Abstract
We develop a new approach to a posteriori error estimation for Galerkin finite element approximations of symmetric and nonsymmetric elliptic eigenvalue problems. The idea is to embed the eigenvalue approximation into the general framework of Galerkin methods for nonlinear variational equations. In this context residual-based a posteriori error representations are available with explicitly given remainder terms. The careful evaluation of these error representations for the concrete situation of an eigenvalue problem results in a posteriori error estimates for the approximations of eigenvalues as well as eigenfunctions. These suggest local error indicators that are used in the mesh refinement process.
Similar content being viewed by others
References
I. Babuška and J. Osborn, A posteriori error estimates for the finite element method, Internat. J. Numer. Methods Engrg. 12 (1978) 1597–1615.
I. Babuška and J.E. Osborn, Eigenvalue problems, in: eds. P.G. Ciarlet and J.L. Lions, Handbook of Numerical Analysis, Vol. 2, Chapter Finite Element Methods (Part 1) (Elsevier, Paris, 1991) pp. 641–792.
I. Babuška and T. Tsuchiya, A posteriori error estimates of finite element solutions of parametrized nonlinear equations, Technical report, University of Maryland (1992).
R. Becker and R. Rannacher, Weighted a posteriori error control in FE methods, in: Proc. of ENUMATH' 97, eds. H.G. Bock et al. (World Scientific, Singapore, 1995) pp. 609–637.
R. Becker and R. Rannacher, A feed-back approach to error control in finite element methods: basic analysis and examples, East-West J. Numer. Math. 4 (1996) 237–264.
R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, in: Acta Numerica 2001, ed. A. Iserles (Cambridge Univ. Press, Cambridge, 2001) pp. 1–102.
J.H. Bramble and J.E. Osborn, Rate of convergence estimates for nonselfadjoint eigenvalue approximations, Math. Comp. 27 (1973) 525–545.
S.C. Brenner and R.L. Scott, The Mathematical Theory of Finite Element Methods (Springer, Berlin, 1994).
P.G. Ciarlet, Finite Element Methods for Elliptic Problems (North-Holland, Amsterdam, 1978).
K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Introduction to adaptive methods for differential equations, in: Acta Numerica 1995, ed. A. Iserles (Cambridge Univ. Press, 1995) pp. 105–158.
V. Heuveline and C. Bertsch, On multigrid methods for the eigenvalue computation of non-selfadjoint elliptic operators, East-West J. Numer. Math. 8 (2000) 275–297.
C. Johnson, A new paradigm for adaptive finite element methods, in: Proc. of MAFELAP '93, ed. J. Whiteman (Wiley, New York, 1994) pp. 105–120.
T. Kato, Perturbation Theory for Linear Operators (Springer, Berlin, 1966).
M.G. Larson, A posteriori and a priori error analysis for finite element approximations of self-adjoint elliptic eigenvalue problems, SIAM J. Numer. Anal. 38 (2000) 608–625.
C. Nystedt, A priori and a posteriori error estimates and adaptive finite element methods for a model eigenvalue problem, Technical Report No. 1995-05, Department of Mathematics, Chalmers University of Technology (1995).
J.E. Osborn, Spectral approximation for compact operators, Math. Comp. 29 (July 1975) 712–725.
R. Verfürth, A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques (Wiley/Teubner, New York/Stuttgart, 1996).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Heuveline, V., Rannacher, R. A posteriori error control for finite element approximations of elliptic eigenvalue problems. Advances in Computational Mathematics 15, 107–138 (2001). https://doi.org/10.1023/A:1014291224961
Issue Date:
DOI: https://doi.org/10.1023/A:1014291224961