A cell problem for the boundary density and approximation of boundary correctors in homogenization theory

Abstract

We study the boundary corrector ${\theta }_{\epsilon }$ in the two-scale expansion for periodic elliptic problems with Dirichlet conditions. By results of Gérard-Varet--Masmoudi and Prange, on smooth uniformly convex domains ${\theta }_{\epsilon }$ converges to the solution ${\theta }^{\ast }$ of a homogenized problem with boundary data given by the boundary layer tail $d(n)$, which is determined by a half-space problem for each normal $n$. In the lifted form of Gérard-Varet and Masmoudi, $d(n)$ is given explicitly by the cell correctors and the Dirichlet-to-Neumann operator $N_n$ of this half-space problem. Our main result is that $N_n$ satisfies a nonlinear equation on the period cell alone, so that $d(n)$ and the boundary density that represents it can be computed on the cell. This equation is a Riccati equation in which the derivative in depth is replaced by a commutator with the derivative along $n$. This cell problem can therefore be solved by a Riccati doubling algorithm, and we can compute $d(n)$ at irrational normals without rational approximation. Dropping the commutator term in the cell equation gives an operator square root approximation for $N_n$ that captures the leading term of $d(n)$ for small contrast, for the truncated problem. Finally, we approximate ${\theta }_{\epsilon }$ at finite $\epsilon$ by the sum of two solutions of constant coefficient problems, one with oscillatory boundary data and the other ${\theta }^{\ast }$, and test it on the disk.