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DIFFERENTIAL COMPLEXES AND STABILITY OF FINITE ELEMENT METHODS II: THE ELASTICITY COMPLEX
 

Summary: DIFFERENTIAL COMPLEXES AND STABILITY OF FINITE
ELEMENT METHODS II: THE ELASTICITY COMPLEX
DOUGLAS N. ARNOLD, RICHARD S. FALK, AND RAGNAR WINTHER
Abstract. A close connection between the ordinary de Rham complex and a cor-
responding elasticity complex is utilized to derive new mixed finite element methods
for linear elasticity. For a formulation with weakly imposed symmetry, this approach
leads to methods which are simpler than those previously obtained. For example, we
construct stable discretizations which use only piecewise linear elements to approximate
the stress field and piecewise constant functions to approximate the displacement field.
We also discuss how the strongly symmetric methods proposed in [8] can be derived in
the present framework. The method of construction works in both two and three space
dimensions, but for simplicity the discussion here is limited to the two dimensional case.
Key words. Mixed finite element method, Hellinger­Reissner principle, elasticity.
AMS(MOS) subject classifications. Primary: 65N30, Secondary: 74S05.
1. Introduction. In this paper we discuss finite element methods
for the equations of linear elasticity derived from the Hellinger­Reissner
variational principle. The equations can be written as a system of the form
A = u, div = f in . (1.1)
The unknowns and u denote the stress and displacement fields engendered
by a body force f acting on a linearly elastic body that occupies a region

  

Source: Arnold, Douglas N. - School of Mathematics, University of Minnesota

 

Collections: Mathematics