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

TL;DR

This work develops a discrete weak duality theory for a prototypical hybrid high-order (HHO) method applied to convex minimization problems with energy $E(v)=\int_\Omega (\Psi(\mathrm{D}v)+\psi(x,v))\,dx$, and its dual $E^*(\tau)=-\int_\Omega (\Psi^*(\tau)+\psi^*(x,\mathrm{div}\tau))\,dx$. By constructing a gradient/divergence/potential reconstruction suite and a dual potential reconstruction within the HHO framework on general polyhedral meshes, the authors prove a discrete weak duality $E_h^*(\tau_h) \le E_h(v_h)$ and derive a priori error estimates with convergence rates under smoothness assumptions. A novel $W^{p'}(\mathrm{div},\Omega;\mathbb{M})$-conforming postprocessing $\sigma_0$ enables an a posteriori error estimate via a primal-dual gap, which is localized on regular triangulations and supports an adaptive mesh-refinement algorithm shown to outperform uniform refinement. Numerical experiments across adaptive refinement, optimal design, Bingham flow, and $p$-Laplace problems demonstrate the practical effectiveness of the discrete duality framework and postprocessing in achieving reliable error control and improved convergence. Overall, the paper provides a robust pathway to reliable a posteriori analysis and adaptive strategies for high-order, polyhedral-HHO discretizations of convex minimization problems.

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.

Figures (5)

• Figure 1: (a) Initial triangulation of the L-shaped domain into 6 triangles and (b) material distribution in the optimal design problem of \ref{['sec:odp']}
• Figure 2: (a) Convergence history plot of $E_{\sigma_0}(v_0) - E^*_{\sigma_0}(\sigma_0)$ for various $k$ and (b) adaptive triangulation into 2013 triangles obtained with $k = 2$ in \ref{['sec:odp']}
• Figure 3: Convergence history plot of $E_{\sigma_0^\varepsilon}(v_0^\varepsilon) - E_{\sigma_0^\varepsilon}^*(\sigma_0^\varepsilon)$ for (a) various $k$ and $\varepsilon = 10^{-4}$ and (b) $k = 2$ and various $\varepsilon$ in \ref{['sec:Bingham-flow']}
• Figure 4: Convergence history plot of (a) RHS and (b) $\|\sqrt{\varrho}\nabla(u - v_0)\|^2_2$ for various $k$ in \ref{['sec:plaplace']}
• Figure 5: Convergence history plot of (a) $\|\sigma - \nabla \Psi(\nabla v_0)\|_{4/3}^2$ and (b) $\|\nabla(u - v_0)\|_4^2$ for various $k$ in \ref{['sec:plaplace']}

Theorems & Definitions (20)

• Lemma 2.1: commuting property
• Lemma 2.2: well-definedness of $\mathcal{R}^*_h$
• proof
• Theorem 3.1: weak duality
• Lemma 3.2: discrete integration by parts
• proof
• proof : Proof of \ref{['thm:weak-duality']}
• Example 3.3: classical stabilization
• Theorem 4.1: a priori
• proof