A simple preconditioner for a Discontinuous Galerkin method for the Stokes problem

Journal of Scientific Computing

Volumn 58, Issue 3, 2014, Pages 517-547

  De Dios, Blanca Ayuso ^a   Brezzi, Franco ^b,c   Marini, L Donatella ^d   Xu, Jinchao ^e   Zikatanov, Ludmil ^e  

^a CENTRE DE RECERCA MATEMÀTICA   (Spain)

^b CNR   (Italy)

^c KING ABDULAZIZ UNIVERSITY   (Saudi Arabia)

^d UNIVERSITY OF PAVIA   (Italy)

^e PENNSYLVANIA STATE UNIVERSITY   (United States)

Author keywords

Auxiliary space; Discontinuous Galerkin; Stokes equation

Indexed keywords

LINEAR SYSTEMS; NAVIER STOKES EQUATIONS; VELOCITY;

AUXILIARY SPACE; DISCONTINUOUS GALERKIN; DISCONTINUOUS GALERKIN METHODS; DIVERGENCE FREE; PRECONDITIONERS; STOKES EQUATIONS; STOKES PROBLEM; VELOCITY FIELD;

GALERKIN METHODS;

EID: 84899453877     PISSN: 08857474     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10915-013-9758-0     Document Type: Article

References (42)

1. Adams, R. A.: Sobolev Spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, vol. 65
2. Agmon, S.: Lectures on Elliptic Boundary Value Problems. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. Van Nostrand Mathematical Studies, No. 2. D. Van Nostrand Co., Inc., Princeton (1965)
3. Amrouche, C., Seloula, N. E. H.: On the Stokes equations with the Navier-type boundary conditions. Differ. Equ. Appl. 3(4), 581-607 (2011)
4. Antil, H., Nochetto, R. H., Sodré, P.: The Stokes problem with slip boundary conditions on fractional Sobolev domains. (in preparation) (2013)
5. Arnold, D. N., Brezzi, F., Cockburn, B., Marini, L. D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39(5), 1749-1779 (electronic) (2001/02)
6. Arnold, D. N., Falk, R. S., Winther, R.: Preconditioning in H (div) and applications. Math. Comp. 66(219), 957-984 (1997)
7. Arnold, D. N., Falk, R. S., Winther, R.: Multigrid in H (div) and H (curl). Numer. Math. 85(2), 197-217 (2000)
8. Arnold, D. N., Brezzi, F., Marini, L. D.: A family of discontinuous Galerkin finite elements for the Reissner-Mindlin plate. J. Sci. Comput. 22/23, 25-45 (2005)
9. Arnold, D. N., Falk, R. S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1-155 (2006)
10. Beirão Da Veiga, H.: Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip-type boundary conditions. Adv. Differ. Equ. 9(9-10), 1079-1114 (2004)
11. Brandt, A., McCormick, S. F., Ruge, J. W.: Algebraic Multigrid (AMG) for Automatic Multigrid Solution with Application to Geodetic Computations. Tech. Rep., Institute for Computational Studies, Colorado State University (1982)
12. 1 functions. SIAM J. Numer. Anal. 41(1), 306-324 (2003). (electronic)
13. 1 vector fields. Math. Comp. 73(247), 1067-1087 (2004)
14. Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8, 129-151 (1974)
15. Brezzi, F., Douglas Jr, J., Marini, L. D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47(2), 217-235 (1985)
16. Brezzi, F., Fortin, M.: Numerical approximation of Mindlin-Reissner plates. Math. Comp. 47(175), 151-158 (1986)
17. Brezzi, F., Douglas Jr, J., Durán, R., Fortin, M.: Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51(2), 237-250 (1987)
18. Brezzi, F., Douglas Jr, J., Fortin, M., Marini, L. D.: Efficient rectangular mixed finite elements in two and three space variables. RAIRO Modél. Math. Anal. Numér. 21(4), 581-604 (1987)
19. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods, volume 15 of Springer Series in Computational Mathematics. Springer, New York (1991)
20. Brezzi, F., Fortin, M., Stenberg, R.: Error analysis of mixed-interpolated elements for Reissner-Mindlin plates. Math. Model. Methods Appl. Sci. 1(2), 125-151 (1991)
21. Cockburn, B., Kanschat, G., Schötzau, D.: A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations. J. Sci. Comput. 31(1-2), 61-73 (2007)
22. Elman, H. C., Silvester, D. J., Wathen, A. J.: Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2005)
23. Galdi, G. P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. vol. I, volume 38 of Springer Tracts in Natural Philosophy.. Springer, New York (1994). Linearized steady problems
24. Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations, volume 5 of Springer Series in Computational Mathematics.. Springer, Berlin (1986). Theory and algorithms
25. Greif, C., Li, D., Schötzau, D., Wei, X.: A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics. Comput. Methods Appl. Mech. Eng. 199(45-48), 2840-2855 (2010)
26. Kanschat, G., Rivière, B.: A strongly conservative finite element method for the coupling of Stokes and Darcy flow. J. Comput. Phys. 229(17), 5933-5943 (2010)
27. Ladyzhenskaya, O. A.: The Mathematical Theory of Viscous Incompressible Flow. Second English edition, Revised and Enlarged. Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications, Vol. 2. Gordon and Breach Science Publishers, New York (1969)
28. Lee, Y.-J., Xu, J.: New formulations, positivity preserving discretizations and stability analysis for non-Newtonian flow models. Comput. Methods Appl. Mech. Engrg. 195(9-12), 1180-1206 (2006)
29. Lewicka, M., Müller, S.: The uniform Korn-Poincaré inequality in thin domains. Ann. Inst. H. Poincaré Anal. Non Linéaire 28(3), 443-469 (2011)
30. Lin, F.-H., Liu, C., Zhang, P.: On hydrodynamics of viscoelastic fluids. Commun. Pure Appl. Math. 58(11), 1437-1471 (2005)
31. 3. Numer. Math. 35(3), 315-341 (1980)
32. 3. Numer. Math. 50(1), 57-81 (1986)
33. Nepomnyaschikh, S. V.: Decomposition and fictitious domains methods for elliptic boundary value problems. In Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations (Norfolk, VA, 1991), pp. 62-72. SIAM, Philadelphia (1992)
34. Nepomnyaschikh, S. V.: Mesh theorems on traces, normalizations of function traces and their inversion. Sov. J. Numer. Anal. Math. Model. 6(3), 223-242 (1991)
35. Oswald, P.: Preconditioners for nonconforming discretizations. Math. Comp. 65(215), 923-941 (1996)
36. Raviart, P.-A., Thomas, J. M.: A mixed finite element method for 2nd order elliptic problems. In: Mathematical Aspects of Finite Element Methods (Proc. Conf., Consiglio Naz. delle Ricerche (C. N. R.), Rome, 1975), pp. 292-315. Lecture Notes in Math., vol. 606. Springer, Berlin (1977)
37. Solonnikov, V. A., Ščadilov, V. E.: A certain boundary value problem for the stationary system of Navier-Stokes equations. Trudy Mat. Inst. Steklov., 125, 196-210, 235, (1973). Boundary value problems of mathematical, physics, 8
38. Temam, R.: Navier-Stokes Equations, volume 2 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam, revised edition, (1979). Theory and numerical analysis, With an appendix by F. Thomasset
39. Temam, R., Miranville, A.: Mathematical Modeling in Continuum Mechanics. Cambridge University Press, Cambridge (2001)
40. Wang, J., Ye, X.: New finite element methods in computational fluid dynamics by H (div) elements. SIAM J. Numer. Anal. 45(3), 1269-1286 (2007). (electronic)
41. Xu, J.: The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. Computing, 56(3):215-235, 1996. International GAMM-Workshop on Multi-level Methods (Meisdorf, 1994)
42. Yoo, J., Joseph, D. D., Beavers, G. S.: Boundary conditions at a naturally impermeable wall. J. Fluid Mech. 30, 197-207 (1967)