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

Kreuzer, Christian

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

Analysis of an adaptive Uzawa finite element method for the nonlinear Stokes problem
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Math. Comp. 81 (2012), 21-55 Request permission

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.

References

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
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, DOI 10.1002/9781118032824
Eberhard Bänsch, Local mesh refinement in $2$ and $3$ dimensions, Impact Comput. Sci. Engrg. 3 (1991), no. 3, 181–191. MR 1141298, DOI 10.1016/0899-8248(91)90006-G
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, DOI 10.1137/S0036142901392134
John W. Barrett and W. B. Liu, Finite element approximation of the $p$-Laplacian, Math. Comp. 61 (1993), no. 204, 523–537. MR 1192966, DOI 10.1090/S0025-5718-1993-1192966-4
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, DOI 10.1007/BF01388698
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, DOI 10.1007/s002110050071
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.
L. Belenki, L. Diening, and C. Kreuzer, Optimality of an adaptive finite element method for the nonlinear $p$-Laplacian equation, Preprint Universität Duisburg-Essen 718 (2010), to appear IMA Journal of Numerical Analysis.
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, DOI 10.1093/imanum/drm017
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
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, DOI 10.1137/07069047X
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
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
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, DOI 10.1137/0520006
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, DOI 10.1515/FORUM.2008.027
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, DOI 10.1051/m2an:2000113
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, DOI 10.1137/070681508
L. Diening and M. Růžička, Interpolation operators in Orlicz-Sobolev spaces, Numer. Math. 107 (2007), no. 1, 107–129. MR 2317830, DOI 10.1007/s00211-007-0079-9
Michael Růžička 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
Lars Diening, Michael Růžička, and Katrin Schumacher, A decomposition technique for John domains, Ann. Acad. Sci. Fenn. Math. 35 (2010), no. 1, 87–114. MR 2643399, DOI 10.5186/aasfm.2010.3506
Ivar Ekeland and Roger Temam, Convex analysis and variational problems, Studies in Mathematics and its Applications, Vol. 1, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1976. Translated from the French. MR 0463994
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, DOI 10.1007/s00211-005-0671-9
Michel Fortin, Old and new finite elements for incompressible flows, Internat. J. Numer. Methods Fluids 1 (1981), no. 4, 347–364. MR 633812, DOI 10.1002/fld.1650010406
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
Vakhtang Kokilashvili and Miroslav Krbec, Weighted inequalities in Lorentz and Orlicz spaces, World Scientific Publishing Co., Inc., River Edge, NJ, 1991. MR 1156767, DOI 10.1142/9789814360302
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, DOI 10.1137/06066566X
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, DOI 10.1016/0377-0427(94)90034-5
M. A. Krasnosel’skij and Ya. B. Rutitskij, Convex functions and Orlicz spaces., Groningen, The Netherlands: P. Noordhoff, Ltd., 1961.
C. Kreuzer, A convergent adaptive Uzawa finite element method for the nonlinear Stokes problem, Ph.D. thesis, Mathematisches Institut, Universität Augsburg, 2008.
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, DOI 10.1137/0916014
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. MR 2192319, DOI 10.1137/04060929X
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, DOI 10.1145/76909.76912
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. MR 1770058, DOI 10.1137/S0036142999360044
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, DOI 10.1142/9789812709394_{0}041
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, DOI 10.1142/S0218202508002838
Julian Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983. MR 724434, DOI 10.1007/BFb0072210
Ricardo H. Nochetto and Jae-Hong Pyo, Optimal relaxation parameter for the Uzawa method, Numer. Math. 98 (2004), no. 4, 695–702. MR 2099317, DOI 10.1007/s00211-004-0522-0
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
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
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.
A. Squillacote, Paraview Guide, Third edition, Kitware, Inc., 2008.
Rob Stevenson, Optimality of a standard adaptive finite element method, Found. Comput. Math. 7 (2007), no. 2, 245–269. MR 2324418, DOI 10.1007/s10208-005-0183-0
Rob Stevenson, The completion of locally refined simplicial partitions created by bisection, Math. Comp. 77 (2008), no. 261, 227–241. MR 2353951, DOI 10.1090/S0025-5718-07-01959-X
Andreas Veeser, Convergent adaptive finite elements for the nonlinear Laplacian, Numer. Math. 92 (2002), no. 4, 743–770. MR 1935808, DOI 10.1007/s002110100377
R. Verfürth, A posteriori error estimators for the Stokes equations, Numer. Math. 55 (1989), no. 3, 309–325. MR 993474, DOI 10.1007/BF01390056