A class of discontinuous Galerkin methods for nonlinear variational problems

Abstract:

In the context of discontinuous Galerkin methods, we study approximations of nonlinear variational problems. We propose element-wise nonconforming finite element methods to discretize the continuous minimisation problem. Using $\Gamma$-convergence arguments we show that for convex energies the discrete minimisers converge to a 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. In addition, we provide error analysis in the case where the solution of the continuous minimization problem is smooth enough. Under appropriate assumptions we show local existence, uniqueness and corresponding optimal order error estimates for the discrete Euler-Lagrange equations of our scheme. We finally substantiate the feasibility of our methods by numerical examples.