SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1492-1514, August 2024.
Abstract. 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 better compared to uniform mesh refinements.

中文翻译：

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凸最小化问题的混合高阶方法的离散弱对偶性


《SIAM 数值分析杂志》，第 62 卷，第 4 期，第 1492-1514 页，2024 年 8 月。

抽象的。本文推导了解决凸最小化问题的典型混合高阶方法的离散对偶问题。离散原问题和对偶问题满足弱凸对偶性，这导致在附加平滑度假设下具有收敛率的先验误差估计。这种对偶性适用于一般的多面体网格和任意多项式的离散化程度。提出了一种新颖的后处理，并允许使用原始对偶技术对规则三角剖分进行后验误差估计。这催生了自适应网格细化算法，与均匀网格细化相比，该算法的性能更好。