Robust Arbitrary Order Mixed Finite Element Methods for the Incompressible Stokes Equations with pressure independent velocity errors
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 1, pp. 289-309.

Standard mixed finite element methods for the incompressible Navier–Stokes equations that relax the divergence constraint are not robust against large irrotational forces in the momentum balance and the velocity error depends on the continuous pressure. This robustness issue can be completely cured by using divergence-free mixed finite elements which deliver pressure-independent velocity error estimates. However, the construction of H 1 -conforming, divergence-free mixed finite element methods is rather difficult. Instead, we present a novel approach for the construction of arbitrary order mixed finite element methods which deliver pressure-independent velocity errors. The approach does not change the trial functions but replaces discretely divergence-free test functions in some operators of the weak formulation by divergence-free ones. This modification is applied to inf-sup stable conforming and nonconforming mixed finite element methods of arbitrary order in two and three dimensions. Optimal estimates for the incompressible Stokes equations are proved for the H 1 and L 2 errors of the velocity and the L 2 error of the pressure. Moreover, both velocity errors are pressure-independent, demonstrating the improved robustness. Several numerical examples illustrate the results.

Reçu le :
DOI : 10.1051/m2an/2015044
Classification : 65N15, 65N30, 76D07
Mots clés : Mixed finite element methods, incompressible Stokes problem, divergence-free methods, conforming and nonconforming FEM, mass conservation
Linke, Alexander 1 ; Matthies, Gunar 2 ; Tobiska, Lutz 3

1 Weierstraß-Institut, Mohrenstraße 39, 10117 Berlin, Germany.
2 Institut für Numerische Mathematik, Technische Universität Dresden, 01062 Dresden, Germany.
3 Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany.
@article{M2AN_2016__50_1_289_0,
     author = {Linke, Alexander and Matthies, Gunar and Tobiska, Lutz},
     title = {Robust {Arbitrary} {Order} {Mixed} {Finite} {Element} {Methods} for the {Incompressible} {Stokes} {Equations} with pressure independent velocity errors},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {289--309},
     publisher = {EDP-Sciences},
     volume = {50},
     number = {1},
     year = {2016},
     doi = {10.1051/m2an/2015044},
     zbl = {1381.76186},
     mrnumber = {3460110},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2015044/}
}
TY  - JOUR
AU  - Linke, Alexander
AU  - Matthies, Gunar
AU  - Tobiska, Lutz
TI  - Robust Arbitrary Order Mixed Finite Element Methods for the Incompressible Stokes Equations with pressure independent velocity errors
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2016
SP  - 289
EP  - 309
VL  - 50
IS  - 1
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2015044/
DO  - 10.1051/m2an/2015044
LA  - en
ID  - M2AN_2016__50_1_289_0
ER  - 
%0 Journal Article
%A Linke, Alexander
%A Matthies, Gunar
%A Tobiska, Lutz
%T Robust Arbitrary Order Mixed Finite Element Methods for the Incompressible Stokes Equations with pressure independent velocity errors
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2016
%P 289-309
%V 50
%N 1
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2015044/
%R 10.1051/m2an/2015044
%G en
%F M2AN_2016__50_1_289_0
Linke, Alexander; Matthies, Gunar; Tobiska, Lutz. Robust Arbitrary Order Mixed Finite Element Methods for the Incompressible Stokes Equations with pressure independent velocity errors. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 1, pp. 289-309. doi : 10.1051/m2an/2015044. http://archive.numdam.org/articles/10.1051/m2an/2015044/

T. Apel and G. Matthies, Nonconforming, anisotropic, rectangular finite elements of arbitrary order for the Stokes problem. SIAM J. Numer. Anal. 46 (2008) 1867–1891. | DOI | MR | Zbl

D.N. Arnold and G. Awanou, Finite element differential forms on cubical meshes. Math. Comput. 83 (2014) 1551–1570. | DOI | MR | Zbl

D. Boffi, F. Brezzi and M. Fortin, Mixed finite element methods and applications. Vol. 44 of Springer Ser. Comput. Math. Springer, Heidelberg (2013). | 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

J.H. Bramble and S.R. Hilbert, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation. SIAM J. Numer. Anal. 7 (1970) 112–124. | DOI | MR | Zbl

C. Brennecke, A. Linke, C. Merdon and J. Schöberl, Optimal and pressure-independent L 2 velocity error estimates for a modified Crouzeix–Raviart Stokes element with BDM reconstructions. J. Comput. Math. 33 (2015) 191–208. | DOI | MR | Zbl

S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Vol. 15 of Texts Appl. Math. 3rd edition. Springer, New York (2008). | MR | Zbl

F. Brezzi, J. Douglas, Jr. and LD. Marini, Recent results on mixed finite element methods for second order elliptic problems. In Vistas in Applied Mathematics, Transl. Ser. Math. Engrg. Optimization Software, New York (1986). | MR | Zbl

F. Brezzi, J. Douglas, Jr., R. Durán and M. Fortin, Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51 (1987) 237–250. | DOI | MR | Zbl

A. Buffa, C. De Falco and G. Sangalli, IsoGeometric Analysis: stable elements for the 2D Stokes equation. Int. J. Numer. Methods Fluids 65 (2011) 1407–1422. | DOI | MR | Zbl

B. Cockburn, G. Kanschat and D. Schötzau, A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations. J. Sci. Comput. 31 (2007) 61–73. | DOI | MR | Zbl

M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973) 33–75. | Numdam | MR | Zbl

O. Dorok, W. Grambow and L. Tobiska, Aspects of finite element discretizations for solving the Boussinesq approximation of the Navier–Stokes Equations. Notes on Numerical Fluid Mechanics: Numerical Methods for the Navier–Stokes Equations 47 (1994) 50–61. | Zbl

C. Druzgalski, M. Andersen and A. Mani, Direct numerical simulation of electroconvective instability and hydrodynamic chaos near an ion-selective surface. Phys. Fluids 25 (2013). | DOI

J.A. Evans and T.J.R. Hughes, Isogeometric divergence-conforming B-splines for the steady Navier-Stokes equations. Math. Models Methods Appl. Sci. 23 (2013) 1421–1478. | DOI | MR | Zbl

R.S. Falk and M. Neilan, Stokes complexes and the construction of stable finite elements with pointwise mass conservation. SIAM J. Numer. Anal. 51 (2013) 1308–1326. | DOI | MR | Zbl

M. Fortin, An analysis of the convergence of mixed finite element methods. RAIRO Anal. Numér. 11 (1977) 341–354. | DOI | Numdam | MR | Zbl

L.P. Franca and T.J.R. Hughes, Two classes of mixed finite element methods. Comput. Methods Appl. Mech. Engrg. 69 (1988) 89–129. | DOI | MR | Zbl

K.J. Galvin, A. Linke, L.G. Rebholz and N.E. Wilson, Stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection. Comput. Methods Appl. Mech. Engrg. 237–240 (2012) 166–176. | DOI | MR | Zbl

S. Ganesan, G. Matthies and L. Tobiska, On spurious velocities in incompressible flow problems with interfaces. Comput. Methods Appl. Mech. Engrg. 196 (2007) 1193–1202. | DOI | MR | Zbl

J.-F. Gerbeau, C. Le Bris and M. Bercovier, Spurious velocities in the steady flow of an incompressible fluid subjected to external forces. Internat. J. Numer. Methods Fluids 25 (1997) 679–695. | DOI | MR | Zbl

V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations. Theory and algorithms. Vol. 5 of Springer Ser. Comput. Math. Springer-Verlag, Berlin (1986). | MR | Zbl

J. Guzmán and M. Neilan, Conforming and divergence-free Stokes elements in three dimensions. IMA J. Numer. Anal. 34 (2014) 1489–1508,. | DOI | MR | Zbl

J. Guzmán and M. Neilan, Conforming and divergence-free Stokes elements on general triangular meshes. Math. Comput. 83 (2014) 15–36. | DOI | MR | Zbl

J.P. Hennart, J. Jaffré and J.E. Roberts, A constructive method for deriving finite elements of nodal type. Numer. Math. 53 (1988) 701–738. | DOI | MR | Zbl

E.W. Jenkins, V. John, A. Linke and L.G. Rebholz, On the parameter choice in grad-div stabilization for the Stokes equations. Adv. Comput. Math. 40 (2014) 491–516. | DOI | MR | Zbl

G. Kanschat and N. Sharma, Divergence-conforming discontinuous Galerkin methods and C 0 interior penalty methods. SIAM J. Numer. Anal. 52 (2014) 1822–1842. | DOI | MR | Zbl

M. Koddenbrock, Effizienz und Genauigkeit einer divergenzfreien Diskretisierung für die stationären inkompressiblen Navier–Stokes-Gleichungen. Master’s thesis, Freie Universität Berlin, Germany (2014).

C. Lehrenfeld, Hybrid discontinuous Galerkin methods for incompressible flow problems. Master’s thesis, RWTH Aachen, Germany (2010).

A. Linke, Collision in a cross-shaped domain – a steady 2d Navier–Stokes example demonstrating the importance of mass conservation in CFD. Comput. Methods Appl. Mech. Engrg. 198 (2009) 3278–3286. | DOI | MR | Zbl

A. Linke, On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime. Comput. Methods Appl. Mech. Engrg. 268 (2014) 782–800. | DOI | MR | Zbl

G. Matthies, Inf-sup stable nonconforming finite elements of higher order on quadrilaterals and hexahedra. ESAIM: M2AN 41 (2007) 855–874. | DOI | Numdam | MR | Zbl

G. Matthies and L. Tobiska, The inf-sup condition for the mapped Q k -P k-1 disc element in arbitrary space dimensions. Computing 69 (2002) 119–139. | DOI | MR | Zbl

G. Matthies and L. Tobiska, Inf-sup stable non-conforming finite elements of arbitrary order on triangles. Numer. Math. 102 (2005) 293–309. | DOI | MR | Zbl

G. Matthies and L. Tobiska, Mass conservation of finite element methods for coupled flow-transport problems. Int. J. Comput. Sci. Math. 1 (2007) 293–307. | DOI | MR | Zbl

J.-C. Nédélec, Mixed finite elements in R 3 . Numer. Math. 35 (1980) 315–341. | DOI | MR | Zbl

M.A. Olshanskii, G. Lube, T. Heister and J. Löwe, Grad-div stabilization and subgrid pressure models for the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 198 (2009) 3975–3988. | DOI | MR | Zbl

M.A. Olshanskii and A. Reusken, Grad-div stabilization for Stokes equations. Math. Comput. 73 (2004) 1699–1718. | DOI | MR | Zbl

P.-A. Raviart and J.M. Thomas, A mixed finite element method for 2nd order elliptic problems. In Mathematical aspects of finite element methods. Proc. Conf., Consiglio Naz. delle Ricerche, C.N.R., Rome, 1975. Vol. 606 of Lect. Notes Math. Springer, Berlin (1977) 292–315. | MR | Zbl

H.-G. Roos, M. Stynes and L. Tobiska, Robust numerical methods for singularly perturbed differential equations. Convection-diffusion-reaction and flow problems. Vol. 24 of Springer Ser. Comput. Math. 2nd edition. Springer-Verlag, Berlin (2008). | MR | Zbl

L.R. Scott and M. Vogelius, Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. RAIRO Modél. Math. Anal. Numér. 19 (1985) 111–143. | DOI | Numdam | MR | Zbl

J. Wang, Y. Wang and X. Ye, A posteriori error estimation for an interior penalty type method employing H(div) elements for the Stokes equations. SIAM J. Sci. Comput. 33 (2011) 131–152. | DOI | MR | Zbl

S. Zhang, A new family of stable mixed finite elements for the 3D Stokes equations. Math. Comput. 74 (2005) 543–554. | DOI | MR | Zbl

Cité par Sources :