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.

本文讨论了凸极小化问题的非协调有限元方法，并系统地推导了双重混合公式。对偶关系导致简单的错误估计，避免了对不合格错误的显式处理。重构公式通过简单的后处理过程提供了对偶问题的离散解决方案，该过程隐含了强对偶关系，并且在后验误差估计中很有用。该框架适用于可​​微和非平滑问题，示例包括p -Laplace，正则化总变化和障碍问题。数值实验说明了不符合项优于标准符合性方法的优点。