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Contemporary Mathematics Mixed Finite Elements for Elasticity in the

Summary: Contemporary Mathematics
Mixed Finite Elements for Elasticity in the
Stress-Displacement Formulation
Douglas N. Arnold and Ragnar Winther
Abstract. We present a family of pairs of finite element spaces for the unal-
tered Hellinger­Reissner 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 or-
der 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 conform-
ing 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


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


Collections: Mathematics