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

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