Does weak convergence imply some kind of approximation?

Question

Let $X$ be a Banach space and $x_n$ be a sequence in $X$ converging weakly to $x \in X$. Then can we say $x$ 'approximates' $x_n$ in some sense? This question is motivated from the homogenization theory. There one has a PDE of the form \begin{equation} A_\epsilon(u_\epsilon)=f_\epsilon, \end{equation} where $A_\epsilon$ is a linear partial differential operator. Here $\epsilon$ can represents the heterogeneity of the domain. Due to heterogeneity, this problems are numerically expensive. Now suppose $u_\epsilon$ weakly converges to $u$ where $u$ is the unique solution of the following PDE \begin{equation} A(u)=f, \end{equation} for some suitable operator $A$ and a function $f$. Then can we say that $u$ is the approximation of $u_\epsilon$ in some sense?