\zeta) \right) $, with a corrector term $ \varepsilon^p K_N^0(\zeta;\varepsilon) $.
• When $ g^0 = \overline{g} $, the corrector vanishes and the error estimate simplifies to $ \| \mathbf{u}_\varepsilon - \mathbf{u}_0 \|_{H^p} \leq \mathfrak{C}_{10} \left( \varepsilon^{1/2} \rho_\flat(\zeta)^{1/2} + \varepsilon |\zeta+1| \rho_\flat(\zeta) \right) \| \mathbf{F} \|_{L_2} $.
• For the flux $ \mathbf{p}_\varepsilon = g^\varepsilon b(\mathbf{D}) \mathbf{u}_\varepsilon $, the error in $ L_2 $-norm is bounded by $ \widetilde{\mathfrak{C}}_{11} \left( \varepsilon^{1/2} \rho_\flat(\zeta)^{1/2} + \varepsilon |\zeta+1| \rho_\flat(\zeta) \right) \| \mathbf{F} \|_{L_2} $.
• When $ g^0 = \underline{g} $, the error estimate becomes $ \| \mathbf{u}_\varepsilon - \mathbf{v}_\varepsilon^0 \|_{H^p} \leq \widehat{\mathfrak{C}}_{10} \varepsilon \left( \rho_\flat(\zeta)^{1/2} + |\zeta+1| \rho_\flat(\zeta) \right) \| \mathbf{F} \|_{L_2} $, showing improved $ \varepsilon $-dependence.
• The estimates are sharp and depend explicitly on the spectral parameter $ \zeta $, with $ \rho_\flat(\zeta) $ encoding the spectral gap and decay properties of the resolvent.