On homogenization of elliptic equations with random coefficients (Q1572754)

The authors investigate the rate of convergence of the solutions \(u_\varepsilon\) of the random elliptic partial difference equation \[ (\nabla^{\varepsilon*}a(x/\varepsilon,\omega)\nabla^\varepsilon+1) u_\varepsilon(x,\omega)=f(x) \] to the corresponding homogenized solution. Here \(x\in \varepsilon\mathbb Z^d\), and \(\omega \in \Omega\) represents the randomness. Under some conditions an upper bound \(\varepsilon^\alpha\) for the rate of convergence is established with a constant \(\alpha\) which depends on \(d\) and the deviation of \(a(x,\omega)\) from the identity matrix.