# Evaluate numerical error estimates

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?

• How did you do your refinement?
– Paul
Sep 20, 2015 at 19:23
• 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). Sep 20, 2015 at 19:58