# Fast methods to solve an elliptic PDE if high accuracy is needed only in part of the domain

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 $$div(\alpha(x) \nabla u) = 0, \quad x \in \Omega \\ u(x) = x,\quad x \in \partial \Omega$$ 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.

• I upvoted Wolfgang's answer, but it might be helpful if you defined "cheap" and said whether this was 2D or 3D. How many $\alpha$s do you need to solve this for, etc.? Oct 22, 2014 at 13:22
• Hi Bill, I have updated the question accordingly. Thank you.
– Tom
Oct 22, 2014 at 15:47
• I was thinking of something along the lines of "I need to compute 1 million different $\alpha$s and need each to run in less than 1 millisecond" or the like. If you can give us some idea of what the $\alpha$s are like and how many there are, we can help you find solutions in reasonable time. Oct 22, 2014 at 23:27
• I think if $\alpha$ spans ten orders of magnitude and is not smooth, then this problem cannot be solved to any kind of accuracy with today's methods. Oct 24, 2014 at 1:42
• Is there any sort of periodicity you can assume about $\alpha$? If so, you could look at homogenization (at least to obtain suitable boundary conditions for local problems). Otherwise you're probably better of treating $\alpha$ as a proper random field and using methods from uncertainty quantification. Oct 25, 2014 at 21:23

• Hi Wolfgang ! I thought about it: in order to produce a coarse-level solution (that you could optionally refine in the area of interest to gain precision), you need to have your diffusion tensor $\alpha$ at a coarse level. However, my diffusion tensor is only defined on the fine scale, so upscaling would be necessary to compute the adapted mesh solution.