Uniformly convergent iterative methods for discontinuous Galerkin discretizations

Journal of Scientific Computing

Volumn 40, Issue 1-3, 2009, Pages 4-36

  Ayuso de Dios, Blanca ^a   Zikatanov, Ludmil ^b  

^a UNIVERSIDAD AUTÓNOMA DE MADRID   (Spain)

^b PENNSYLVANIA STATE UNIVERSITY   (United States)

Author keywords

Discontinuous Galerkin finite element methods; Interior Penalty methods; Iterative methods for non symmetric problems; Subspace correction methods

Indexed keywords

CONSTRAINED OPTIMIZATION; FINITE ELEMENT METHOD; INCOMPRESSIBLE FLOW; ITERATIVE METHODS; LINEAR SYSTEMS;

CONVERGENCE PROPERTIES; CORRECTION METHOD; DISCONTINUOUS GALERKIN; DISCONTINUOUS GALERKIN FINITE-ELEMENT METHOD; FINITE ELEMENT SPACE; INTERIOR PENALTY METHOD; NONCONFORMING FINITE ELEMENT; PRECONDITIONING TECHNIQUES;

GALERKIN METHODS;

EID: 67349258011     PISSN: 08857474     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10915-009-9293-1     Document Type: Article

References (40)

1. Adams, R.A.: Sobolev spaces. Pure and Applied Mathematics, vol. 65. Academic Press [Harcourt Brace Jovanovich], San Diego (1975)
2. Agmon, S.: Lectures on elliptic boundary value problems. Van Nostrand Mathematical Studies, vol. 2. Van Nostrand, Princeton (1965). Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr.
3. P.F. Antonietti B. Ayuso 2007 Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case Math. Model. Numer. Anal. 41 1 21 54
4. P.F. Antonietti B. Ayuso 2008 Multiplicative Schwarz methods for discontinuous Galerkin approximations of elliptic problems Math. Model. Numer. Anal. 42 3 443 469
5. P.F. Antonietti B. Ayuso 2009 Two-level Schwarz preconditioners for super penalty discontinuous Galerkin methods Commun. Comput. Phys. 5 398 412
6. D.N. Arnold 1982 An interior penalty finite element method with discontinuous elements SIAM J. Numer. Anal. 19 4 742 760
7. D.N. Arnold F. Brezzi B. Cockburn L.D. Marini 2001/02 Unified analysis of discontinuous Galerkin methods for elliptic problems SIAM J. Numer. Anal. 39 5 1749 1779 (electronic)
8. Ayuso, B., Georgiev, I., Kraus, J., Zikatanov, L.: Preconditioning techniques for discontinuous Galerkin methods discretizations for linear elasticity equations (2009, in preparation)
9. B. Ayuso L.D. Marini 2009 Discontinuous Galerkin methods for advection-diffussion-reaction problems SIAM J. Numer. Anal. 47 2 1391 1420
10. S.C. Brenner 1996 Two-level additive Schwarz preconditioners for nonconforming finite element methods Math. Comput. 65 215 897 921 (Pubitemid 126352000)
11. S.C. Brenner 1999 Convergence of nonconforming multigrid methods without full elliptic regularity Math. Comput. 68 225 25 53
12. S.C. Brenner 2004 Convergence of nonconforming V-cycle and F-cycle multigrid algorithms for second order elliptic boundary value problems Math. Comput. 73 247 1041 1066 (electronic) (Pubitemid 38782485)
13. S.C. Brenner L. Owens 2007 A weakly over-penalized non-symmetric interior penalty method JNAIAM J. Numer. Anal. Ind. Appl. Math. 2 1-2 35 48
14. S.C. Brenner L. Owens L.-Y. Sung 2008 A weakly over-penalized symmetric interior penalty method Electron. Trans. Numer. Anal. 30 107 127
15. Brenner, S.C., Ridgway Scott, L.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, vol. 15. Springer, New York (1994)
16. S.C. Brenner J. Zhao 2005 Convergence of multigrid algorithms for interior penalty methods Appl. Numer. Anal. Comput. Math. 2 1 3 18
17. F. Brezzi B. Cockburn L.D. Marini E. Süli 2006 Stabilization mechanisms in discontinuous Galerkin finite element methods Comput. Methods Appl. Mech. Eng. 195 25-28 3293 3310
18. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991)
19. E. Burman B. Stamm 2008 Low order discontinuous Galerkin methods for second order elliptic problems SIAM J. Numer. Anal. 47 1 508 533
20. Ciarlet, P.G.: Basic error estimates for elliptic problems. In: Handbook of Numerical Analysis, vol. II, pp. 17-351. North-Holland, Amsterdam (1991)
21. M. Crouzeix P.-A. Raviart 1973 Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I Rev. Fr. Autom. Inform. Rech. Opér. Sér. Rouge 7 R-3 33 75
22. C. Dawson S. Sun M.F. Wheeler 2004 Compatible algorithms for coupled flow and transport Comput. Methods Appl. Mech. Eng. 193 23-26 2565 2580 (Pubitemid 38503368)
23. V.A. Dobrev R.D. Lazarov P.S. Vassilevski L.T. Zikatanov 2006 Two-level preconditioning of discontinuous Galerkin approximations of second-order elliptic equations Numer. Linear Algebra Appl. 13 9 753 770
24. Douglas Jr., J., Dupont, T.: Interior penalty procedures for elliptic and parabolic Galerkin methods. In: Computing Methods in Applied Sciences, Second Internat. Sympos., Versailles, 1975. Lecture Notes in Phys., vol. 58, pp. 207-216. Springer, Berlin (1976)
25. S.C. Eisenstat H.C. Elman M.H. Schultz 1983 Variational iterative methods for nonsymmetric systems of linear equations SIAM J. Numer. Anal. 20 2 345 357
26. R.D. Falgout P.S. Vassilevski L.T. Zikatanov 2005 On two-grid convergence estimates Numer. Linear Algebra Appl. 12 5-6 471 494 (Pubitemid 40977207)
27. X. Feng O.A. Karakashian 2001 Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems SIAM J. Numer. Anal. 39 4 1343 1365 (electronic)
28. J. Gopalakrishnan G. Kanschat 2003 A multilevel discontinuous Galerkin method Numer. Math. 95 3 527 550 (Pubitemid 37513718)
29. M.R. Hestenes E. Stiefel 1953 Methods of conjugate gradients for solving linear systems J. Res. Natl. Bur. Stand. 49 409 436
30. J. Kraus S. Margenov J. Synka 2008 On the multilevel preconditioning of Crouzeix-Raviart elliptic problems Numer. Linear Algebra Appl. 15 5 395 416
31. Luenberger, D.G.: Linear and Nonlinear Programming, 2nd edn. Kluwer Academic, Boston (2003)
32. B. Rivière M.F. Wheeler V. Girault 2000 Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. I Comput. Geosci. 3 3-4 337 360 1999
33. B. Rivière M.F. Wheeler V. Girault 2001 A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems SIAM J. Numer. Anal. 39 3 902 931 (electronic) (Pubitemid 34757163)
34. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2003)
35. Strang, G.: Variational crimes in the finite element method. In: The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Proc. Sympos. Univ. Maryland, Baltimore, Md., 1972, pp. 689-710. Academic Press, San Diego (1972)
36. Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Prentice-Hall Series in Automatic Computation. Prentice-Hall, Englewood Cliffs (1973)
37. S. Sun M.F. Wheeler 2005 Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media SIAM J. Numer. Anal. 43 1 195 219 (electronic) (Pubitemid 43304498)
38. M.F. Wheeler 1978 An elliptic collocation-finite element method with interior penalties SIAM J. Numer. Anal. 15 1 152 161
39. J. Xu 1992 Iterative methods by space decomposition and subspace correction SIAM Rev. 34 4 581 613
40. J. Xu L. Zikatanov 2002 The method of alternating projections and the method of subspace corrections in Hilbert space J. Am. Math. Soc. 15 3 573 597 (electronic)