Numerical validation of probabilistic laws to evaluate finite element error estimates

We propose a numerical validation of a probabilistic approach applied to estimate the relative accuracy between two Lagrange finite elements P_k and P_m, (k < m). In particular, we show practical cases where finite element P_k gives more accurate results than finite element P_m. This illustrates the theoretical probabilistic framework we recently derived in order to evaluate the actual accuracy. This also highlights the importance of the extra caution required when comparing two numerical methods, since the classical results of error estimates concerns only the asymptotic convergence rate.