Krylov-based Uzawa algorithms for the solution of the stokes equations using discontinuous-pressure tetrahedral finite elements

Journal of Computational Physics

Volumn 181, Issue 2, 2002, Pages 617-638

  Bertrand, Francçois ^a   Tanguy, Philippe A ^a  

^a ÉCOLE POLYTECHNIQUE DE MONTRÉAL   (Canada)

Author keywords

3D; Finite element method; Iterative solver; Krylov subspace methods; Stokes equations; Uzawa algorithm

Indexed keywords

EQUATIONS OF STATE; ITERATIVE METHODS; NAVIER STOKES EQUATIONS;

3D; DISCONTINUOUS PRESSURES; ITERATIVE SOLVERS; KRYLOV BASED UZAWA ALGORITHM; KRYLOV-SUBSPACE METHODS; STEADY-STATE STATE; STOKES EQUATIONS; TETRAHEDRAL FINITE ELEMENTS; UNSTEADY STATE; UZAWA'S ALGORITHMS;

FINITE ELEMENT METHOD;

EID: 0037144921     PISSN: 00219991     EISSN: None     Source Type: Journal    
DOI: 10.1006/jcph.2002.7144     Document Type: Article

References (44)

1. P. A. Tanguy, M. Fortin, and L. Choplin, Finite element simulation of dip coating, II: Non-Newtonian fluids, Int. J. Numer. Methods Fluids 4, 459 (1984).
2. R. Glowinski and O. Pironneau, Finite element methods for Navier-Stokes equations, Annu. Rev. Fluid. Mech. 24, 167 (1992).
3. O. Axelsson and V. A. Barker, Finite Element Solution of Boundary Value Problems (Academic Press, San Diego 1984).
4. R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Soc. for Industr. & Appl. Math., Philadelphia, 1984).
5. H. P. Langtangen, Conjugate gradient methods and ILU preconditioning of non-symmetric matrix systems with arbitrary sparsity patterns, Int. J. Numer. Methods Fluids 9, 213 (1989).
6. O. Dahl and S. Wille, An ILU preconditioner with coupled node fill-in for iterative solution of the mixed finite element formulation of the 2D and 3D Navier-Stokes equations, Int. J. Numer. Methods Fluids 15, 525 (1992).
7. F. F. Campos and N. R. C. Birkett, An efficient solver for multi-right-hand-side linear systems based on the CCCG(η) method with applications to implicit time-dependent partial differential equations, SIAM J. Sci. Comput. 19(1), 126 (1998).
8. W. Hackbush, Multigrid Methods and Applications (Springer-Verlag, Berlin, 1985).
9. O. Axelsson and P. Vassilevski, Algebraic multilevel preconditioning methods, I, Numer. Math. 56,157 (1989).
10. O. Axelsson and P. Vassilevski, Algebraic multilevel preconditioning methods, II, Numer. Math. 57, 1569 (1990).
11. G. Poole and Y. C. Liu, Advancing analysis capabilities in ANSYS through solver technology, presented at the 10th Copper Mountain Conf. on Multigrid Methods, Copper Mountain, Colorado, 2001.
12. N. Nachtigal, S. Reddy, and L. Trefethen, How fast are nonsymmetric matrix iterations, SIAM J. Matrix Anal. Appl. 13, 778 (1992).
13. P. Sonneveld, CGS, a fast Lanczos-type solver for non-symmetric linear systems, SIAM J. Sci. Stat. Comput. 10, 36 (1989).
14. H. Van der Vorst, A fast and smoothly converging variant of bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 13, 631 (1992).
15. R. W. Freund, A transpose-free quasi-minimum residual algorithm for non-Hermitian linear systems, SIAM J. Sci. Comput. 14, 470 (1993).
16. Y. Saad and M. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 7, 856 (1986).
17. M. Robichaud and P. A. Tanguy, Finite element solution of three-dimensional incompressible fluid flow problems by a preconditioned conjugate residual method, Int. J. Numer. Methods Eng. 24, 447 (1987).
18. T. Rusten and R. Winther, A preconditioned iterative method for saddle point problems, SIAM J. Matrix Anal. Appl. 13(3), 887 (1992).
19. J. Atanga and A. Wathen, Iterative methods for stabilized mixed velocity-pressure finite elements, Int. J. Numer. Methods Fluids 14, 71 (1992).
20. A. Ramage and A. Wathen, Iterative solution techniques for the Stokes and Navier-Stokes equations, Int. J. Numer. Methods Fluids 19, 67 (1994).
21. J. N. Reddy, On penalty function methods in the finite element analysis of fluid flow, Int. J. Numer. Methods Fluids 2, 151 (1982).
22. M. Fortin and R. Glowinski, Augmented Lagrangian Methods (North-Holland, Amsterdam, 1983).
23. J. Atanga and D. Silvester, Iterative methods for stabilized mixed velocity-pressure finite elements, Int. J. Numer. Methods Fluids 14, 71 (1992).
24. U. Langer and W. Queck, On the convergence factor of Uzawa's algorithm, J. Comput. Appl. Math. 15, 191 (1986).
25. D. J. Silvester and N. Kechkar, Stabilized bilinear-constant velocity-pressure finite elements for the conjugate gradient solution of the Stokes problem, Comput. Methods Appl. Mech. Eng. 79, 71 (1990).
26. R. Verfürth, A combined conjugate gradient-multigrid algorithm for the numerical solution of the Stokes problem, IMA J. Numer. Anal. 4, 441 (1984).
27. J. Cahouet and J. P. Chabard, Some fast 3D finite element solvers for the generalized Stokes problem, Int. J. Numer. Methods Fluids 8, 869 (1988).
28. P. Carriere and D. Jeandel, A 3D finite element method for the simulation of thermoconvective flows and its performances on a vector-parallel computer, Int. J. Numer Methods Fluids 12, 929 (1991).
29. R. Q. N. Zhou, Preconditioned finite element algorithms for 3D Stokes flow, Int. J. Numer. Methods Fluids 17, 667 (1993).
30. C. Vincent, The influence of the stabilization parameter on the convergence factor of iterative methods for the solution of the discretized Stokes problem, Int. J. Numer. Methods Fluids 20, 1237 (1995).
31. D. Pelletier, A. Fortin, and R. Camarero, Are FEM solutions of incompressible flows really incompressible? (Or how simple flows can cause headaches!), Int. J. Numer. Methods Fluids 9, 99 (1989).
32. M. P. Robichaud, P. A. Tanguy, and M. Fortin, An iterative implementation of the Uzawa algorithm for 3D fluid flow problems, Int. J. Numer. Methods Fluids 10, 429 (1990).
33. F. Bertrand, M. Gadbois, and P. A. Tanguy, Tetrahedral elements for fluid flow problems, Int. J. Numer. Methods Fluids 33, 1251 (1992).
34. M. Fortin and A. Fortin, Experiments with several elements for viscous incompressible flows, Int. J. Numer. Methods Fluids 5, 911 (1985).
35. W. L. Briggs, A Multigrid Tutorial (Soc. for Industr. & Appl. Math., Philadelphia, 1987).
36. G. Labadie and P. Lasbleiz, Quelques méthodes de résolution du problème de Stokes en éléments finis, EDF Report HE41/83.01 (1983).
37. S. O. Wille, A preconditioned alternating inner-outer iterative solution method for the mixed finite element formulation of the Navier-Stokes equations, Int. J. Numer. Methods Fluids 18, 1135 (1994).
38. H. C. Elman and G. H. Golub, Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal. 31(6), 1645 (1994).
39. J. H. Bramble, J. E. Pasciak, and A. T. Vassilev, Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal. 34(3), 1072 (1997).
40. W. Queck, The convergence factor of preconditioned algorithms of the Arrow-Hurwicz type, SIAM J. Numer. Anal. 26(4), 1016 (1989).
41. T. J. R. Hughes, The Finite Element Method (Prentice Hall, New York, 1987).
42. M. Fortin and A. Fortin, Newer and newer elements for incompressible flow, in Finite Elements in Fluids 6, edited by R. H. Gallagher, G. F. Carey, J. T. Oden, and O. C. Zienckiewicz (Wiley, New York, 1985).
43. A. Van der Sluis and H. A. Van der Vorst, The rate of convergence of conjugate gradients, Numer. Math. 48, 543 (1986).
44. G. M. Kobelkov and M. A. Olshanskii, Effective preconditioning of Uzawa type schemes for a generalized Stokes problem, Numer. Math. 86, 443 (2000).