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：本文推导了用于凸极小化问题的原型混合高阶方法的离散对偶问题。离散原始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阶数。本文提出一种新型后处理方法，通过原始-对偶技术允许在单纯形网格上进行后验误差估计。这促使了一种自适应网格细化算法，其性能优于均匀网格细化。