We prove the stability of a weighted projection operator onto piecewise linear finite elements spaces in a weighted Sobolev norm. Namely, we consider the orthogonal projections from to , where is the unit disk, and the spaces consist of piecewise linear functions on a family of shape-regular and quasi-uniform triangulations of . We show that is stable in a weighted Sobolev norm, and prove an upper bound on the stability constant that does not depend on . The result also holds when the disk is replaced by a more general surface , replacing the weight by , the square root of the distance from to the manifold boundary of .
On démontre la stabilité dans une norme de Sobolev à poids, de la projection orthogonale par rapport au produit scalaire d’un espace à poids, sur une famille d’éléments finis linéaires par morceaux. Plus précisément, soit , de dans , où est le disque unité, et les espaces sont des espaces de fonctions continues et linéaires par morceaux sur une famille de triangulations régulière de . On montre que est stable dans une norme de Sobolev à poids, avec une borne supérieure sur la constante de stabilité qui ne dépend pas de . Le résultat s’étend au cas de surfaces plus générales , en remplaçant le poids par , la racine carrée de la distance de à , le bord de .
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Martin Averseng 1
@article{CRMATH_2023__361_G4_757_0, author = {Martin Averseng}, title = {Stability of a weighted {L2} projection in a weighted {Sobolev} norm}, journal = {Comptes Rendus. Math\'ematique}, pages = {757--766}, publisher = {Acad\'emie des sciences, Paris}, volume = {361}, year = {2023}, doi = {10.5802/crmath.426}, language = {en}, }
Martin Averseng. Stability of a weighted L2 projection in a weighted Sobolev norm. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 757-766. doi : 10.5802/crmath.426. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.426/
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