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. The key is to design a finite element
discretization for the space of square-integrable symmetric tensor
fields with square-integrable divergence, which enjoys a variety of
properties. Although there are a number of well-known discretizations
of H(div) vector fields with the analogous properties, such finite
elements for symmetric tensor fields have proven very hard to design.
We will present a new family of such elements, one for each polynomial
degree quadratic and above, and also analyze the obstructions to the
construction of such elements which account for the paucity of
elements available. A star supporting role will be played by the de
Rham sequence and related sequences of partial differential operators,
and their discrete analogues.
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