  In this work we introduce a dG framework for nonlinear elasticity based on a
Bassi-Rebay (BR2) formulation. The framework encompasses compressible and
incompressible hyperelastic materials and is capable of dealing with large
deformations. In order to achieve stability, we combine higher-order lifting
operators for the BR2 stabilization term with an adaptive stabilization
strategy which relies on the BR2 Laplace operator stabilization and a penalty
parameter based on the spectrum of the fourth-order elasticity tensor.
Dirichlet boundary conditions for the displacement can be imposed by means of
Lagrange multipliers and Nitsche method. Efficiency of the solution strategy is
achieved by means of state-of-the-art agglomeration based $h$-multigrid
preconditioners and the code implementation supports distributed memory
execution on modern parallel architectures. Several benchmark test cases are
proposed in order to investigate some relevant computational aspects, namely
the performance of the $h$-multigrid iterative solver varying the stabilization
parameters and the influence of Dirichlet boundary conditions on Newton's
method globalisation strategy.
