Local projection stabilization of equal order interpolation applied to the Stokes problem

Ganesan, Sashikumaar; Matthies, Gunar; Tobiska, Lutz

doi:10.1090/S0025-5718-08-02130-3

Local projection stabilization of equal order interpolation applied to the Stokes problem
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by Sashikumaar Ganesan, Gunar Matthies and Lutz Tobiska PDF

Math. Comp. 77 (2008), 2039-2060 Request permission

Abstract:

The local projection stabilization allows us to circumvent the Babuška–Brezzi condition and to use equal order interpolation for discretizing the Stokes problem. The projection is usually done in a two-level approach by projecting the pressure gradient onto a discontinuous finite element space living on a patch of elements. We propose a new local projection stabilization method based on (possibly) enriched finite element spaces and discontinuous projection spaces defined on the same mesh. Optimal order of convergence is shown for pairs of approximation and projection spaces satisfying a certain inf-sup condition. Examples are enriched simplicial finite elements and standard quadrilateral/hexahedral elements. The new approach overcomes the problem of an increasing discretization stencil and, thus, is simple to implement in existing computer codes. Numerical tests confirm the theoretical convergence results which are robust with respect to the user-chosen stabilization parameter.

References

Thomas Apel, Anisotropic finite elements: local estimates and applications, Advances in Numerical Mathematics, B. G. Teubner, Stuttgart, 1999. MR 1716824
D. N. Arnold, F. Brezzi, and M. Fortin, A stable finite element for the Stokes equations, Calcolo 21 (1984), no. 4, 337–344 (1985). MR 799997, DOI 10.1007/BF02576171
Wen Bai, The quadrilateral “Mini” finite element for the Stokes problem, Comput. Methods Appl. Mech. Engrg. 143 (1997), no. 1-2, 41–47. MR 1442388, DOI 10.1016/S0045-7825(96)01146-2
R. Becker and M. Braack, A finite element pressure gradient stabilization for the Stokes equations based on local projections, Calcolo 38 (2001), no. 4, 173–199. MR 1890352, DOI 10.1007/s10092-001-8180-4
Roland Becker and Malte Braack, A two-level stabilization scheme for the Navier-Stokes equations, Numerical mathematics and advanced applications, Springer, Berlin, 2004, pp. 123–130. MR 2121360
Pavel Bochev and Max Gunzburger, An absolutely stable pressure-Poisson stabilized finite element method for the Stokes equations, SIAM J. Numer. Anal. 42 (2004), no. 3, 1189–1207. MR 2113682, DOI 10.1137/S0036142903416547
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 (2006), no. 6, 2544–2566. MR 2206447, DOI 10.1137/050631227
D. Braess and R. Sarazin, An efficient smoother for the Stokes problem, Appl. Numer. Math. 23 (1997), no. 1, 3–19. Multilevel methods (Oberwolfach, 1995). MR 1438078, DOI 10.1016/S0168-9274(96)00059-1
Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
F. Brezzi and J. Pitkäranta, On the stabilization of finite element approximations of the Stokes equations, Efficient solutions of elliptic systems (Kiel, 1984) Notes Numer. Fluid Mech., vol. 10, Friedr. Vieweg, Braunschweig, 1984, pp. 11–19. MR 804083
Philippe G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics, vol. 40, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]. MR 1930132, DOI 10.1137/1.9780898719208
Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. 9 (1975), no. R-2, 77–84 (English, with Loose French summary). MR 0400739
Ramon Codina and Jordi Blasco, A finite element formulation for the Stokes problem allowing equal velocity-pressure interpolation, Comput. Methods Appl. Mech. Engrg. 143 (1997), no. 3-4, 373–391. MR 1445157, DOI 10.1016/S0045-7825(96)01154-1
Jim Douglas Jr. and Jun Ping Wang, An absolutely stabilized finite element method for the Stokes problem, Math. Comp. 52 (1989), no. 186, 495–508. MR 958871, DOI 10.1090/S0025-5718-1989-0958871-X
Leopoldo P. Franca and Sérgio L. Frey, Stabilized finite element methods. II. The incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 99 (1992), no. 2-3, 209–233. MR 1186727, DOI 10.1016/0045-7825(92)90041-H
Leopoldo P. Franca, Saulo P. Oliveira, and Marcus Sarkis, Continuous Q1-Q1 Stokes elements stabilized with non-conforming null edge average velocity functions, Math. Models Methods Appl. Sci. 17 (2007), no. 3, 439–459. MR 2311926, DOI 10.1142/S021820250700198X
Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
Peter Hansbo and Anders Szepessy, A velocity-pressure streamline diffusion finite element method for the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 84 (1990), no. 2, 175–192. MR 1087615, DOI 10.1016/0045-7825(90)90116-4
Thomas J. R. Hughes and Michel Mallet, A new finite element formulation for computational fluid dynamics. III. The generalized streamline operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg. 58 (1986), no. 3, 305–328. MR 865671, DOI 10.1016/0045-7825(86)90152-0
Volker John and Gunar Matthies, MooNMD—a program package based on mapped finite element methods, Comput. Vis. Sci. 6 (2004), no. 2-3, 163–169. MR 2061275, DOI 10.1007/s00791-003-0120-1
Petr Knobloch and Lutz Tobiska, Stabilization methods of bubble type for the $Q_1/Q_1$-element applied to the incompressible Navier-Stokes equations, M2AN Math. Model. Numer. Anal. 34 (2000), no. 1, 85–107. MR 1735975, DOI 10.1051/m2an:2000132
Gunar Matthies, Mapped finite elements on hexahedra. Necessary and sufficient conditions for optimal interpolation errors, Numer. Algorithms 27 (2001), no. 4, 317–327. MR 1880917, DOI 10.1023/A:1013860707381
G. Matthies, P. Skrzypacz, and L. Tobiska, A unified convergence analysis for local projection stabilisations applied to the Oseen problem, M2AN Math. Model. Numer. Anal. 41 (2007), no. 4, 713–742.
G. Matthies and L. Tobiska, The inf-sup condition for the mapped $Q_k$-$P^\textrm {disc}_{k-1}$ element in arbitrary space dimensions, Computing 69 (2002), no. 2, 119–139. MR 1954791, DOI 10.1007/s00607-002-1451-3
P. Mons and G. Rogé, L’élément $Q_1$-bulle/$Q_1$, RAIRO Modél. Math. Anal. Numér. 26 (1992), no. 4, 507–521 (French, with English summary). MR 1163979, DOI 10.1051/m2an/1992260405071
Maxim A. Olshanskii and Arnold Reusken, Grad-div stabilization for Stokes equations, Math. Comp. 73 (2004), no. 248, 1699–1718. MR 2059732, DOI 10.1090/S0025-5718-03-01629-6
F. Schieweck, Uniformly stable mixed hp-finite elements on multilevel adaptive grids with hanging nodes, Otto-von-Guericke-Universität Magdeburg, Preprint Nr. 2, 2007.
L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. MR 1011446, DOI 10.1090/S0025-5718-1990-1011446-7
Lutz Tobiska and Gert Lube, A modified streamline diffusion method for solving the stationary Navier-Stokes equation, Numer. Math. 59 (1991), no. 1, 13–29. MR 1103751, DOI 10.1007/BF01385768
Lutz Tobiska and Rüdiger Verfürth, Analysis of a streamline diffusion finite element method for the Stokes and Navier-Stokes equations, SIAM J. Numer. Anal. 33 (1996), no. 1, 107–127. MR 1377246, DOI 10.1137/0733007