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 -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 and errors of the velocity and the error of the pressure. Moreover, both velocity errors are pressure-independent, demonstrating the improved robustness. Several numerical examples illustrate the results.
DOI: 10.1051/m2an/2015044
Mots-clés : Mixed finite element methods, incompressible Stokes problem, divergence-free methods, conforming and nonconforming FEM, mass conservation
@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://www.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://www.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://www.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 , Volume 50 (2016) no. 1, pp. 289-309. doi : 10.1051/m2an/2015044. http://www.numdam.org/articles/10.1051/m2an/2015044/
Nonconforming, anisotropic, rectangular finite elements of arbitrary order for the Stokes problem. SIAM J. Numer. Anal. 46 (2008) 1867–1891. | DOI | MR | Zbl
and ,Finite element differential forms on cubical meshes. Math. Comput. 83 (2014) 1551–1570. | DOI | MR | Zbl
and ,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
Stabilized finite element methods for the generalized Oseen problem. Comput. Methods Appl. Mech. Engrg. 196 (2007) 853–866. | DOI | MR | Zbl
, , and ,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
and ,Optimal and pressure-independent velocity error estimates for a modified Crouzeix–Raviart Stokes element with BDM reconstructions. J. Comput. Math. 33 (2015) 191–208. | DOI | MR | Zbl
, , and ,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
Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51 (1987) 237–250. | DOI | MR | Zbl
, , Jr., and ,IsoGeometric Analysis: stable elements for the 2D Stokes equation. Int. J. Numer. Methods Fluids 65 (2011) 1407–1422. | DOI | MR | Zbl
, and ,A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations. J. Sci. Comput. 31 (2007) 61–73. | DOI | MR | Zbl
, and ,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
and ,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
, and ,Direct numerical simulation of electroconvective instability and hydrodynamic chaos near an ion-selective surface. Phys. Fluids 25 (2013). | DOI
, and ,Isogeometric divergence-conforming B-splines for the steady Navier-Stokes equations. Math. Models Methods Appl. Sci. 23 (2013) 1421–1478. | DOI | MR | Zbl
and ,Stokes complexes and the construction of stable finite elements with pointwise mass conservation. SIAM J. Numer. Anal. 51 (2013) 1308–1326. | DOI | MR | Zbl
and ,An analysis of the convergence of mixed finite element methods. RAIRO Anal. Numér. 11 (1977) 341–354. | DOI | Numdam | MR | Zbl
,Two classes of mixed finite element methods. Comput. Methods Appl. Mech. Engrg. 69 (1988) 89–129. | DOI | MR | Zbl
and ,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
, , and ,On spurious velocities in incompressible flow problems with interfaces. Comput. Methods Appl. Mech. Engrg. 196 (2007) 1193–1202. | DOI | MR | Zbl
, and ,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
, and ,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
Conforming and divergence-free Stokes elements in three dimensions. IMA J. Numer. Anal. 34 (2014) 1489–1508,. | DOI | MR | Zbl
and ,Conforming and divergence-free Stokes elements on general triangular meshes. Math. Comput. 83 (2014) 15–36. | DOI | MR | Zbl
and ,A constructive method for deriving finite elements of nodal type. Numer. Math. 53 (1988) 701–738. | DOI | MR | Zbl
, and ,On the parameter choice in grad-div stabilization for the Stokes equations. Adv. Comput. Math. 40 (2014) 491–516. | DOI | MR | Zbl
, , and ,Divergence-conforming discontinuous Galerkin methods and interior penalty methods. SIAM J. Numer. Anal. 52 (2014) 1822–1842. | DOI | MR | Zbl
and ,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).
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
,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
,Inf-sup stable nonconforming finite elements of higher order on quadrilaterals and hexahedra. ESAIM: M2AN 41 (2007) 855–874. | DOI | Numdam | MR | Zbl
,The inf-sup condition for the mapped - element in arbitrary space dimensions. Computing 69 (2002) 119–139. | DOI | MR | Zbl
and ,Inf-sup stable non-conforming finite elements of arbitrary order on triangles. Numer. Math. 102 (2005) 293–309. | DOI | MR | Zbl
and ,Mass conservation of finite element methods for coupled flow-transport problems. Int. J. Comput. Sci. Math. 1 (2007) 293–307. | DOI | MR | Zbl
and ,Mixed finite elements in . Numer. Math. 35 (1980) 315–341. | DOI | MR | Zbl
,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
, , and ,Grad-div stabilization for Stokes equations. Math. Comput. 73 (2004) 1699–1718. | DOI | MR | Zbl
and ,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
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
and ,A posteriori error estimation for an interior penalty type method employing elements for the Stokes equations. SIAM J. Sci. Comput. 33 (2011) 131–152. | DOI | MR | Zbl
, and ,A new family of stable mixed finite elements for the 3D Stokes equations. Math. Comput. 74 (2005) 543–554. | DOI | MR | Zbl
,Cited by Sources: