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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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/

Bibliographie
Cité par

C. Amrouche and V. Girault, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimensions. Czeschoslovak Math. J. 44 (1994) 109–140. | DOI | MR | Zbl

C. Bernardi, Y. Maday and F. Rapetti, Discretisations variationnelles de problèmes aux limites elliptiques. Springer-Verlag, Berlin (2004). | MR | Zbl

J. Blasco and R. Codina, Stabilized finite element method for the transient Navier-Stokes equations based on a pressure gradient projection. Comput. Methods Appl. Mech. Engrg. 182 (2000) 277–300. | DOI | MR | Zbl

M. Braack and E. Burman, Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method. SIAM J. Numer. Anal. 43 (2000) 2544–2566. | DOI | MR | Zbl

M. Braack, E. Burman, V. John and G. Lube, Stabilized finite element methods for the generalized Oseen problem. Comput. Methods Appl. Mech. Engrg. 196 (2007) 853–866. | DOI | MR | Zbl

S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods, 3rd edition. Springer-Verlag, Berlin (2008). | MR | Zbl

E. Burman, M.A. Fernández and P. Hansbo, Continuous interior penalty finite element method for Oseen equations. SIAM J. Numer. Anal. 44 (2006) 1248–1274. | DOI | MR | Zbl

E. Burman and M.A. Fernández, Continuous interior penalty finite element method for the time-dependent Navier−Stokes equations: space discretization and convergence. Numer. Math. 107 (2007) 39–77. | DOI | MR | Zbl

E. Burman, Interior penalty variational multiscale method for the incompressible Navier-Stokes equations: Monitoring artificial dissipation. Comput. Methods Appl. Mech. Engrg. 196 (2007) 4045–4058. | DOI | MR | Zbl

R. Codina, Stabilization of incompressibility and convection through orthogonal sub-scales in finite element methods. Comput. Methods Appl. Mech. Engrg. 190 (2000) 1579–1599. | DOI | MR | Zbl

T. Chacón Rebollo, An analysis technique for stabilized finite element solution of incompressible flows. ESAIM: M2AN 35 (2001) 57–89. | DOI | Numdam | MR | Zbl

T. Chacón Rebollo and F. Guillén González, An intrinsic analysis of existence of solutions for the hydrostatic approximation of Navier-Stokes equations. C. R. Acad. Sci. Paris, Série I 330 (2000) 841–846. | DOI | MR | Zbl

T. Chacón Rebollo, R. Lewandowski and E. Chacón Vera, Analysis of the hydrostatic approximation in oceanography with compression term. ESAIM: M2AN 34 (2000) 525–537. | DOI | Numdam | MR | Zbl

T. Chacón Rebollo, M. Gómez Mármol and I. Sánchez Muñoz, Numerical solution of the Primitive equations of the ocean by the Orthogonal Sub-Scales VMS method. Appl. Numer. Math. 62 (2012) 342–356. | DOI | MR | Zbl

T. Chacón Rebollo, M. Gómez Mármol, V. Girault, and I. Sánchez Muñoz, A high order term-by-term stabilization solver for incompressible flow problems. IMA J. Numer. Anal. 33-3 (2013) 974–1007. | DOI | MR | Zbl

Ph. Ciarlet, The Finite Element Method for Elliptic Problems. Siamm (2002). | MR | Zbl

S. Ganesan, G. Matthies and L. Tobiska, Local projection stabilization with equal order interpolation applied to the Stokes problem. Math. Comput. 77 (2008) 2039–2060. | DOI | MR | Zbl

V. Girault and J.L. Lions, Two-grid finite-element schemes for the transient Navier-Stokes equations. ESAIM: M2AN 35 (2001) 945–980. | DOI | Numdam | MR | Zbl

P. Knobloch, A generalization of the local projection stabilization for convection-diffusion-reaction equations. SIAM J. Numer. Anal. 48 (2010) 659–680. | DOI | MR | Zbl

J.L. Lions, R. Temman and S. Wang, New formulation of the primitive equations of the atmosphere and applications. Nonlinearity 5 (1992) 237–288. | DOI | MR | Zbl

G. Matthies, P. Skrypacz and L. Tobiska, A unified convergence analysis for local projection stabilisations applied to the Oseen problem. ESAIM: M2AN 41 (2007) 713–742. | DOI | Numdam | MR | Zbl

P. Oswald, On a BPX preconditioner for $P_{1}$ elements. Computing 51 (1993) 125–133. | DOI | MR | Zbl

H.G. Roos, M. Stynes and L. Tobiska, Robust numerical methods for singularly perturbed differential equations. 2nd edition. Springer Series Comput. Math. 24 (2008). | Zbl

R. Verfürth, Error estimates for a mixed finite element approximation of the Stokes equations. RAIRO Anal. Numer. 18 (1984) 175–182. | DOI | Numdam | MR | Zbl

L.B. Wahlbin, Local behavior in finite element methods. Elsevier Science, North Holland (1991). | MR | Zbl

Cité par Sources :