Abstract
In this paper, we investigate the rate of convergence of the solution $u_\varepsilon$ 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 $x\in\varepsilon Z^d$, and $\omega\in\Omega$ represents the randomness. Assuming that $a(x)$'s are independent and uniformly elliptic, we shall obtain an upper bound $\varepsilon^\alpha$ for the rate of convergence, where $\alpha$ is a constant which depends on the dimension $d\ge 2$ and the deviation of $a(x,\omega)$ from the identity matrix. We will also show that the (statistical) average of $u_\varepsilon(x,\omega)$ and its derivatives decay exponentially for large $x$.
Highlights
In this paper we shall be concerned with the problem of homogenization of elliptic equations in divergence form
To carry this out we introduce the Legendre polynomials Pl(z) to give us an approximately orthogonal basis for the space generated by the entries of a(·)
Observe that (4.16) cannot necessarily be solved in L2(Ω) since the norms of Tj,ε, as bounded operators on L2(Ω), can become arbitrarily large as ε → 0. To get around this we define a norm on L2(Ω) which depends on ε
Summary
In this paper we shall be concerned with the problem of homogenization of elliptic equations in divergence form. We first prove Theorems 2 and 3 under the assumption that the a(x, ω), x ∈ Zd, ω ∈ Ω, are given by independent Bernoulli variables. Naddaf and Spencer prove that the results of Theorem 1.3 hold under the assumption that φ is a massive field theory and A has bounded derivative. They further prove that if γ is sufficiently small one can take β = d. Dx f (x)ψε(x, ·) = 0, we turn to the variational formulation of the diffusion matrix q in (1.3) To do this we use the translation operators τx on Ω to define discrete differentiation of a function on Ω. The first term in the last expression can be rewritten as d d dx ∇εi ψε(x, ·)∇εj u(x) aij (τx/ε ·) + aik(τx/ε ·)Ψjk(τx/ε ·)
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