Abstract
We show how to postprocess the approximate displacement given by the continuous Galerkin method for compressible linearly elastic materials to obtain an optimally convergent approximate stress that renders the method locally conservative. The postprocessing is extremely efficient as it requires the inversion of a symmetric, positive definite matrix whose condition number is independent of the mesh size. Although the new stress is not symmetric, its asymmetry can be controlled by a small term which is of the same order as that of the error of the approximate stress itself. The continuous Galerkin method is thus competitive with mixed methods providing stresses with similar convergence properties.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002)
Auricchio, F., Brezzi, F., Lovadina, C.: Mixed finite element methods. In: de Borst, R., Stein, E., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, vol. 1, pp. 235–278. Wiley, England (2004)
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)
Brezzi, F., Arnold, D.N., Douglas, J. Jr.: Peers: a new mixed finite element for plane elasticity. Jpn. J. Appl. Math. 1, 347–367 (1984)
Brezzi, F., Douglas, J. Jr., Marini, L.D.: Recent results on mixed finite element methods for second order elliptic problems. In: Vistas in Applied Mathematics. Numerical Analysis, Atmospheric Science and Immunology, pp. 25–43. Springer, Heidelberg (1986)
Cockburn, B., Gopalakrishnan, J., Wang, H.: Locally conservative fluxes for the continuous Galerkin method. SIAM J. Numer. Anal. 45, 1742–1776 (2007)
Stein, E., Rüter, M.: Finite element methods for elasticity with error-controlled discretization and model adaptivity. In: de Borst, R., Stein, E., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, vol. 2, pp. 5–58. Wiley, England (2004)
Stenberg, R.: A family of mixed finite elements for the elasticity problem. Numer. Math. 53, 513–538 (1988)
Winther, R., Arnold, D.N.: Mixed finite elements for elasticity. Numer. Math. 92, 401–419 (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author was supported in part by the National Science Foundation (Grant DMS-0411254) and by the University of Minnesota Supercomputing Institute.
Rights and permissions
About this article
Cite this article
Cockburn, B., Wang, H. The Computation of a Locally Conservative Stress for the Continuous Galerkin Method for Compressible Linearly Elastic Materials. J Sci Comput 36, 151–163 (2008). https://doi.org/10.1007/s10915-007-9182-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-007-9182-4