Uniform convergence of multigrid V-cycle on adaptively refined finite element meshes for second order elliptic problems

Science in China, Series A: Mathematics

Volumn 49, Issue 10, 2006, Pages 1405-1429

  Wu, Haijun ^a   Chen, Zhiming ^a  

^a INSTITUTE OF GEOLOGY AND GEOPHYSICS   (China)

Author keywords

Adaptive finite element meshes; Local relaxation; Multigrid V cycle algorithm; Scott Zhang interpolation

Indexed keywords

EID: 33845438715     PISSN: 10069283     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11425-006-2005-5     Document Type: Article

References (40)

1. Babuška I, Rheinboldt C. Error estimates for adaptive finite element computations. SIAM J Numer Anal, 1978, 15: 736-754
2. Morin P, Nochetto R H, Siebert K G. Data oscillation and convergence of adaptive FEM. SIAM J Numer Anal, 2000, 38: 466-488
3. Chen Z, Dai S. On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients. SIAM J Sci Comput, 2002, 24: 443-462
4. Mitchell W F. Optimal multilevel iterative methods for adaptive grids. SIAM J Sci Stat Comput, 1992, 13: 146-167
5. Bank R E, Dupont T F, Yserentant H. The hierarchical basis multigrid method. Numer Math, 1988, 52: 427-458
6. Bänsch E. Local mesh refinement in 2 and 3 dimensions. Impact of Computing in Science and Engineering, 1991, 3: 181-191
7. Rivara M C. Algorithms for refining triangular grids suitable for adaptive and multigrid techniques. J Numer Meth Engrg, 1984, 20: 745-756
8. Stals L. Implementation of multigrid on parallel machines using adaptive finite element methods. Bjorstad P E, Espedal M S, Keyes D E, eds. In: Ninth International Conference on Domain Decomposition Methods. Bergen: DDM Org, 1997, 488-496
9. Chen Z, Dai S. Adaptive Galerkin methods with error control for a dynamical Ginzburg-Landau model in superconductivity. SIAM J Numer Anal, 2001, 38: 1961-1985
10. Chen Z, Jia F. An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems. Math Comp, 2004, 73: 1163-1197
11. Chen Z, Nochetto R H, Schmidt A. A characteristic Galerkin method with adaptive error control for continuous casting problem. Comput Methods Appl Mech Engrg, 2000, 189: 249-276
12. Chen Z, Wu H. An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures. SIAM J Numer Anal, 2003, 41: 799-826
13. Schmidt A, Siebert K G. ALBERT: An adaptive hierarchical finite element toolbox. IAM, University of Freiburg. 2000, http://www.mathematik.uni-freiburg.de/IAM/Research/projectsdz/albert.
14. Bramble J H, Ewing R E, Pasciak J E, et al. A preconditioneding technique for the efficient solution of problems with local grid refinement. Comput Methods Appl Mech Engrg, 1988, 67: 149-159
15. McCormick S. Fast adaptive composite grid (FAC) methods. Böhmer K, Stetter H J, eds. In: Defect Correction Methods: Theory and Applications, Computing Supplementum 5. Wien: Sringer-Verlag, 1984, 115-121
16. McCormick S, Thomas J. The fast adaptive composite grid (FAC) method for elliptic equations. Math Comp, 1986, 46: 439-456
17. Widlund O B. Optimal iterative refinement methods. In: Chan T, Glowinski R, Périaux J, Widlund O, eds. Domain Decomposition Methods. Philadelphia: SIAM, 1989, 114-125
18. Bai D, Brandt A. Local mesh refinement multilevel techniques. SIAM J Sci Stat Comput, 1987, 8: 109-134
19. Bramble J H, Pasicak J E. New estimates for multigrid algorithms including the V-cycle. Math Comp, 1993, 60: 447-471
20. Bramble J H, Pasicak J E, Wang J, et al. Convergence estimates for product iterative methods with applications to domain decomposition. Math Comp, 1991, 57: 23-45
21. Oswald P. Multilevel Finite Element Approximation, Theory and Applications. Stuttgart: Teubner Verlag, 1994
22. Xu J. An introduction to multigrid convergence theory. In: Chan R, Chan T, Golub G, eds. Iterative Methods in Scientific Computing. Singapore: Springer-Verlag, 1997
23. Ainsworth M, Mclean W. Multilevel diagonal scaling preconditioners for boundary element equations on locally refined meshes. Numer Math, 2003, 93: 387-413
24. Aksoylu B, Holst M. An odyssey into local refinement and multilevel preconditioning I: Optimality of the BPX preconditioned, Institute for Computational Engineering and Sciences, The University of Texas at Austin, ICES Technical Report 05-03, 2005
25. Dahmen W, Kunoth A. Multilevel preconditioning. Numer Math, 1992, 63: 315-344
26. Oswald P. Multilevel solvers for elliptic problems on domains. In: Dahmen W, Kurdila A J, Oswald P, eds. Multiscale Wavelet Methods for PDEs. New York: Academic Press, 1996, 3-58
27. Bank R E, Dupont T. An optimal order process for solving elliptic finite element equations. Math Comp, 1981, 36: 35-51
28. Bastian P, Wittum G. On robust and adaptive multigrid methods. In: Hemker P W, Wesseling P, eds. Multigrid Methods IV, Proceedings of the Fourth European Multigrid Conference, Amsterdam, July 6-9, 1993. Basel: Birkhäuser, 1994, 1-17
29. Bornemann F A, Yserentant H. A basic norm equivalence for the theory of multilevel methods. Numer Math, 1993, 64: 455-476
30. Brandt A. Multilevel adaptive finite element methods I: Variational problems. In: Frehes J, Pallaschke D, Trotenberg U, eds. Special Topics of Applied Mathematics. Amsterdam: North-Holland, 1979, 91-128
31. Griebel M, Oswald P. On the abstract theory of additive and multiplicative Schwarz algorithms. Numer Math, 1995, 70: 163-180
32. Rüde U. Mathematical and Computational Techniques for Multilevel Adaptive Method, Frontiers in Applied Mathematics 13. Philadelphia: SIAM, 1993
33. Yserentant H. Old and new convergence proofs for multigrid methods. Acta Numerica, 1993, 3: 285-326
34. Xu J. Iterative methods by space decomposition and subspace correction. SIAM Review, 1992, 34: 581-613
35. Xu J, Zikatanov L. The method of alternating projections and the method of subspace corrections in Hilbert space. J Amer Math Soc, 2002, 15: 573-597
36. Bramble J H. Multigrid Methods, Pitman Research Notes in Mathematical Sciences 294. Essex: Longman, 1993
37. Scott L R, Zhang S. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math Comp, 1990, 54: 483-493
38. Xu J. Multigrid convergence theory, Lecture Notes of the 2001 summer program on Multigrid Methods and Computational Electromagnetics in the Morningside Center of Mathematics. Chinese Academy of Sciences, Beijing, China
39. Wu H. Uniform Convergence of a multigrid V-cycle for adaptive linear finite element methods, Postdoctoral Report. Institute of Computational Mathematics, Chinese Academy of Sciences, Beijing, 2002
40. Brandt A. Multigrid Techniques: 1984 Guide, with applications to fluid dynamics, monograph. Weizmann Institute of Science, Rehovot, Israel, 1984