A reduced discrete inf-sup condition in $L^{p}$ for incompressible flows and application

Rebollo, Tomás Chacón; Girault, Vivette; Mármol, Macarena Gómez; Muñoz, Isabel Sánchez

doi:10.1051/m2an/2015008

A reduced discrete inf-sup condition in

L^{p}

for incompressible flows and application

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

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 4, pp. 1219-1238.

Résumé

In this work, we introduce a discrete specific inf-sup condition to estimate the $L^{p}$ norm, $1 < p < + \infty$ , 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.

MR Zbl

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

Affiliations des auteurs :

Rebollo, Tomás Chacón ¹ ; Girault, Vivette ² ; Mármol, Macarena Gómez ³ ; Muñoz, Isabel Sánchez ⁴

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     author = {Rebollo, Tom\'as Chac\'on and Girault, Vivette and M\'armol, Macarena G\'omez and Mu\~noz, Isabel S\'anchez},
     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},
     publisher = {EDP-Sciences},
     volume = {49},
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     year = {2015},
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     url = {http://www.numdam.org/articles/10.1051/m2an/2015008/}
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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://www.numdam.org/articles/10.1051/m2an/2015008/

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