Title:         Mixed finite elements for elasticity in the stress-displacement formulation

Authors:       Douglas N. Arnold and Ragnar Winther

Source:        in: "Current Trends in Scientific Computing" (Z. Chen, R.
Glowinski, and K. Li, eds), Contemporary Mathematics, AMS, 2003

Status:        Published

Abstract:      We present a family of pairs of finite element spaces for
the unaltered Hellinger--Reissner variational principle using
polynomial shape functions on a single triangular mesh for stress and
displacement. There is a member of the family for each polynomial
degree, beginning with degree two for the stress and degree one for the
displacement, and each is stable and affords optimal order
approximation. The simplest element pair involves 24 local degrees of
freedom for the stress and 6 for the displacement.  We also construct a
lower order element involving 21 stress degrees of freedom and 3
displacement degrees of freedom which is, we believe, likely to be the
simplest possible conforming stable element pair with polynomial shape
functions. For all these conforming elements the approximate stress not
only belongs to H(div), but is also continuous at element vertices,
which is more continuity than may be desired.  We show that for
conforming finite elements with polynomial shape functions, this
additional continuity is unavoidable.  To overcome this obstruction, we
construct as well some non-conforming stable mixed finite elements,
which we show converge with optimal order as well.  The simplest of
these involves only 12 stress and 6 displacement degrees of freedom on
each triangle.

Keywords:      mixed method, finite element, elasticity

Subj. class.:  65N30, 74S05

URL:           http://ima.umn.edu/~arnold/papers/icmsc.pdf