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I would like your help understanding this article. 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 !

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  • $\begingroup$ This is very far from my area, but might some more tags help elicit attention? I'm guessing "elliptic pde" and "finite element method" might be appropriate tags, but you or someone else should make that call. $\endgroup$
    – Todd Trimble
    Commented Aug 10, 2015 at 12:20
  • $\begingroup$ The answer to 1. is simple: you have probably seen the Céa lemma with a $\forall v_h \in V_h$, for instance here. But if you prove that $something \leq f(v_h)$ for each $v_h \in V_h$, then this also implies $something \leq \inf_{v_h\in V_h} f(v_h)$. $\endgroup$
    – Federico Poloni
    Commented Aug 10, 2015 at 12:32
  • $\begingroup$ @ToddTrimble You are right, but the "finite element method" tag was not available, and I cannot create any yet. I will try to find other ones $\endgroup$
    – Extan
    Commented Aug 10, 2015 at 13:03
  • $\begingroup$ @FedericoPoloni Ok perfect, I was affraid that I took a shortcut by saying that. Thank you ! $\endgroup$
    – Extan
    Commented Aug 10, 2015 at 13:05

1 Answer 1

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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...

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