  We introduce a new class of mixed finite element methods for 2D and 3D
compressible nonlinear elasticity. The independent unknowns of these conformal
methods are displacement, displacement gradient, and the first Piola-Kirchhoff
stress tensor. The so-called edge finite elements of the curl operator is
employed to discretize the trial space of displacement gradients. Motivated by
the differential complex of nonlinear elasticity, this choice guarantees that
discrete displacement gradients satisfy the Hadamard jump condition for the
strain compatibility. We study the stability of the proposed mixed finite
element methods by deriving some inf-sup conditions. By considering 32 choices
of simplicial conformal finite elements of degrees 1 and 2, we show that 10
choices are not stable as they do not satisfy the inf-sup conditions. We
numerically study the stable choices and conclude that they can achieve optimal
convergence rates. By solving several 2D and 3D numerical examples, we show
that the proposed methods are capable of providing accurate approximations of
strain and stress.
