Hierarchical bases and the finite element method

Acta Numerica

Volumn 5, Issue , 1996, Pages 1-43

  Bank, Randolph E ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85022608074     PISSN: 09624929     EISSN: 14740508     Source Type: Journal    
DOI: 10.1017/S0962492900002610     Document Type: Article

References (41)

1. Ainsworth M. and Oden J. T. (1992), ‘A procedure for a posteriori error estimation for h-p finite element methods’, Comp. Meth. Appl. Mech. Engrg. 101, 73–96.
2. Ainsworth M. and Oden J. T. (1993), ‘A unified approach to a posteriori error estimation using element residual methods’, Numer. Math. 65, 23–50.
3. Aziz A. K. and Babuška I. (1972), ‘Survey lectures on the mathematical foundations of the finite element method’, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (Aziz A. K., ed.), Academic Press, New York, 1–362.
4. Babuška I. and Gui W. (1986), ‘Basic principles of feedback and adaptive approaches in the finite element method’, Comp. Meth. Appl. Mech. Engrg. 55, 27–42.
5. Babuška I., Zienkiewicz O. C., Gago J. P. d. S.R., and de Arantes e Oliveira E. R., eds (1986) Accuracy Estimates and Adaptive Refinements in Finite Element Computations, Wiley, New York.
6. Babuška I. and Rheinboldt W. C. (1978), ‘Error estimates for adaptive finite element computations’, SIAM J. Numer. Anal. 15, 736–754.
7. Babuška I. and Rheinboldt W. C. (1978), ‘ A posteriori error estimates for the finite element method’, Internat. J. Numer. Methods Engrg. 12, 1597–1615.
8. Bank R. E. (1994), PLTMG: A Software Package for Solving Elliptic Partial Differential Equations, Users' Guide 7.0. Frontiers in Applied Mathematics 15, SIAM, Philadelphia.
9. Bank R. E. and Dupont T. F. (1979), ‘Notes on the k-level iteration’, unpublished notes.
10. Bank R. E. and Dupont T. F. (1980), ‘Analysis of a two level scheme for solving finite element equations’, Technical Report CNA-159, Center for Numerical Analysis, University of Texas at Austin.
11. Bank R. E., Dupont T. F., and Yserentant H. (1988), ‘The hierarchical basis multigrid method’, Numer. Math. 52, 427–458.
12. Bank R. E. and Scott L. R. (1989), ‘On the conditioning of finite element equations with highly refined meshes’, SIAM J. Numer. Anal. 26, 1383–1394.
13. Bank R. E. and Smith R. K. (1993), ‘ A posteriori error estimates based on hierarchical bases, SIAM J. Numer. Anal. 30, 921–935.
14. Bank R. E. and Weiser A. (1985), ‘Some a posteriori error estimates for elliptic partial differential equations’, Math. Comp. 44, 283–301.
15. Bornemann F. and Yserentant H. (1993), ‘A basic norm equivalence for the theory of multilevel methods’, Numer. Math. 64, 455–476.
16. Braess D. (1981), ‘The contraction number of a multigrid method for solving the Poisson equation’, Numer. Math. 37, 387–404.
17. Bramble J. H. (1993), Multigrid Methods, Pitman Research Notes in Mathematical Sciences 294, Longman Sci. & Techn., Harlow, UK.
18. Bramble J. H., Pasciak J. E., Wang J., and Xu J. (1991), ‘Convergence estimate for product iterative methods with application to domain decomposition and multigrid’, Math. Comp. 57, 1–21.
19. Bramble J. H., Pasciak J. E., and Xu J. (1990), ‘Parallel multilevel preconditioners’, Math. Comp. 55, 1–22.
20. Brenner S. C. and Scott L. R. (1994), The Mathematical Theory of Finite Element Methods, Springer, Heidelberg.
21. Ciarlet P. G. (1980), The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam.
22. Deuflhard P., Leinen P., and Yserentant H. (1989), ‘Concepts of an adaptive hierarchical finite element code’, IMPACT of Comput. in Sci. and Eng. 1, 3–35.
23. Dupont T. F., Kendall R. P., and Rachford H. H. (1968), ‘An approximate factorization procedure for self-adjoint elliptic difference equations’, SIAM J. Numer. Anal. 5, 559–573.
24. Durán R., Muschietti M. A., and Rodríguez R. (1991), ‘On the asymptotic exactness of error estimators for linear triangular finite elements’, Numer. Math. 59, 107–127.
25. Durán R. and Rodríguez R. (1992), ‘On the asymptotic exactness of Bank–Weiser's estimator’, Numer. Math. 62, 297–303.
26. Eijkhout V. and Vassilevski P. (1991), ‘The role of the strengthened Cauchy–Buniakowskii–Schwarz inequality in multilevel methods’, SIAM Review 33, 405–419.
27. Golub G. H. and Van Loan C. F. (1983), Matrix Computations, Johns Hopkins University Press, Baltimore.
28. Golub G. H. and O'Leary D. P. (1989), ‘Some history of the conjugate gradient and Lanczos algorithms: 1948–1976’, SIAM Review 31, 50–102.
29. Griebel M. (1994), ‘Multilevel algorithms considered as iterative methods on semi-definite systems’, SIAM J. Sci. Comput. 15, 547–565.
30. Hackbusch W. (1985), Multigrid Methods and Applications, Springer, Berlin.
31. Maitre J. F. and Musy F. (1982), ‘The contraction number of a class of two level methods; an exact evaluation for some finite element subspaces and model problems’, in Multigrid Methods: Proceedings, Cologne 1981, Lecture Notes in Mathematics 960, Springer, Heidelberg, 535–544.
32. Ong E. (1989), ‘Hierarchical basis preconditioners for second order elliptic problems in three dimensions’, PhD thesis, University of Washington.
33. Oswald P. (1994), Multilevel Finite Element Approximation: Theory and Applications, Teubner Skripten zur Numerik, B. G. Teubner, Stuttgart.
34. Rüde U. (1993), Mathematical and Computational Techniques for Multilevel Adaptive Methods, Frontiers in Applied Mathematics 13, SIAM, Philadelphia.
35. Verfürth R. (1995), A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques, Teubner Skripten zur Numerik, B. G. Teubner, Stuttgart.
36. Weiser A. (1981), ‘Local-mesh, local-order, adaptive finite element methods with a posteriori error estimators for elliptic partial differential equations’, PhD thesis, Yale University.
37. Xu J. (1989), ‘Theory of multilevel methods’, PhD thesis, Cornell University.
38. Xu J. (1992), ‘Iterative methods by space decomposition and subspace correction’, SIAM Review 34, 581–613.
39. Yserentant H. (1986), ‘On the multi-level splitting of finite element spaces’, Numer. Math. 49, 379–412.
40. Yserentant H. (1992), ‘Old and new convergence proofs for multigrid methods’, in Acta Numerica, Cambridge University Press.
41. Zienkiewicz O. C., Kelley D. W., Gago J. P. d. S. R., and Babuška I. (1982), ‘Hierarchical finite element approaches, adaptive refinement, and error estimates’, in The Mathematics of Finite Elements and Applications, Academic Press, New York, 313–346.