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I am developing a finite element simulation and want to evaluate the errors in $H^1$ and $L_2$ norms. The problem is the classical Poisson equation, with Dirichlet B.C.: $$-\Delta u=f\mbox{ in }\Omega,$$ $$u=g_{D}\mbox{ on }\Gamma_{D}.$$

I am using Lagrange bilinear 2/3D elements in the Galerkin approach. The error estimates are the standard $$\left\Vert u-u_{h}\right\Vert _{H^{1}\left(\Omega\right)}\leq C\,h\,\left\Vert u\right\Vert _{H^{1}\left(\Omega\right)}$$ and $$\left\Vert u-u_{h}\right\Vert _{L_{2}\left(\Omega\right)}\leq C\,h^{2}\,\left\Vert u\right\Vert _{L_{2}\left(\Omega\right)}.$$

After computing the numerical errors and calculating the experimental order of convergence, how can I evaluate how good is the experimental order? I mean, if I have an order of $2.30$ for the $L_2$ norm, is that good enough? What if it is $1.70$? What criteria can I use?

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    $\begingroup$ How did you do your refinement? $\endgroup$ – Paul Sep 20 '15 at 19:23
  • $\begingroup$ The elements are squares in 2D and cubes in 3D and all the elements of the triangulation have the same size. At each refinement step the element is split in 4 (in 2D) or in 8 (in 3D). $\endgroup$ – Robert Sep 20 '15 at 19:58
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The theory says that the error follows the estimates you state asymptotically. It doesn't actually say very much about what the error would be for any given mesh and in relation to the next mesh.

So what you need to do is to compute on finer and finer meshes. You will (or at least should) observe that the convergence rate will approach 2 if you did everything right, and that's all you typically want to demonstrate.

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  • $\begingroup$ Thanks for the answer. So the best way to show this would be to analyze the rate of convergence between two consecutive meshes rather than the rate considering all points? What I have difficulty is to analyze quantitatively whether a higher or lower order of convergence obtained is good enough. But if I should analyze it approaching the theoretical order it makes more sense. $\endgroup$ – Robert Sep 27 '15 at 14:21
  • $\begingroup$ Yes, use your last two points to determine the convergence rate. If the last several points are not on a straight line in a log-log plot, then your mesh is not fine enough and you need a finer mesh. $\endgroup$ – Wolfgang Bangerth Sep 29 '15 at 13:23

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