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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 $[0,1]$, ...

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