Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-26T20:56:50.796Z Has data issue: false hasContentIssue false
  • English
  • Français

Hierarchical bases and the finite element method

Published online by Cambridge University Press:  07 November 2008

Randolph E. Bank
Randolph E. Bank
Affiliation:
Department of MathematicsUniversity of California at San DiegoLa Jolla, CA 92093, USA E-mail: rbank@ucsd. edu
Get access Rights & Permissions [Opens in a new window]

Extract

In this work we present a brief introduction to hierarchical bases, and the important part they play in contemporary finite element calculations. In particular, we examine their role in a posteriori error estimation, and in the formulation of iterative methods for solving the large sparse sets of linear equations arising from finite element discretization.

Type
Research Article
Information
Acta Numerica , Volume 5 , January 1996 , pp. 1 - 43
Copyright
Copyright © Cambridge University Press 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

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.CrossRefGoogle Scholar
Ainsworth, M. and Oden, J. T. (1993), ‘A unified approach to a posteriori error estimation using element residual methods’, Numer. Math. 65, 23–50.CrossRefGoogle Scholar
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.Google Scholar
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.CrossRefGoogle Scholar
Babuška, I., Zienkiewicz, O. C., Gago, J. P. de S.R., and de Arantes e Oliveira, E. R., eds (1986) Accuracy Estimates and Adaptive Refinements in Finite Element Computations, Wiley, New York.Google Scholar
Babuška, I. and Rheinboldt, W. C. (1978), ‘Error estimates for adaptive finite element computations’, SIAM J. Numer. Anal. 15, 736–754.CrossRefGoogle Scholar
Babuška, I. and Rheinboldt, W. C. (1978), ‘A posteriori error estimates for the finite element method’, Internat. J. Numer. Methods Engrg. 12, 1597–1615.CrossRefGoogle Scholar
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.Google Scholar
Bank, R. E. and Dupont, T. F. (1979), ‘Notes on the k-level iteration’, unpublished notes.Google Scholar
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.CrossRefGoogle Scholar
Bank, R. E., Dupont, T. F., and Yserentant, H. (1988), ‘The hierarchical basis multigrid method’, Numer. Math. 52, 427–458.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Bank, R. E. and Smith, R. K. (1993), ‘A posteriori error estimates based on hierarchical bases, SIAM J. Numer. Anal. 30, 921–935.CrossRefGoogle Scholar
Bank, R. E. and Weiser, A. (1985), ‘Some a posteriori error estimates for elliptic partial differential equations’, Math. Comp. 44, 283–301.CrossRefGoogle Scholar
Bornemann, F. and Yserentant, H. (1993), ‘A basic norm equivalence for the theory of multilevel methods’, Numer. Math. 64, 455–476.CrossRefGoogle Scholar
Braess, D. (1981), ‘The contraction number of a multigrid method for solving the Poisson equation’, Numer. Math. 37, 387–404.CrossRefGoogle Scholar
Bramble, J. H. (1993), Multigrid Methods, Pitman Research Notes in Mathematical Sciences 294, Longman Sci. & Techn., Harlow, UK.Google Scholar
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.CrossRefGoogle Scholar
Bramble, J. H., Pasciak, J. E., and Xu, J. (1990), ‘Parallel multilevel preconditioners’, Math. Comp. 55, 1–22.CrossRefGoogle Scholar
Brenner, S. C. and Scott, L. R. (1994), The Mathematical Theory of Finite Element Methods, Springer, Heidelberg.CrossRefGoogle Scholar
Ciarlet, P. G. (1980), The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam.Google Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Durán, R. and Rodríguez, R. (1992), ‘On the asymptotic exactness of Bank–Weiser's estimator’, Numer. Math. 62, 297–303.CrossRefGoogle Scholar
Eijkhout, V. and Vassilevski, P. (1991), ‘The role of the strengthened Cauchy–Buniakowskii–Schwarz inequality in multilevel methods’, SIAM Review 33, 405–419.CrossRefGoogle Scholar
Golub, G. H. and Van Loan, C. F. (1983), Matrix Computations, Johns Hopkins University Press, Baltimore.Google Scholar
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.CrossRefGoogle Scholar
Griebel, M. (1994), ‘Multilevel algorithms considered as iterative methods on semi-definite systems’, SIAM J. Sci. Comput. 15, 547–565.CrossRefGoogle Scholar
Hackbusch, W. (1985), Multigrid Methods and Applications, Springer, Berlin.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Ong, E. (1989), ‘Hierarchical basis preconditioners for second order elliptic problems in three dimensions’, PhD thesis, University of Washington.Google Scholar
Oswald, P. (1994), Multilevel Finite Element Approximation: Theory and Applications, Teubner Skripten zur Numerik, B. G. Teubner, Stuttgart.CrossRefGoogle Scholar
Rüde, U. (1993), Mathematical and Computational Techniques for Multilevel Adaptive Methods, Frontiers in Applied Mathematics 13, SIAM, Philadelphia.CrossRefGoogle Scholar
Verfürth, R. (1995), A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques, Teubner Skripten zur Numerik, B. G. Teubner, Stuttgart.Google Scholar
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.Google Scholar
Xu, J. (1989), ‘Theory of multilevel methods’, PhD thesis, Cornell University.Google Scholar
Xu, J. (1992), ‘Iterative methods by space decomposition and subspace correction’, SIAM Review 34, 581–613.CrossRefGoogle Scholar
Yserentant, H. (1986), ‘On the multi-level splitting of finite element spaces’, Numer. Math. 49, 379–412.CrossRefGoogle Scholar
Yserentant, H. (1992), ‘Old and new convergence proofs for multigrid methods’, in Acta Numerica, Cambridge University Press.Google Scholar
Zienkiewicz, O. C., Kelley, D. W., Gago, J. P. de 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.Google Scholar