Analysis of an adaptive Uzawa finite element method for the nonlinear Stokes problem

Kreuzer, Christian

doi:10.1090/S0025-5718-2011-02524-X

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ISSN 1088-6842(online) ISSN 0025-5718(print)

Analysis of an adaptive Uzawa finite element method for the nonlinear Stokes problem

Author: Christian Kreuzer
Journal: Math. Comp. 81 (2012), 21-55
MSC (2010): Primary 65N30, 65N12, 35J60
Posted: May 11, 2011
MathSciNet review: 2833486
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Abstract: We design and study an adaptive algorithm for the numerical solution of the stationary nonlinear Stokes problem. The algorithm can be interpreted as a disturbed steepest descent method, which generalizes Uzawa's method to the nonlinear case. The outer iteration for the pressure is a descent method with fixed step-size. The inner iteration for the velocity consists of an approximate solution of a nonlinear Laplace equation, which is realized with adaptive linear finite elements. The descent direction is motivated by the quasi-norm which naturally arises as distance between velocities. We establish the convergence of the algorithm within the framework of descent direction methods.

[AG94] Chérif Amrouche and Vivette Girault, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J. 44(119) (1994), no. 1, 109–140. MR 1257940 (95c:35190)
[AO00] Mark Ainsworth and J. Tinsley Oden, A posteriori error estimation in finite element analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2000. MR 1885308 (2003b:65001)
[Bän91] Eberhard Bänsch, Local mesh refinement in 2 and 3 dimensions, Impact Comput. Sci. Engrg. 3 (1991), no. 3, 181–191. MR 1141298 (92h:65150), http://dx.doi.org/10.1016/0899-8248(91)90006-G
[BMN02] Eberhard Bänsch, Pedro Morin, and Ricardo H. Nochetto, An adaptive Uzawa FEM for the Stokes problem: convergence without the inf-sup condition, SIAM J. Numer. Anal. 40 (2002), no. 4, 1207–1229. MR 1951892 (2004e:65127), http://dx.doi.org/10.1137/S0036142901392134
[BL93a] John W. Barrett and W. B. Liu, Finite element approximation of the 𝑝-Laplacian, Math. Comp. 61 (1993), no. 204, 523–537. MR 1192966 (94c:65129), http://dx.doi.org/10.1090/S0025-5718-1993-1192966-4
[BL93b] John W. Barrett and W. B. Liu, Finite element error analysis of a quasi-Newtonian flow obeying the Carreau or power law, Numer. Math. 64 (1993), no. 4, 433–453. MR 1213411 (94b:65144), http://dx.doi.org/10.1007/BF01388698
[BL94] John W. Barrett and W. B. Liu, Quasi-norm error bounds for the finite element approximation of a non-Newtonian flow, Numer. Math. 68 (1994), no. 4, 437–456. MR 1301740 (95h:65078), http://dx.doi.org/10.1007/s002110050071
[BRS04] J. W. Barrett, J. A. Robson, and E. Süli, A posteriori error analysis of mixed finite element approximations to quasi-newtonian incompressible flows., Research Reports from the Numerical Analysis Group of the computing laboratory at Oxford University, UK NA-04/13 (2004), 1-16.
[BDK10] L. Belenki, L. Diening, and C. Kreuzer, Optimality of an adaptive finite element method for the nonlinear -Laplacian equation, Preprint Universität Duisburg-Essen 718 (2010), to appear IMA Journal of Numerical Analysis.
[BS08] Stefano Berrone and Endre Süli, Two-sided a posteriori error bounds for incompressible quasi-Newtonian flows, IMA J. Numer. Anal. 28 (2008), no. 2, 382–421. MR 2401203 (2009b:76091), http://dx.doi.org/10.1093/imanum/drm017
[BF91] 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 (92d:65187)
[CKNS08] J. Manuel Cascon, Christian Kreuzer, Ricardo H. Nochetto, and Kunibert G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 46 (2008), no. 5, 2524–2550. MR 2421046 (2009h:65174), http://dx.doi.org/10.1137/07069047X
[Cia88] Philippe G. Ciarlet, Introduction to numerical linear algebra and optimisation, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1989. With the assistance of Bernadette Miara and Jean-Marie Thomas; Translated from the French by A. Buttigieg. MR 1015713 (90k:65001)
[Clé75] Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. \jname RAIRO Analyse Numérique 9 (1975), no. R-2, 77–84 (English, with Loose French summary). MR 0400739 (53 #4569)
[Dau89] Monique Dauge, Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. I. Linearized equations, SIAM J. Math. Anal. 20 (1989), no. 1, 74–97. MR 977489 (90b:35191), http://dx.doi.org/10.1137/0520006
[DE08] Lars Diening and Frank Ettwein, Fractional estimates for non-differentiable elliptic systems with general growth, Forum Math. 20 (2008), no. 3, 523–556. MR 2418205 (2009h:35101), http://dx.doi.org/10.1515/FORUM.2008.027
[DHU00] Stephan Dahlke, Reinhard Hochmuth, and Karsten Urban, Adaptive wavelet methods for saddle point problems, M2AN Math. Model. Numer. Anal. 34 (2000), no. 5, 1003–1022. MR 1837765 (2002e:65078), http://dx.doi.org/10.1051/m2an:2000113
[DK08] Lars Diening and Christian Kreuzer, Linear convergence of an adaptive finite element method for the 𝑝-Laplacian equation, SIAM J. Numer. Anal. 46 (2008), no. 2, 614–638. MR 2383205 (2009a:65313), http://dx.doi.org/10.1137/070681508
[DR07a] L. Diening and M. R\ocirc{u}žička, Interpolation operators in Orlicz-Sobolev spaces, Numer. Math. 107 (2007), no. 1, 107–129. MR 2317830 (2010g:65208), http://dx.doi.org/10.1007/s00211-007-0079-9
[DR07b] Michael R\ocirc{u}žička and Lars Diening, Non-Newtonian fluids and function spaces, NAFSA 8—Nonlinear analysis, function spaces and applications. Vol. 8, Czech. Acad. Sci., Prague, 2007, pp. 94–143. MR 2657118 (2011k:35171)
[DRS10] Lars Diening, Michael R\ocirc{u}žička, and Katrin Schumacher, A decomposition technique for John domains, Ann. Acad. Sci. Fenn. Math. 35 (2010), no. 1, 87–114. MR 2643399 (2011i:26020), http://dx.doi.org/10.5186/aasfm.2010.3506
[ET76] Ivar Ekeland and Roger Temam, Convex analysis and variational problems, North-Holland Publishing Co., Amsterdam, 1976. Translated from the French; Studies in Mathematics and its Applications, Vol. 1. MR 0463994 (57 #3931b)
[FV06] Francesca Fierro and Andreas Veeser, A posteriori error estimators, gradient recovery by averaging, and superconvergence, Numer. Math. 103 (2006), no. 2, 267–298. MR 2222811 (2007a:65178), http://dx.doi.org/10.1007/s00211-005-0671-9
[For81] Michel Fortin, Old and new finite elements for incompressible flows, Internat. J. Numer. Methods Fluids 1 (1981), no. 4, 347–364. MR 633812 (83a:76015), http://dx.doi.org/10.1002/fld.1650010406
[GR86] 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 (88b:65129)
[KK91] Vakhtang Kokilashvili and Miroslav Krbec, Weighted inequalities in Lorentz and Orlicz spaces, World Scientific Publishing Co. Inc., River Edge, NJ, 1991. MR 1156767 (93g:42013)
[KS08] Yaroslav Kondratyuk and Rob Stevenson, An optimal adaptive finite element method for the Stokes problem, SIAM J. Numer. Anal. 46 (2008), no. 2, 747–775. MR 2383210 (2009b:65312), http://dx.doi.org/10.1137/06066566X
[Kos94] Igor Kossaczký, A recursive approach to local mesh refinement in two and three dimensions, J. Comput. Appl. Math. 55 (1994), no. 3, 275–288. MR 1329875 (95m:65207), http://dx.doi.org/10.1016/0377-0427(94)90034-5
[KR61] M. A. Krasnosel'skij and Ya. B. Rutitskij, Convex functions and Orlicz spaces., Groningen, The Netherlands: P. Noordhoff, Ltd., 1961.
[Kre08] C. Kreuzer, A convergent adaptive Uzawa finite element method for the nonlinear Stokes problem, Ph.D. thesis, Mathematisches Institut, Universität Augsburg, 2008.
[Mau95] Joseph M. Maubach, Local bisection refinement for 𝑛-simplicial grids generated by reflection, SIAM J. Sci. Comput. 16 (1995), no. 1, 210–227. MR 1311687 (95i:65128), http://dx.doi.org/10.1137/0916014
[MN05] Khamron Mekchay and Ricardo H. Nochetto, Convergence of adaptive finite element methods for general second order linear elliptic PDEs, SIAM J. Numer. Anal. 43 (2005), no. 5, 1803–1827 (electronic). MR 2192319 (2006i:65201), http://dx.doi.org/10.1137/04060929X
[Mit89] William F. Mitchell, A comparison of adaptive refinement techniques for elliptic problems, ACM Trans. Math. Software 15 (1989), no. 4, 326–347 (1990). MR 1062496, http://dx.doi.org/10.1145/76909.76912
[MNS00] Pedro Morin, Ricardo H. Nochetto, and Kunibert G. Siebert, Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal. 38 (2000), no. 2, 466–488 (electronic). MR 1770058 (2001g:65157), http://dx.doi.org/10.1137/S0036142999360044
[MSV07] Pedro Morin, Kunibert G. Siebert, and Andreas Veeser, Convergence of finite elements adapted for weaker norms, Applied and industrial mathematics in Italy II, Ser. Adv. Math. Appl. Sci., vol. 75, World Sci. Publ., Hackensack, NJ, 2007, pp. 468–479. MR 2367592, http://dx.doi.org/10.1142/9789812709394_0041
[MSV08] Pedro Morin, Kunibert G. Siebert, and Andreas Veeser, A basic convergence result for conforming adaptive finite elements, Math. Models Methods Appl. Sci. 18 (2008), no. 5, 707–737. MR 2413035 (2009d:65178), http://dx.doi.org/10.1142/S0218202508002838
[Mus83] Julian Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983. MR 724434 (85m:46028)
[NP04] Ricardo H. Nochetto and Jae-Hong Pyo, Optimal relaxation parameter for the Uzawa method, Numer. Math. 98 (2004), no. 4, 695–702. MR 2099317 (2005h:65052), http://dx.doi.org/10.1007/s00211-004-0522-0
[RR91] M. M. Rao and Z. D. Ren, Theory of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 146, Marcel Dekker Inc., New York, 1991. MR 1113700 (92e:46059)
[SS05] Alfred Schmidt and Kunibert G. Siebert, Design of adaptive finite element software, Lecture Notes in Computational Science and Engineering, vol. 42, Springer-Verlag, Berlin, 2005. The finite element toolbox ALBERTA; With 1 CD-ROM (Unix/Linux). MR 2127659 (2005i:65003)
[Sie09] K. G. Siebert, A convergence proof for adaptive finite elements without lower bound, IMA Journal of Numerical Analysis, doi:10.1093/imanum/drq001, May 2010.
[Squ08] A. Squillacote, Paraview Guide, Third edition, Kitware, Inc., 2008.
[Ste07] Rob Stevenson, Optimality of a standard adaptive finite element method, Found. Comput. Math. 7 (2007), no. 2, 245–269. MR 2324418 (2008i:65272), http://dx.doi.org/10.1007/s10208-005-0183-0
[Ste08] Rob Stevenson, The completion of locally refined simplicial partitions created by bisection, Math. Comp. 77 (2008), no. 261, 227–241 (electronic). MR 2353951 (2008j:65219), http://dx.doi.org/10.1090/S0025-5718-07-01959-X
[Vee02] Andreas Veeser, Convergent adaptive finite elements for the nonlinear Laplacian, Numer. Math. 92 (2002), no. 4, 743–770. MR 1935808 (2003j:65121), http://dx.doi.org/10.1007/s002110100377
[Ver89] R. Verfürth, A posteriori error estimators for the Stokes equations, Numer. Math. 55 (1989), no. 3, 309–325. MR 993474 (90d:65187), http://dx.doi.org/10.1007/BF01390056

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