Does someone know a method to get cheap approximation of harmonic problems  (and possibly local approximations)?Let me explain: I need to compute the solution of an harmonic problem\begin{equation} div(\alpha(x) \nabla u) = 0, \quad x \in \Omega \\ u(x) = x,\quad x \in \partial \Omega\end{equation}on a very fine mesh $\mathcal T_f$. However I might only be interested by local approximations, ie on convex subdomains of $\mathcal T_f$ of $u(x)$ without the agonizing pain to solve for $u$ on the whole domain. Does some Green functions exists for such problem ? Is there any method to compute cheap approximations? The diffusion tensor $\alpha(x)$ can be any integrable function (no smoothness property assumed here, the equation must be interpreted in its weak formulation).With that respect, cheap can mean many thing, but what I mainly want is to have something cheap to compute that still retains an acceptable accuracy in my subdomains of interest. Since I will have to compute such approximations in many subdomains (in fact the collection of subdomains covers the domain itself), this is why I want it to be cheap. An analytical formulation, or hybrid one would be fine, mesh refinement in the vicinity of my subdomain as well (and this is well known I agree) while keeping a coarse resolution elsewhere. 
Fast methods to solve an elliptic PDE if high accuracy is needed only in part of the domain
adaptive mesh refinement;elliptic pde