Stabilised finite element methods for the Oseen problem on anisotropic quadrilateral meshes

Barrenechea, Gabriel R.; Wachtel, Andreas

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 1, pp. 99-122.

In this work we present and analyse new inf-sup stable, and stabilised, finite element methods for the Oseen equation in anisotropic quadrilateral meshes. The meshes are formed of closed parallelograms, and the analysis is restricted to two space dimensions. Starting with the lowest order ${\mathbb{Q}}_1^2\times {\mathbb{P}}_0$ pair, we first identify the pressure components that make this finite element pair to be non-inf-sup stable, especially with respect to the aspect ratio. We then propose a way to penalise them, both strongly, by directly removing them from the space, and weakly, by adding a stabilisation term based on jumps of the pressure across selected edges. Concerning the velocity stabilisation, we propose an enhanced grad-div term. Stability and optimal a priori error estimates are given, and the results are confirmed numerically.

Classification : 65N30, 65N12, 65N50
Mots clés : Oseen equation, stabilised finite element method, anisotropic quadrilateral mesh

[1] R. Araya, G.R. Barrenechea and F. Valentin, Stabilized finite element methods based on multiscale enrichment for the Stokes problem. SIAM J. Numer. Anal. 44 (2006) 322–348 | DOI | MR | Zbl

[2] M. Ainsworth, G.R. Barrenechea and A. Wachtel, Stabilization of high aspect ratio mixed finite elements for incompressible flow. SIAM J. Numer. Anal. 53 (2015) 1107–1120 | DOI | MR | Zbl

[3] M. Ainsworth and P. Coggins, The stability of mixed hp-finite element methods for Stokes flow on high aspect ratio elements. SIAM J. Numer. Anal. 38 (2000) 1721–1761 | DOI | MR | Zbl

[4] T. Apel, T. Knopp and G. Lube, Stabilized finite element methods with anisotropic mesh refinement for the Oseen problem. Appl. Numer. Math. 58 (2008) 1830–1843 | DOI | MR | Zbl

[5] R. Becker and M. Braack, A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38 (2001) 173–199 | DOI | MR | Zbl

[6] C. Baiocchi, F. Brezzi and L.P. Franca, Virtual bubbles and Galerkin-least-squares methods (Ga.L.S.). Comput. Methods Appl. Mech. Engrg. 105 (1993) 125–141 | DOI | MR | Zbl

[7] D. Boffi, F. Brezzi and M. Fortin, Mixed finite element methods and applications. Springer Series Comput. Math., vol. 44. Springer, Heidelberg (2013) | MR | Zbl

[8] 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

[9] P.B. Bochev, C.R. Dohrmann and M.D. Gunzburger, Stabilization of low-order mixed finite elements for the Stokes equations. SIAM J. Numer. Anal. 44 (2006) 82–101 | DOI | MR | Zbl

[10] F. Brezzi and M. Fortin, A minimal stabilisation procedure for mixed finite element methods. Numer. Math. 89 (2001) 457–491 | DOI | MR | Zbl

[11] A.N. Brooks and T.J.R. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 32 (1982) 199–259 | DOI | MR | Zbl

[12] J. Blasco, An anisotropic GLS-stabilized finite element method for incompressible flow problems. Comput. Methods Appl. Mech. Engrg. 197 (2008) 3712–3723 | DOI | MR | Zbl

[13] M. Braack, G. Lube and L. Röhe, Divergence preserving interpolation on anisotropic quadrilateral meshes. Comput. Methods Appl. Math. 12 (2012) 123–138 | DOI | MR | Zbl

[14] M. Braack, A stabilized finite element scheme for the Navier-Stokes equations on quadrilateral anisotropic meshes. ESAIM: M2AN 42 (2008) 903–924 | DOI | Numdam | MR | Zbl

[15] E. Burman, Pressure projection stabilizations for Galerkin approximations of Stokes’ and Darcy’s problem. Numer. Methods Partial Differ. Equ. 24 (2008) 127–143 | DOI | MR | Zbl

[16] G.R. Barrenechea and F. Valentin, A residual local projection for the Oseen equation. Comput. Methods Appl. Mech. Engrg. 199 (2010) 1906–1921 | DOI | MR | Zbl

[17] G.R. Barrenechea and A. Wachtel, A note on the stabilised Q1-P0 method on quadrilaterals with high aspect ratios, Boundary and Interior Layers, Proceedings of BAIL 2014, edited by P. Knobloch. Lect. Notes Comput. Sci. Eng. 108 (2015) 1–11 | MR

[18] R. Codina, Analysis of a stabilized finite element approximation of the Oseen equations using orthogonal subscales. Appl. Numer. Math. 58 (2008) 264–283 | DOI | MR | Zbl

[19] Z.-H. Cao and L.-H. Feng, A note on variational Representation for singular values of matrix. Appl. Math. Comput. 143 (2003) 559–563 | MR | Zbl

[20] L.P. Franca and S. Frey, Stabilized finite element methods. II. The incompressible Navier -Stokes equation.Comput. Methods Appl. Mech. Engrg. 99 (1992) 209–233 | DOI | MR | Zbl

[21] 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

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

[23] V. Girault and P.-A. Raviart, Finite element methods for Navier Stokes equations. Springer, Berlin (1986) | DOI | MR | Zbl

[24] T.J.R. Hughes, L.P. Franca and M. Balestra, A new finite element formulation for computational fluid dynamics. V. Circumventing the Babuška-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comput. Methods Appl. Mech. Engrg. 59 (1986) 85–99 | DOI | MR | Zbl

[25] T.J.R. Hughes, L.P. Franca and G. Hulbert, A new finite element formulation for computational fluid dynamics. VIII. The Galerkin/least-squares method for advective-diffusive equations. Comput. Methods Appl. Mech. Engrg. 73 (1989) 173–189 | DOI | MR | Zbl

[26] V. John, Finite element methods for incompressible flow problems. Springer Series in Computational Mathematics, vol. 51. Springer Verlag (2017) | MR

[27] 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

[28] 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

[29] N. Kechkar and D.J. Silvester, Analysis of locally stabilized mixed finite element methods for the Stokes problem. Math. Comput. 58 (1992) 1–10 | DOI | MR | Zbl

[30] W. Layton, C. Manica, M. Neda, M.A. Olshanskii and L. Rebholtz, On the accuracy of the rotation form in simulations of the Navier -Stokes equations. J. Comput. Phys. 228 (2009) 3433–3447 | DOI | MR | Zbl

[31] Q. Liao and D. Silvester, Robust stabilized Stokes approximation methods for highly stretched grids. IMA J. Numer. Anal. 33 (2013) 413–431 | DOI | MR | Zbl

[32] S. Micheletti, S. Perotto and M. Picasso, Stabilized finite elements on anisotropic meshes: a priori error estimates for the advection-diffusion and the Stokes problems. SIAM J. Numer. Anal. 41 (2003) 1131–1162 | DOI | MR | Zbl

[33] G. Matthies, P. Skrzypacz 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

[34] G. Matthies and L. Tobiska, Local projection type stabilization applied to inf-sup stable discretizations of the Oseen problem. IMA J. Numer. Anal. 35 (2015) 239–269 | DOI | MR | Zbl

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

[36] J. Pitkäranta and T. Saarinen, A multigrid version of a simple finite element method for the Stokes problem. Math. Comput. 45 (1985) 1–14 | DOI | MR | Zbl

[37] H.-G. Roos, M. Stynes and L. Tobiska, Robust numerical methods for singularly perturbed differential equations, 2nded. Springer Series in Computational Mathematics, vol. 24. Springer Verlag, Berlin (2008) | MR | Zbl