A reduced discrete inf-sup condition in L p for incompressible flows and application
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 4, pp. 1219-1238.

In this work, we introduce a discrete specific inf-sup condition to estimate the L p norm, 1<p<+, of the pressure in a number of fluid flows. It applies to projection-based stabilized finite element discretizations of incompressible flows, typically when the velocity field has a low regularity. We derive two versions of this inf-sup condition: The first one holds on shape-regular meshes and the second one on quasi-uniform meshes. As an application, we derive reduced inf-sup conditions for the linearized Primitive equations of the Ocean that apply to the surface pressure in weighted L p norm. This allows to prove the stability and convergence of quite general stabilized discretizations of these equations: SUPG, Least Squares, Adjoint-stabilized and OSS discretizations.

DOI : 10.1051/m2an/2015008
Classification : 35Q35, 65N12, 76D05
Mots clés : Inf-sup condition, Finite element method, Stabilized method, Incompressible flows, Primitive equations of the Ocean
Rebollo, Tomás Chacón 1 ; Girault, Vivette 2 ; Mármol, Macarena Gómez 3 ; Muñoz, Isabel Sánchez 4

1 Departamento de Ecuaciones Diferenciales y Análisis Numérico and Instituto de Matemáticas de la Universidad de Sevilla (IMUS), Apdo. de correos 1160, Universidad de Sevilla, 41080 Sevilla, Spain
2 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie & C.N.R.S, UMR 7598, Paris 6, 4 Place Jussieu, 75252 Paris cedex 05, France
3 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Apdo. de correos 1160, Universidad de Sevilla, 41080 Sevilla, Spain
4 Departamento de Matemática Aplicada I, Carretera de Utrera Km 1, Universidad de Sevilla, 41013 Sevilla, Spain
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     title = {A reduced discrete inf-sup condition in $L^{p}$ for incompressible flows and application},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1219--1238},
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Rebollo, Tomás Chacón; Girault, Vivette; Mármol, Macarena Gómez; Muñoz, Isabel Sánchez. A reduced discrete inf-sup condition in $L^{p}$ for incompressible flows and application. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 4, pp. 1219-1238. doi : 10.1051/m2an/2015008. http://archive.numdam.org/articles/10.1051/m2an/2015008/

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