  In the context of Discontinuous Galerkin methods, we study approximations of
nonlinear variational problems associated with convex energies. We propose
element-wise nonconforming finite element methods to discretize the continuous
minimisation problem. Using $\Gamma$-convergence arguments we show that the
discrete minimisers converge to the unique minimiser of the continuous problem
as the mesh parameter tends to zero, under the additional contribution of
appropriately defined penalty terms at the level of the discrete energies. We
finally substantiate the feasibility of our methods by numerical examples.
