\begin{algorithmic}[1]
\State Solve \eqref{intrphi} for first-order correctors $\phi_i$.
\State Determine the homogenized coefficients $a_h$ via \eqref{intrhomcoeff}.
\State Solve \eqref{intruhtilde} for $\tilde{u}_h$ on $\partial Q_L$ by $\tilde{u}_h = \int G_h*(\nabla\cdot g)$.
\State Solve \eqref{eqn:intrsig} for first-order flux correctors $\sigma_{ijk}$ and \eqref{eqn:2ndcordef} for second-order correctors $\psi_{ij}$.
\State Obtain $u_h$ via \eqref{eqn:effectivequadp}.
\State Solve \eqref{eqn:coruhat} for $\hat{u}$, which is the approximation we desire.
\end{algorithmic}

\begin{algorithmic}
[1]
\State Solve \eqref{intrphi} for first-order correctors $\phi_i$.
\State Determine the homogenized coefficients $a_h$ via \eqref{intrhomcoeff}.
\State Solve \eqref{intruhtilde} for $\tilde{u}_h$ on $\partial Q_L$ by $\tilde{u}_h = \int G_h*(\nabla\cdot g)$.
\State Solve \eqref{eqn:intrsig} for first-order flux correctors $\sigma_{ijk}$ and \eqref{eqn:2ndcordef} for second-order correctors $\psi_{ij}$.
\State Obtain $u_h$ via \eqref{eqn:effectivequadp}.
\State Solve \eqref{eqn:coruhat} for $\hat{u}$, which is the approximation we desire.
\end{algorithmic}
{
    "title": "optimal artificial boundary conditions based on second-order correctors   for three dimensional random elliptic media",
    "id": "2109.01616",
    "abstract": "we are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale $\\ell$ in an infinite heterogeneous medium, in a situation where the medium is only known in a box of diameter $l\\gg\\ell$ around the support of the charge. we propose a boundary condition that with overwhelming probability is (near) optimal with respect to scaling in terms of $\\ell$ and $l$, in the setting where the medium is a sample from a stationary ensemble with a finite range of dependence (set to be unity and with the assumption that $\\ell \\gg 1$). the boundary condition is motivated by quantitative stochastic homogenization that allows for a multipole expansion [bgo20].   this work extends [lo21], the algorithm in which is optimal in two dimension, and thus we need to take quadrupoles, next to dipoles, into account. this in turn relies on stochastic estimates of second-order, next to first-order, correctors. these estimates are provided for finite range ensembles under consideration, based on an extension of the semi-group approach of [go15].",
    "categories": "math.ap cs.na math.na math.pr",
    "doi": "",
    "created": "2021-09-03",
    "updated": "2024-01-11",
    "authors": [
        "jianfeng lu",
        "felix otto",
        "lihan wang"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2109.01616"
}
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