Numerische Mathematik

, Volume 92, Issue 3, pp 401–419 | Cite as

Mixed finite elements for elasticity

  • Douglas N. Arnold
  • Ragnar Winther
Original article


There have been many efforts, dating back four decades, to develop stable mixed finite elements for the stress-displacement formulation of the plane elasticity system. This requires the development of a compatible pair of finite element spaces, one to discretize the space of symmetric tensors in which the stress field is sought, and one to discretize the space of vector fields in which the displacement is sought. Although there are number of well-known mixed finite element pairs known for the analogous problem involving vector fields and scalar fields, the symmetry of the stress field is a substantial additional difficulty, and the elements presented here are the first ones using polynomial shape functions which are known to be stable. We present a family of such pairs of finite element spaces, one for each polynomial degree, beginning with degree two for the stress and degree one for the displacement, and show stability and optimal order approximation. We also analyze some obstructions to the construction of such finite element spaces, which account for the paucity of elements available.

Mathematics Subject Classification (1991): 65N30, 74S05 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Douglas N. Arnold
    • 1
  • Ragnar Winther
    • 2
  1. 1.Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455, USA; e-mail: US
  2. 2.Department of Informatics, University of Oslo, Oslo, Norway; e-mail: NO

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