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?