[Paper Review] An embedded corrector problem to approximate the homogenized coefficients of an elliptic equation

Éric Cancès, Virginie Ehrlacher+2 more|arXiv (Cornell University)|Dec 19, 2014

Advanced Mathematical Modeling in Engineering16 references18 citations

TL;DR

This paper introduces an embedded corrector problem for approximating homogenized coefficients in elliptic PDEs with highly oscillatory coefficients. By solving a diffusion equation over the whole space with a uniform diffusion matrix outside a ball of radius R, the authors define three convergent approximations of the homogenized matrix $A^\star$ as $R \to \infty$, offering numerically efficient alternatives to standard supercell methods.

ABSTRACT

We consider a diffusion equation with highly oscillatory coefficients that admits a homogenized limit. As an alternative to standard corrector problems, we introduce here an embedded corrector problem, written as a diffusion equation in the whole space in which the diffusion matrix is uniform outside some ball of radius $R$. Using that problem, we next introduce three approximations of the homogenized coefficients. These approximations, which are variants of the standard approximations obtained using truncated (supercell) corrector problems, are shown to converge when $R \ o \\infty$. We also discuss efficient numerical methods to solve the embedded corrector problem.

Motivation & Objective

• To address the challenge of computing homogenized coefficients $A^\star$ in elliptic PDEs with highly oscillatory coefficients, especially in non-periodic or random stationary settings.
• To overcome the difficulty of solving corrector problems over the entire space by introducing a novel embedded corrector formulation.
• To develop three new approximations of $A^\star$ that converge to the true homogenized matrix as the domain radius $R \to \infty$.
• To provide numerically efficient alternatives to standard supercell (truncated corrector) methods, particularly through boundary integral formulations.
• To establish theoretical convergence of the proposed approximations and prove existence of solutions in special cases, such as isotropic homogenization.

Proposed method

• Formulate an embedded corrector problem as a diffusion equation over $\mathbb{R}^d$, with a heterogeneous diffusion matrix $\mathbb{A}$ inside a ball $B_R$ and a constant matrix $A^R_1$ outside.
• Define the first approximation $A^R_1$ as the average of the flux over $B_R$ using the solution to the embedded corrector problem.
• Define the second approximation $A^R_2$ via a formal energy equality that equates the energy of the heterogeneous system to that of a homogeneous system with matrix $A^R_2$.
• Introduce a third approximation $A^R_3$ through a self-consistent equation $A^R_3 = G^{R,\mathbb{A}}(A^R_3)$, where $G^{R,\mathbb{A}}$ maps a matrix to the average flux from the embedded corrector solution.
• Use boundary integral formulations to enable efficient numerical solution of the embedded corrector problem.
• Prove convergence of all three approximations to the true homogenized matrix $A^\star$ as $R \to \infty$ under Assumption 2.1.

Experimental results

Research questions

• RQ1 Can a novel embedded corrector problem be formulated to approximate homogenized coefficients in elliptic PDEs with highly oscillatory coefficients?
• RQ2 Do the three proposed approximations—$A^R_1$, $A^R_2$, and $A^R_3$—converge to the true homogenized matrix $A^\star$ as the domain radius $R \to \infty$?
• RQ3 Can the embedded corrector problem be solved efficiently using numerical methods such as boundary integral formulations?
• RQ4 Under what conditions does the self-consistent equation $A^R_3 = G^{R,\mathbb{A}}(A^R_3)$ admit a solution, and does it yield a convergent approximation?
• RQ5 How do the proposed approximations compare to standard supercell methods in terms of accuracy and computational efficiency?

Key findings

• The approximation $A^R_1$ defined via the flux average over $B_R$ converges to $A^\star$ as $R \to \infty$ under Assumption 2.1.
• The approximation $A^R_2$, defined by an energy equality, also converges to $A^\star$ in the limit $R \to \infty$.
• The self-consistent approximation $A^R_3$ satisfies $A^R_3 = G^{R,\mathbb{A}}(A^R_3)$ and converges to $A^\star$ when the sequence $R_k \to \infty$ and solutions exist.
• In the isotropic case with $A^\star = a^\star I_d$, a scalar solution $a^R_3$ exists for every $R > 0$ and satisfies $a^R_3 \to a^\star$ as $R \to \infty$.
• For one-dimensional problems, explicit expressions of $A^R_1$, $A^R_2$, and $A^R_3$ can be derived and shown to converge to $A^\star$.
• When $\mathbb{A}$ is constant in $B_R$, all three approximations coincide with the constant matrix: $A^R_1 = A^R_2 = A^R_3 = A$.