Estimate of the L2 projection error

Question

https://pdfs.semanticscholar.org/097d/1f110ae8d39e67545bffca0e7835e4a2679d.pdf

This paper (3.5) says, for any $\nu\in[0,1]$, the $L^2$ orthogonal projection operator $\pi_h$ has the stability: $\forall \psi \in H^{\nu}(\Gamma_C)$,

$$ \left\|\pi_{h} \psi\right\|_{H^{\nu}\left(\Gamma_{C}\right)} \leq C\|\psi\|_{H^{\nu}\left(\Gamma_{C}\right)} $$

It also says the following approximation result holds: for any $\nu\in[0,1]$ and $\mu\in[0,\nu+\frac12]$ we have, $\forall \psi\in H^{\frac12+\nu}(\Gamma_C)$,

$$ h^{-\frac{1}{2}}\left\|\psi-\pi_{h} \psi\right\|_{H^{-\frac{1}{2}}\left(\Gamma_{C}\right)}+h^{\mu}\left\|\psi-\pi_{h} \psi\right\|_{H^{\mu}\left(\Gamma_{C}\right)} \leq C h^{\nu+\frac{1}{2}}\|\psi\|_{H^{\frac{1}{2}+\nu}\left(\Gamma_{C}\right)}$$

It mentions that the estimate in $H^\nu(\Gamma_C)$ is a direct consequence of the stability result and the boundedness of the error w.r.t the $H^{-\frac12}(\Gamma_C)$-norm is obtained by duality.

Could somebody explain how we get the error estimate result from the stability?

Answer 1

I'm assume that $\pi_h$ is the $L^2$-projection onto $\mathcal{P}_p$, the set of piecewise polynomials of degree $p$ over $\Gamma_C$, and that $\Gamma_C$ is some shape-regular union of simplices.

The result follows from scaling and the Bramble--Hilbert Lemma, but the main trick is to realise that since $\pi_h(q)=q$ for any $q\in \mathcal{P}_p$, and so using the stability estimate, $$ \|\psi-\pi_h\psi\|_{H^\mu(\Gamma_C)}=\|(\psi-q)-(\pi_h\psi-q)\|_{H^\mu(\Gamma_C)}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\|(\psi-q)-\pi_h(\psi-q)\|_{H^\mu(\Gamma_C)}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\le\|\psi-q\|_{H^\mu(\Gamma_C)}+\|\pi_h(\psi-q)\|_{H^\mu(\Gamma_C)} $$ $$~~~~~~~~~~~~~~~~\le(1+C)\|\psi-q\|_{H^\mu(\Gamma_C)}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~\le(1+C)\inf_{q\in\mathcal{P}_p}\|\psi-q\|_{H^\mu(\Gamma_C)},$$ where the inf holds due to the fact that we can pick any $q\in \mathcal{P}_p$. Now we can apply the Bramble--Hilbert Lemma and scaling on each simplex to get the error estimate. Essentially the stability of the projection in the $H^\mu$-norm implies that the $L^2$ projection is (up to a constant) as accurate as the best approximation in the $H^\mu$-norm.

Answer 2

For the completeness and for the convenience of those who are curious about the same problem on Mathstack, I answer the dual part and fractional norm part of my own question.

For the dual part, I am mostly interesting in $$h^{-\frac{1}{2}}\left\|\psi-\pi_{h} \psi\right\|_{H^{-\frac{1}{2}}\left(\Gamma_{C}\right)}\leq C h^{\frac{1}{2}}\|\psi\|_{H^{\frac{1}{2}}\left(\Gamma_{C}\right)}.$$

Since $(\psi-\pi_h \psi,v)=(\psi-\pi_h \psi, v-v_h)$ for any $v_h\in \mathcal{P}_p,$ we have $$|(\psi-\pi_h\psi,v)|\leq \|\psi-\pi_h \psi\|_{L^2(\Gamma_C)}\|v-v_h\|_{L^2(\Gamma_C)}\leq h^{\frac{1}{2}}\|\psi\|_{H^{\frac12}(\Gamma_C)}h^{\frac12}\|v\|_{H^{\frac12}(\Gamma_C)}$$ when we choose $v_h $ to be $\pi_h v$.

From this equality and the definition of the dual norm, we get the desired result. I guess this is similar to duality argument. A reference for this is (error estimate in negative Sobolev spaces $^{-s}$ instead of 2) in Theorem 5.8.3 (together with Theorem 5.4.8) in

Susanne C. Brenner and L. Ridgway Scott, MR 2373954 The mathematical theory of finite element methods, Texts in Applied Mathematics ISBN: 978-0-387-75933-3.

For the fractional part, we can make use of the Riesz–Thorin theorem to get the result.

There are several papers about the $L^2$ projection error. I refer the interested readers to the works below:

Weighted L2 projection by J.H.Bramble and J. Xu

The Stability in- Lp and $W^1_p$ of the L2-Projection onto Finite Element Function Spaces by by M. Crouzeix and V. Thomée