Evaluate numerical error estimates

Question

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?

Answer 1

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.