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$\DeclareMathOperator\diam{diam}$Looking for a proof in the literature of the following lemma:

Let $K\subset\mathbb{R}^d$ be a bounded domain. Let $P_X$ be a finite dimensional subspace of $\mathcal{H}^k(K)$ then

$$\left\|v\right\|_{\mathcal{H}^k(K)} \leq C \diam(K)^{-k} \left\|v\right\|_{L_2(K)}$$

for all $v\in P_X$ and where the constant $C$ does not depend on $\diam(K)$.

There is a proof in The mathematical theory of finite element method by Susanne C. Brenner, L. Ridgway Scott (p111), but they do not check if the constant $C$ is independent of $\diam(K)$.

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  • $\begingroup$ Having trouble getting the main equation to Latex... $\endgroup$
    – alext87
    Commented Feb 23, 2011 at 14:56
  • $\begingroup$ Theorem II.6.8 in Braess' book is quite closely related to your question, but the proof is worked out only for very special domains coming from finite element methods. But maybe you can use the idea... $\endgroup$
    – András Bátkai
    Commented Feb 23, 2011 at 22:26

1 Answer 1

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You may take a look at Theorem 3.2.6 p140 in the book Philippe G. Ciarlet, The Finite Element Method for Elliptic Problems.

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