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

1. Problem Statement

The paper addresses the numerical approximation of convex minimization problems of the form:
$$\min_{v \in V_D} E(v) := \int_{\Omega} \left( \Psi(Dv) + \psi(x, v) \right) dx$$
where:

• $\Omega \subset \mathbb{R}^n$ is a bounded polyhedral domain.
• $V_D$ is a Sobolev space with homogeneous Dirichlet boundary conditions.
• $\Psi$ is a proper, lower semicontinuous, convex energy density (e.g., related to $p$-Laplace or plasticity).
• $\psi$ is a convex potential term.

The goal is to develop a Hybrid High-Order (HHO) discretization that preserves the weak duality between the primal (minimization) and dual (maximization) problems at the discrete level. This duality is crucial for rigorous error analysis (both a priori and a posteriori) and adaptive mesh refinement.

2. Methodology

2.1. Discretization Framework

The author employs the Hybrid High-Order (HHO) method on general polyhedral meshes.

• Primal Space ($V(M)$): Defined as $P_{k+1}(M; \mathbb{R}^m) \times P_k(\Sigma; \mathbb{R}^m)$, consisting of cell-based polynomials of degree $k+1$ and face-based polynomials of degree $k$ on the mesh skeleton $\Sigma$.
• Dual Space ($W(M)$): Defined as $P_k(M; \mathbb{M}) \times P_k(\Sigma; \mathbb{R}^m)$, where $\mathbb{M} = \mathbb{R}^{m \times n}$.
• Reconstruction Operators:
  • Gradient Reconstruction ($D_h$): Maps primal variables to piecewise polynomials of degree $k$.
  • Divergence Reconstruction ($\text{div}_h$): Maps dual variables to piecewise polynomials of degree $k$.
  • Potential Reconstruction ($R_h$ and $R_h^*$): Crucial for the duality proof. A novel dual potential reconstruction $R_h^*$ is introduced on the dual level to map dual variables back to a space compatible with the primal gradient reconstruction.

2.2. Stabilization

To handle the non-trivial kernel of the reconstruction operators, a stabilization term is required.

• The paper utilizes a nonlinear Lehrenfeld-Schöberl stabilization ($s_h$) for the primal problem and a corresponding dual stabilization ($\gamma_h$).
• Unlike classical HHO stabilizations that penalize jumps directly, this formulation is designed specifically to facilitate the integration by parts required for duality.

2.3. Discrete Weak Duality

The core theoretical achievement is proving the Discrete Weak Duality:
$$\max_{\tau_h \in W_N(M)} E_h^*(\tau_h) \leq \min_{v_h \in V_D(M)} E_h(v_h)$$
where $E_h$ and $E_h^*$ are the discrete primal and dual energies.

• Proof Mechanism: The proof relies on a discrete integration by parts formula involving the three reconstruction operators ($D_h$, $\text{div}_h$, $R_h^*$).
• Key Identity: The duality gap is bounded by the sum of the primal and dual stabilization terms, derived via Fenchel-Young inequalities and careful manipulation of boundary integrals on the mesh skeleton.

3. Key Contributions

1. First Discrete Weak Duality for HHO on Polyhedral Meshes:
   Previous methods preserving duality (like Crouzeix-Raviart or Raviart-Thomas) were restricted to regular simplicial meshes. This paper extends the duality property to general polyhedral meshes and arbitrary polynomial degrees using HHO methods.

2. Novel Dual Potential Reconstruction:
   The introduction of the operator $R_h^*$ allows for the construction of a discrete dual problem that satisfies the weak duality inequality. This operator is defined via a variational formulation involving the divergence reconstruction and a projection onto the orthogonal complement of the gradient space.

3. A Priori Error Estimates:
   The established duality leads to a priori error estimates with optimal convergence rates ($O(h^{k+1})$) under suitable smoothness assumptions. The analysis covers general convex energies, including the $p$-Laplace problem.

4. A Posteriori Error Estimation and Postprocessing:

   • A novel conforming postprocessing $\sigma_0$ (in $RT_k \cap W_N$) is constructed on a simplicial submesh.
   • This postprocessing is based on the discrete Euler-Lagrange equations, ensuring "equilibrated tractions" (normal jumps of the stress match the stabilization terms).
   • This allows for the computation of a primal-dual gap error estimator, which provides guaranteed upper bounds on the error without requiring additional consistency terms.

5. Adaptive Mesh Refinement:
   The paper proposes an adaptive algorithm driven by the primal-dual error estimator. Numerical results demonstrate that this approach significantly outperforms uniform mesh refinement, particularly for problems with singularities.

4. Results and Numerical Benchmarks

The paper validates the theory through three numerical benchmarks on an L-shaped domain:

• Optimal Design Problem:

  • Involves a non-smooth energy density (two-phase material).
  • Adaptive refinement successfully captures the transition layers and singularities.
  • Adaptive rates are superior to uniform rates ($O(\text{ndof}^{-5/4})$ vs $O(\text{ndof}^{-2/3})$ for $k=2$).

• Bingham Flow in a Pipe:

  • Models viscoplastic flow with a yield stress (non-differentiable energy).
  • Uses a regularization technique to handle non-differentiability.
  • Adaptive refinement recovers optimal convergence rates ($O(\text{ndof}^{-(k+1)})$) for the error estimator, whereas uniform meshes suffer from reduced rates due to the singularity at the origin.

• $p$-Laplace Problem ($p=4$):

  • Tests the method on a problem with a known singular solution ($r^{7/8}$).
  • The study highlights the sensitivity of convergence rates to the stabilization parameters ($r$ and $s$).
  • With optimal parameter choices ($s \approx (p-2)(k+1) + p-1$), the method achieves optimal convergence rates for both the primal-dual gap and the displacement error.

5. Significance

• Theoretical Advancement: This work bridges a gap in the mathematical theory of HHO methods by establishing a rigorous duality framework previously unavailable for high-order methods on general polyhedral meshes.
• Practical Utility: The derived a posteriori error estimator is computable and reliable. It enables efficient adaptive mesh refinement, which is critical for solving complex engineering problems (like plasticity and fluid flow) where solutions exhibit singularities or sharp gradients.
• Flexibility: The methodology is robust across different polynomial degrees and mesh types, making it a powerful tool for high-fidelity simulations of convex minimization problems.

In summary, Ngoc Tien Tran successfully demonstrates that Hybrid High-Order methods can be formulated to preserve the fundamental weak duality of convex optimization, unlocking powerful error estimation and adaptive capabilities for a wide class of non-linear problems on complex geometries.