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 polyhedral meshes and arbitrary polynomial degrees of the discretization. A novel postprocessing is proposed and allows for a~posteriori error estimates on regular triangulations into simplices using primal-dual techniques. This motivates an adaptive mesh-refining algorithm, which performs superiorly compared to uniform mesh refinements.

翻译：本文针对凸极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得到具有收敛率的先验误差估计。该对偶性适用于一般多面体网格及任意离散多项式次数。本文提出一种新颖的后处理方法，利用原-对偶技术可在单纯形正则三角剖分上实现后验误差估计。这激发了自适应网格细化算法，其性能显著优于均匀网格细化。