Mixed finite elements for elasticity in the stress-displacement formulation

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.