Rate of convergence in homogenization of parabolic PDEs
Rates of convergence (\(n\rightarrow\infty\)) are established in homogenization of a simple heat equation \(\partial u^{(n)} / \partial t =a^{(n)}\Delta u^{(n)}\) with a random coefficient \(a^{(n)}\). Let \(u_0\) be a compactly supported \(C^2\) function on \(\mathbb R^d\) and \(a_0\) a nonzero real number, convergence is considered to the solution \(u\) of the homogenized equation \[ u(0,x)=u_0(x),\qquad \frac{\partial}{\partial t}u(t,x) =a_0\Delta u(t,x). \] The \(u^{(n)}\)'s are solutions of \[ u^{(n)}(0,x,\omega)=u_0(x),\qquad \frac{\partial}{\partial t}u^{(n)}(t,x,\omega) =a^{(n)}(x,\omega)\Delta u^{(n)}(t,x,\omega). \] The main result of the paper, under a condition of so-called well-mixing of \(1/a^{(n)}\) around a mean value of \(1/a_0\), is to quantify the convergence \(u^{(n)}\rightarrow u\) with an estimation of the expected value, for \(t\) fixed, \(\mathbb E\left(\int[u(t,x)-u^{(n)}(t,x,\omega)]^2 \,dx\right).\) This result is remarkable since, apart from the non-random periodic case when \(a^{(n)}(x)=a(nx)\) for \(a\) periodic, most known results prove convergence without estimation. There may be a confusion in Example 2. I would expect \(b^{(n)}\) well-mixing with mean \((b_1+b_2)/2\) and rate \(q(m,n)=0\) for \(m<n\). This would give in Corollary 2: \(c_4 t^{1/2} 2^{-c_1 n}\).
scientific article; zbMATH DE number 1968873