[Paper Review] Homogenization of hyperbolic equations with periodic coefficients

Mark Dorodnyi, T. A. Suslina|arXiv (Cornell University)|Jun 19, 2016

Advanced Mathematical Modeling in Engineering11 references3 citations

TL;DR

This paper establishes operator error estimates for the cosine of the square root of self-adjoint, second-order, periodic differential operators in $L_2(\mathbb{R}^d;\mathbb{C}^n)$, focusing on hyperbolic equations with oscillatory coefficients. Using spectral theory and Floquet-Bloch decomposition, it derives $(H^s \to L_2)$-norm approximations for $\cos(\mathcal{A}_\varepsilon^{1/2}\tau)$ with $O(\varepsilon)$ error, applicable to acoustics and elasticity systems.

ABSTRACT

In $L_2(\mathbb{R}^d;\mathbb{C}^n)$ we consider selfadjoint strongly elliptic second order differential operators ${\mathcal A}_\varepsilon$ with periodic coefficients depending on ${\mathbf x}/ \varepsilon$, $\varepsilon>0$. We study the behavior of the operator cosine $\cos( {\mathcal A}^{1/2}_\varepsilon τ)$, $τ\in \mathbb{R}$, for small $\varepsilon$. Approximations for this operator in the $(H^s o L_2)$-operator norm with a suitable $s$ are obtained. The results are used to study the behavior of the solution ${\mathbf v}_\varepsilon$ of the Cauchy problem for the hyperbolic equation $\partial^2_τ{\mathbf v}_\varepsilon = - \mathcal{A}_\varepsilon {\mathbf v}_\varepsilon +\mathbf{F}$. General results are applied to the acoustics equation and the system of elasticity theory.

Motivation & Objective

• To analyze the asymptotic behavior of the operator cosine $\cos(\mathcal{A}_\varepsilon^{1/2}\tau)$ as $\varepsilon \to 0$ for hyperbolic equations with periodic coefficients.
• To derive operator norm error estimates in the $(H^s \to L_2)$-topology for approximations of $\cos(\mathcal{A}_\varepsilon^{1/2}\tau)$ by its homogenized counterpart.
• To apply the results to the Cauchy problem for hyperbolic equations $\partial_\tau^2 \mathbf{v}_\varepsilon = -\mathcal{A}_\varepsilon \mathbf{v}_\varepsilon + \mathbf{F}$, ensuring convergence of solutions.
• To validate the theory on physical models, including the acoustics equation and linear elasticity system.
• To establish sharpness of error estimates via spectral analysis of the associated Floquet-Bloch operators.

Proposed method

• Utilizes the spectral approach based on Floquet-Bloch theory to decompose the operator $\mathcal{A}_\varepsilon$ into a family of fiber operators parameterized by the quasi-momentum $\boldsymbol{\theta}$.
• Employs the factorization $\mathcal{A}_\varepsilon = f^\varepsilon^* b(\mathbf{D})^* g^\varepsilon b(\mathbf{D}) f^\varepsilon$ to represent the operator in a form amenable to homogenization.
• Applies analytic perturbation theory to study the convergence of the resolvent and semigroups, extending to the cosine operator via functional calculus.
• Derives error estimates by analyzing the difference between the original and effective operators in the $L_2$ and Sobolev norms, incorporating first-order correctors.
• Uses the effective operator $\mathcal{A}^0 = b(\mathbf{D})^* g^0 b(\mathbf{D})$ with constant effective matrix $g^0$, computed via volume averages.
• Establishes sharpness of estimates by analyzing the spectrum of the Bloch fiber operator $\widehat{N}(\boldsymbol{\theta})$ and verifying non-vanishing eigenvalues at critical points $\boldsymbol{\theta}^{(j)}$.

Experimental results

Research questions

• RQ1 What is the rate of convergence of $\cos(\mathcal{A}_\varepsilon^{1/2}\tau)$ to its homogenized limit in the $(H^s \to L_2)$-operator norm as $\varepsilon \to 0$?
• RQ2 How do the error estimates for the hyperbolic cosine operator depend on the regularity of initial data in Sobolev spaces $H^s$?
• RQ3 Can the operator-theoretic homogenization framework developed for elliptic and parabolic problems be extended to hyperbolic equations with periodic coefficients?
• RQ4 What is the sharpness of the $O(\varepsilon)$ error estimate in the $(H^s \to L_2)$-norm for the hyperbolic cosine operator?
• RQ5 How do the results apply to concrete physical systems such as the acoustics equation and linear elasticity in periodic media?

Key findings

• The operator cosine $\cos(\mathcal{A}_\varepsilon^{1/2}\tau)$ converges to $\cos(\mathcal{A}^{0,1/2}\tau)$ in the $(H^s \to L_2)$-operator norm with error $O(\varepsilon)$ for $0 \leq s \leq 3/2$.
• For the Cauchy problem $\partial_\tau^2 \mathbf{v}_\varepsilon = -\mathcal{A}_\varepsilon \mathbf{v}_\varepsilon + \mathbf{F}$, the solution $\mathbf{v}_\varepsilon$ converges to the homogenized solution $\mathbf{v}_0$ with error $\|\mathbf{v}_\varepsilon - \mathbf{v}_0\|_{L_2} \leq \varepsilon^{2s/3} \left( \widehat{\mathfrak{C}}_3(s;\tau) \|\boldsymbol{\phi}\|_{H^s} + \widehat{\mathfrak{C}}_4(s;\tau) \|\boldsymbol{\psi}\|_{H^s} \right)$.
• The error estimate is sharp: non-vanishing eigenvalues $\pm \widehat{\mu} \approx \pm 0.0985$ of the Bloch fiber operator $\widehat{N}(\boldsymbol{\theta}^{(j)})$ confirm the $O(\varepsilon)$ rate cannot be improved.
• For the Hill body (isotropic elasticity with constant shear modulus), the effective matrix $g^0_{\wedge}$ coincides with the average $\underline{g_{\wedge}}$, and the error estimate improves to $O(\varepsilon^2)$ in $L_2$-norm when $\widehat{N}(\boldsymbol{\theta}) = 0$.
• The corrector terms in the approximation are derived from the solution $\Lambda_{22}$ of a periodic boundary value problem, with explicit expressions involving $\arctan$ and complex logarithmic terms.
• The results are extended to the acoustics equation and linear elasticity system, where the factorization $\mathcal{A}_\varepsilon = b(\mathbf{D})^* g^\varepsilon b(\mathbf{D})$ applies directly, and the homogenized operator is explicitly computable.