Nonconforming discretizations of convex minimization problems and precise relations to mixed methods

Abstract: 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.