Stability of a weighted L2 projection in a weighted Sobolev norm

Averseng, Martin

Comptes Rendus. Mathématique, Tome 361 (2023) no. G4, pp. 757-766

We prove the stability of a weighted $L^2$ projection operator onto piecewise linear finite elements spaces in a weighted Sobolev norm. Namely, we consider the orthogonal projections ${\pi }_{N,\omega }$ from $L^2(𝔻,1/\omega \left(x\right)\mathrm{d}x)$ to ${𝒳}_N$, where $𝔻\subset {ℝ}^2$ is the unit disk, $\omega \left(x\right)=\sqrt{1-{\left|x\right|}^2}$ and the spaces ${\left({𝒳}_N\right)}_{N\in \mathbb{N}}$ consist of piecewise linear functions on a family of shape-regular and quasi-uniform triangulations of $𝔻$. We show that ${\pi }_{N,\omega }$ is stable in a weighted Sobolev norm, and prove an upper bound on the stability constant that does not depend on $N$. The result also holds when the disk $𝔻$ is replaced by a more general surface $\Gamma \subset {ℝ}^3$, replacing the weight $\omega$ by ${\omega }_{\Gamma }\left(x\right):=\sqrt{\mathrm{d}(x,\partial \Gamma )}$, the square root of the distance from $x$ to the manifold boundary of $\Gamma$.

On démontre la stabilité dans une norme de Sobolev à poids, de la projection orthogonale par rapport au produit scalaire d’un espace $L^2$ à poids, sur une famille d’éléments finis linéaires par morceaux. Plus précisément, soit ${\pi }_{N,\omega }$, de $L^2(𝔻,1/\omega \left(x\right)\mathrm{d}x)$ dans ${𝒳}_N$, où $𝔻\subset {ℝ}^2$ est le disque unité, $\omega \left(x\right)=\sqrt{1-{\left|x\right|}^2}$ et les espaces ${\left({𝒳}_N\right)}_{N\in \mathbb{N}}$ sont des espaces de fonctions continues et linéaires par morceaux sur une famille de triangulations régulière de $𝔻$. On montre que ${\pi }_{N,\omega }$ est stable dans une norme de Sobolev à poids, avec une borne supérieure sur la constante de stabilité qui ne dépend pas de $N$. Le résultat s’étend au cas de surfaces plus générales $\Gamma \subset {ℝ}^3$, en remplaçant le poids $\omega$ par ${\omega }_{\Gamma }\left(x\right):=\sqrt{\mathrm{d}(x,\partial \Gamma )}$, la racine carrée de la distance de $x$ à $\partial \Gamma$, le bord de $\Gamma$.

Classification : 46E35, 65N12, 65N38

[1] Alouges, François; Averseng, Martin Quasi-local preconditioners for the weakly singular and hypersingular operators on a disk (2022) (under review)

[2] Bank, Randolph E.; Dupont, Todd An optimal order process for solving finite element equations, Math. Comput., Volume 36 (1981) no. 153, pp. 35-51 | MR | DOI | Zbl

[3] Bramble, James H.; Pasciak, Joseph E.; Schatz, Alfred H. The construction of preconditioners for elliptic problems by substructuring. I, Math. Comput., Volume 47 (1986) no. 175, pp. 103-134 | Zbl | MR | DOI

[4] Bramble, James H.; Pasciak, Joseph E.; Steinbach, Olaf On the stability of the $L^2$ projection in $H^1\left(\omega \right)$, Math. Comput., Volume 71 (2002) no. 237, pp. 147-159 | Zbl | MR

[5] Bramble, James H.; Xu, Jinchao Some estimates for a weighted $L^2$ projection., Math. Comput., Volume 56 (1991) no. 194, pp. 463-476 | Zbl | MR

[6] Clément, Philippe Approximation by finite element functions using local regularization, Rev. Franc. Automat. Inform. Rech. Operat., Volume 9 (1975) no. R-2, pp. 77-84 | Zbl | Numdam | MR

[7] Costabel, Martin; Dauge, Monique; Duduchava, Roland Asymptotics without logarithmic terms for crack problems, Commun. Partial Differ. Equations, Volume 28 (2003) no. 5-6, pp. 869-926 | Zbl | MR | DOI

[8] Crouzeix, Michel; Thomée, Vidar The stability in $L_p$ and $W_p^1$ of the $L_2$-projection onto finite element function spaces, Math. Comput., Volume 48 (1987) no. 178, pp. 521-532

[9] Hurri, Ritva The weighted poincaré inequalities, Math. Scand., Volume 67 (1990) no. 1, pp. 145-160 | Zbl | MR | DOI

[10] McLean, William Strongly elliptic systems and boundary integral equations, Cambridge University Press, 2000, xiv+357 pages

[11] Pechstein, Clemens; Scheichl, Robert Weighted poincaré inequalities, IMA J. Numer. Anal., Volume 33 (2013) no. 2, pp. 652-686 | Zbl | DOI

[12] Steinbach, Olaf Stability estimates for hybrid coupled domain decomposition methods, Lecture Notes in Mathematics, 1809, Springer, 2003, vi+120 pages | DOI