Discrete weak duality of hybrid high-order methods for convex minimization problems

TL;DR

This paper establishes a discrete duality framework for hybrid high-order methods in convex minimization, enabling error estimation and adaptive refinement on general meshes.

Contribution

It introduces a novel discrete dual problem and postprocessing technique for hybrid high-order methods, enhancing error analysis and adaptivity.

Findings

The duality holds for general polyhedral meshes and polynomial degrees.

A new postprocessing method provides a posteriori error estimates.

Adaptive mesh refinement outperforms uniform refinement.

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 superiorly compared to uniform mesh refinements.