I would like your help understanding this [article](http://web.mit.edu/kjb/www/Principal_Publications/A_posteriori_Error_Estimation_Techniques_in_Practical_Finite_Element_Analysis.pdf). 
Page 239 (3.2 A priori error estimates), I am quickly getting lost because of the type of norm that is always changing. 


Things I do not understand:  

1. I never saw Cea's lemma written with an "inf", but I am guessing this is due to Galerkin's procedure.  

2. For equation 14 and 15, I simply do not understand how they change the norms in the inequalities, and how they jumped from the H-norm result to the L2-norm result. 


Thank you for your help !



Cea's lemma and norms
About the different norms in (14) and (15) : only the inequality in the *energy* norm $||.||\_V$ stems from Céa's lemma.


The one involving $||.||\_{L^2}$ is said to be proven in Bathe's textbook. It is classical in finite element analysis for the model problem considered here.


In the simplest case of the Dirichlet problem in *one* dimension $-u''=f$ on $[0,1]$, $u(0)=u(1)=0$, with $V\_h$ made of piecewise linear continuous functions with mesh size $h$, the solution $u\_h$ (defined as the projection of $u$ onto $V\_h$ in the $V$-norm $(\int v'^2)^\frac12$) is in fact the unique *interpolant* of $u$ : $u\_h(kh)=u(kh)$, $k=1,\ldots,1/h$. Then $||u-u\_h||\_{L^2}\leq h||u-u\_h||\_V$.


The reason for (15) in the more general model problem considered in the paper is not that simple, of course...


