  This article discusses nonconforming finite element methods for convex
minimization problems and systematically derives dual mixed formulations.
Duality relations lead to simple error estimates that avoid an explicit
treatment of nonconformity errors. A reconstruction formula provides the
discrete solution of the dual problem via a simple postprocessing procedure
which implies a strong duality relation and is of interest in a posteriori
error estimation. The framework applies to differentiable and nonsmooth
problems, examples include $p$-Laplace, total-variation regularized, and
obstacle problems. Numerical experiments illustrate advantages of nonconforming
over standard conforming methods.
