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I am using FEM to do an assignment on a heat conduction problem on a complex domain, which needs me to get the variation of the temparature distribution subject to the variation of boundary conditions, and its exact solution is unknown. Also I have to show the correctness and convergence of my solutions numerically.

I am considering to compute the mean temperature over the whole domain, and then with the time variation of this mean value, I can get a curve denoting the evolvement of mean temparature. Then using different refinement levels, I can get a bounch of these curves.

In order to show its convergence, could I just put these curves on the paper with some description like this:"Obviously, it can be seen from the figure that with the increasing refinement level, the mean temparature variation is converging." Or do I have to compute the mean differences of these curve?

In addition, I am not sure whether it's appropriate to use this mean temparature variation as a way to verify the correctness of my solution. Even though I can show the convergence, it is still not necessary to be correct because I don't have the exact solution. Also, even if the mean value is correct, I still can't show the correctness on the whole domain. Maybe the values at some points are larger while smaller elsewhere. Who knows? (After all, no correct solution can be refered to.)

Anyone have some good suggestions?

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You should definitely do a mesh refinement study (some conferences and journals require this). You should compare the results of this study to a known, analytical solution. You can either get this solution from the literature or generate one using the Method of Manufactured Solutions (PDF warning). You should demonstrate that your solution is converging at the expected rate as $h$ decreases as given by the a priori error estimate for your problem if one exists, or you should compute the rate and state it if such an estimate does not (say in a hard nonlinear problem). There may be a nearby problem that gives a rate that you can compare to. If an estimate for the convergence rate of the average of the solution exists or you can derive it, then you can give a result based on that as well. Nevertheless, you need a known solution to compare to.

Sometimes, when no analytic solution exists and an MMS one is hard to compute, people compute a solution on a so-called "overkill" mesh (i.e., as fine as you can manage to compute) and pretend like that one is the analytic one and compare to that. This approach is considered reasonable in some branches of PDE simulation, but better to have an answer at hand in my opinion.

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  • $\begingroup$ Hi @Bill, thank you for your detailed answer. I am also wondering is it appropriate to use the mean temparature value as a measurement? $\endgroup$ – user123 Mar 4 '16 at 14:31
  • $\begingroup$ I should say is it enough compute just the mean value, do I also have to compute the error L^2 norm and H^1 norm $\endgroup$ – user123 Mar 4 '16 at 14:37
  • $\begingroup$ I don't think it's appropriate to use the mean unless you have a proof of the expected convergence rate of the mean which I can't recall if there is for the solution of the heat equation (for example) with, say, the Galerkin method. There probably is, but someone who's got the math closer to brain should chime in on that. $\endgroup$ – Bill Barth Mar 4 '16 at 16:15

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