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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.

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    $\begingroup$ 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.? $\endgroup$ – Bill Barth Oct 22 '14 at 13:22
  • $\begingroup$ Hi Bill, I have updated the question accordingly. Thank you. $\endgroup$ – Tom Oct 22 '14 at 15:47
  • $\begingroup$ 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. $\endgroup$ – Bill Barth Oct 22 '14 at 23:27
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    $\begingroup$ 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. $\endgroup$ – Wolfgang Bangerth Oct 24 '14 at 1:42
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    $\begingroup$ 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. $\endgroup$ – Christian Clason Oct 25 '14 at 21:23
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This would call for adaptive finite element methods, and if you are only interested in specific regions, for the use of goal oriented error estimators to drive the adaptive mesh refinement.

The problem you want to solve is a pretty standard one and you can find my own contribution to this in the step-6 tutorial program of the deal.II library for the general case of solving this equation (http://www.dealii.org/developer/doxygen/deal.II/step_6.html) and in step-14 for the goal oriented adaptivity (http://www.dealii.org/developer/doxygen/deal.II/step_14.html).

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  • $\begingroup$ 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. $\endgroup$ – Tom Oct 23 '14 at 13:15

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