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

A Conformal Three-Field Formulation for Nonlinear Elasticity: From Differential Complexes to Mixed Finite Element Methods