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.

翻译：在间断伽辽金方法的框架下，我们研究了与凸能量相关的非线性变分问题的近似求解方法。我们提出了单元层面的非协调有限元方法来离散化连续极小化问题。通过$\Gamma$-收敛性论证，我们证明了在离散能量层面引入适当定义的罚项后，当网格参数趋于零时，离散极小化子收敛于连续问题的唯一极小化子。最后，我们通过数值算例验证了所提方法的可行性。