\begin{algorithmic}[1]
\State For $\beta\in (1,\frac{3}{2})$ set $\varepsilon = \frac{3}{2}-\beta$ and $T = L^{2(1-\varepsilon)}$. For $i=1,\cdots,d$, solve for the approximate first-order corrector $\phi_{i,T}^{(L)}$:
\begin{equation}\label{eqn:phiTL}
 \dfrac{1}{T}\phi_{i,T}^{(L)}-\nabla  \cdot a \nabla \phi_{i,T}^{(L)} =\nabla\cdot ae_i \,  \mbox{ in }Q_{2L}, \hspace{0.3in} \phi_{i,T}^{(L)}=0 \, \mbox{ on }\partial Q_{2L}. 
\end{equation}
\State Calculate the approximate homogenized coefficients via \begin{equation}\label{eqn:algahL}
    a_h^{(L)}e_i=\int \omega q_{i,T}^{(L)},
\end{equation} where \begin{equation}\label{eqn:defqiTL} q_{i,T}^{(L)}:=a(e_i+\nabla\phi_{i,T}^{(L)})\end{equation} and $\omega(x)=\frac{1}{L^d}\hat{\omega}(\frac{x}{L})$ with $\hat{\omega}$ as in Theorem \ref{thm:luottooptimal}. 
\State Find $\tilde{u}_h^{(L)}$ on $\partial Q_L$: 
\begin{equation}\label{eqn:alguhtildeL}
    \tilde{u}_h^{(L)} =\int G_h^{(L)}* (\nabla\cdot g),
\end{equation}where $G_h^{(L)}(x) := \frac{1}{4\pi\left|(a_h^{(L)})^{-1/2}x\right|}$ is the Green function for the constant-coefficient operator $-\nabla\cdot a_h^{(L)} \nabla$.
\State Solve for approximate first-order flux correctors $\sigma_{i,T}^{(L)}=\{\sigma_{ijk,T}^{(L)}\}_{j,k}$:  \begin{equation}\label{eqn:algsigma}
    \dfrac{1}{T}\sigma_{ijk,T}^{(L)}-\Delta \sigma_{ijk,T}^{(L)} =\partial_j q_{ik,T}^{(L)}-\partial_k q_{ij,T}^{(L)} \, \mbox{ in }Q_{\frac{7}{4}L}, \hspace{0.3in} \sigma_{ijk,T}^{(L)}=0 \, \mbox{ on }\partial Q_{\frac{7}{4}L}. 
\end{equation}
\State Solve for approximate second-order correctors $\psi_{ij,T}^{(L)}$:  \begin{equation}\label{eqn:2ndcorapprox} 
     \dfrac{1}{T}\psi_{ij,T}^{(L)} - \nabla\cdot a \nabla \psi_{ij,T}^{(L)} = \nabla\cdot (\phi_{i,T}^{(L)}a-\sigma_{i,T}^{(L)})e_j \, \mbox{ in }Q_{\frac{3}{2}L}, \hspace{0.3in} \psi_{ij,T}^{(L)}=0 \,\mbox{ on }\partial Q_{\frac{3}{2}L}. 
\end{equation}
\State For the indices \begin{equation}\label{eqn:calJ}(i,j)\in \mathcal{J}=\{(1,2),(1,3),(2,3),(2,2),(3,3)\},\end{equation} calculate  \begin{equation}\label{eqn:cijlt}
    c_{ij,T}^{(L)}=-\int g\cdot  \nabla \Bigl(\sum_{k=1}^3\phi_{k,T}^{(L)}\partial_k v_{h,ij}^{(L)}+(2-\delta_{ij})(\psi_{ij,T}^{(L)}-\dfrac{a_{hij}^{(L)}}{a_{h11}^{(L)}}\psi_{11,T}^{(L)})\Bigr) ,
\end{equation} where $v_{h,ij}^{(L)}$ denote the $a_h^{(L)}$-harmonic polynomials \begin{equation}\label{eqn:harmpolL}
    v_{h,ij}^{(L)}=(1-\dfrac{1}{2}\delta_{ij})(x_ix_j-\dfrac{a_{hij}^{(L)}}{a_{h11}^{(L)}}x_1^2).
\end{equation}
\State Obtain $u_h^{(L)}$ as \begin{equation}\label{eqn:algapproxbdry}
    u_h^{(L)}=\tilde{u}_h^{(L)}+ \sum_{i=1}^3(\int  g \cdot\nabla \phi_{i,T}^{(L)})\partial_i G_h^{(L)} +\sum_{(i,j)\in\mathcal{J}}c_{ij,T}^{(L)}\partial_{ij} G_h^{(L)}.
\end{equation} 
\State Solve for $u^{(L)}$ (here and for the rest of the paper we adopt Einstein's summation convention for repeated indices): \begin{equation}\label{eqn:finalapprox}
   -\nabla \cdot a \nabla u^{(L)}=\nabla \cdot g\text{ in }Q_L,\hspace{0.3in} u^{(L)}=(1+\phi_{i,T}^{(L)}\partial_i+\psi_{ij,T}^{(L)}\partial_{ij}) u_h^{(L)}\text{ on }\partial Q_L.
\end{equation} 
\end{algorithmic}

\begin{algorithmic}
[1]
\State For $\beta\in (1,\frac{3}{2})$ set $\varepsilon = \frac{3}{2}-\beta$ and $T = L^{2(1-\varepsilon)}$. For $i=1,\cdots,d$, solve for the approximate first-order corrector $\phi_{i,T}^{(L)}$:
\begin{equation*}
\dfrac{1}{T}\phi_{i,T}^{(L)}-\nabla \cdot a \nabla \phi_{i,T}^{(L)} =\nabla\cdot ae_i \, \mbox{ in }Q_{2L}, \hspace{0.3in} \phi_{i,T}^{(L)}=0 \, \mbox{ on }\partial Q_{2L}.
\end{equation*}
\State Calculate the approximate homogenized coefficients via \begin{equation*}
a_h^{(L)}e_i=\int \omega q_{i,T}^{(L)},
\end{equation*} where \begin{equation*}
q_{i,T}^{(L)}:=a(e_i+\nabla\phi_{i,T}^{(L)})\end{equation*} and $\omega(x)=\frac{1}{L^d}\hat{\omega}(\frac{x}{L})$ with $\hat{\omega}$ as in Theorem \ref{thm:luottooptimal}.
\State Find $\tilde{u}_h^{(L)}$ on $\partial Q_L$:
\begin{equation*}
\tilde{u}_h^{(L)} =\int G_h^{(L)}* (\nabla\cdot g),
\end{equation*}where $G_h^{(L)}(x) := \frac{1}{4\pi\left|(a_h^{(L)})^{-1/2}x\right|}$ is the Green function for the constant-coefficient operator $-\nabla\cdot a_h^{(L)} \nabla$.
\State Solve for approximate first-order flux correctors $\sigma_{i,T}^{(L)}=\{\sigma_{ijk,T}^{(L)}\}_{j,k}$: \begin{equation*}
\dfrac{1}{T}\sigma_{ijk,T}^{(L)}-\Delta \sigma_{ijk,T}^{(L)} =\partial_j q_{ik,T}^{(L)}-\partial_k q_{ij,T}^{(L)} \, \mbox{ in }Q_{\frac{7}{4}L}, \hspace{0.3in} \sigma_{ijk,T}^{(L)}=0 \, \mbox{ on }\partial Q_{\frac{7}{4}L}.
\end{equation*}
\State Solve for approximate second-order correctors $\psi_{ij,T}^{(L)}$: \begin{equation*}
\dfrac{1}{T}\psi_{ij,T}^{(L)} - \nabla\cdot a \nabla \psi_{ij,T}^{(L)} = \nabla\cdot (\phi_{i,T}^{(L)}a-\sigma_{i,T}^{(L)})e_j \, \mbox{ in }Q_{\frac{3}{2}L}, \hspace{0.3in} \psi_{ij,T}^{(L)}=0 \,\mbox{ on }\partial Q_{\frac{3}{2}L}.
\end{equation*}
\State For the indices \begin{equation*}
(i,j)\in \mathcal{J}=\{(1,2),(1,3),(2,3),(2,2),(3,3)\},\end{equation*} calculate \begin{equation*}
c_{ij,T}^{(L)}=-\int g\cdot \nabla \Bigl(\sum_{k=1}^3\phi_{k,T}^{(L)}\partial_k v_{h,ij}^{(L)}+(2-\delta_{ij})(\psi_{ij,T}^{(L)}-\dfrac{a_{hij}^{(L)}}{a_{h11}^{(L)}}\psi_{11,T}^{(L)})\Bigr) ,
\end{equation*} where $v_{h,ij}^{(L)}$ denote the $a_h^{(L)}$-harmonic polynomials \begin{equation*}
v_{h,ij}^{(L)}=(1-\dfrac{1}{2}\delta_{ij})(x_ix_j-\dfrac{a_{hij}^{(L)}}{a_{h11}^{(L)}}x_1^2).
\end{equation*}
\State Obtain $u_h^{(L)}$ as \begin{equation*}
u_h^{(L)}=\tilde{u}_h^{(L)}+ \sum_{i=1}^3(\int g \cdot\nabla \phi_{i,T}^{(L)})\partial_i G_h^{(L)} +\sum_{(i,j)\in\mathcal{J}}c_{ij,T}^{(L)}\partial_{ij} G_h^{(L)}.
\end{equation*}
\State Solve for $u^{(L)}$ (here and for the rest of the paper we adopt Einstein's summation convention for repeated indices): \begin{equation*}
-\nabla \cdot a \nabla u^{(L)}=\nabla \cdot g\text{ in }Q_L,\hspace{0.3in} u^{(L)}=(1+\phi_{i,T}^{(L)}\partial_i+\psi_{ij,T}^{(L)}\partial_{ij}) u_h^{(L)}\text{ on }\partial Q_L.
\end{equation*}
\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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