This paper derives a discrete dual problem for a prototypical hybrid high-order method for convex minimization problems. The discrete primal and dual problem satisfy a weak convex duality that leads to a priori error estimates with convergence rates under additional smoothness assumptions. This duality holds for general polytopal meshes and arbitrary polynomial degree of the discretization. A nouvelle postprocessing is proposed and allows for a~posteriori error estimates on simplicial meshes using primal-dual techniques. This motivates an adaptive mesh-refining algorithm, which performs superiorly compared to uniform mesh refinements.

翻译：本文针对凸最小化问题的典型混合高阶方法，推导了离散对偶问题。离散原问题与对偶问题满足弱凸对偶性，在额外光滑性假设下可导出具有收敛速率的先验误差估计。该对偶性适用于一般多面体网格及任意离散多项式次数。本文提出一种新型后处理方法，利用原-对偶技术在单纯形网格上实现后验误差估计，并据此设计自适应网格细化算法。数值实验表明，该算法性能显著优于均匀网格细化。