Unstabilized Hybrid High-Order method for a class of degenerate convex minimization problems

TL;DR

The paper develops and analyzes an unstabilized Hybrid High-Order (HHO) method for a class of degenerate convex minimization problems with two-sided $p$-growth, where the primal minimizers may be non-unique but the stress $\sigma=DW(Du)$ is unique in $H(\mathrm{div})$. By reconstructing gradients into Raviart-Thomas or BDM spaces and using a no-stabilization approach, the method yields a conforming stress approximation $\sigma_h$ and a computable lower energy bound (LEB). The authors establish rigorous a priori and a posteriori estimates for the stress error $\|\sigma-\sigma_h\|_{L^{p'}}$ and the energy gap, with a convergence rate $\|\sigma-\sigma_h\|_{L^{p'}} + |E(u)-E_h(u_h)| \lesssim h_{\max}^{(k+1)/r}$ for smooth data, and demonstrate superlinear LEB convergence under refinement. Numerical benchmarks on $p$-Laplace, topology optimization, and relaxed double-well problems confirm the theoretical rates and show that adaptive mesh refinement yields robust, higher-order convergence, especially for higher polynomial degrees $k$.

Abstract

The relaxation in the calculus of variation motivates the numerical analysis of a class of degenerate convex minimization problems with non-strictly convex energy densities with some convexity control and two-sided $p$-growth. The minimizers may be non-unique in the primal variable but lead to a unique stress $σ\in H(\operatorname{div},Ω;\mathbb{M})$. Examples include the p-Laplacian, an optimal design problem in topology optimization, and the convexified double-well problem. The approximation by hybrid high-order methods (HHO) utilizes a reconstruction of the gradients with piecewise Raviart-Thomas or BDM finite elements without stabilization on a regular triangulation into simplices. The application of this HHO method to the class of degenerate convex minimization problems allows for a unique $H(\operatorname{div})$ conforming stress approximation $σ_h$. The main results are a~priori and a posteriori error estimates for the stress error $σ-σ_h$ in Lebesgue norms and a computable lower energy bound. Numerical benchmarks display higher convergence rates for higher polynomial degrees and include adaptive mesh-refining with the first superlinear convergence rates of guaranteed lower energy bounds.

Figures (11)

• Figure 1: Initial triangulation $\mathcal{T}_0$ of the square (left) and of the L-shaped domain (right)
• Figure 2: Polynomial degrees $k = 0,\dots,4$ in the numerical benchmarks of \ref{['sec:numericalExamples']}
• Figure 3: Convergence history plot of RHS (solid line), $\|\mathop{\mathrm{D}}\nolimits u - R u_h\|^2_{L^4(\Omega)}$ (dashed line), and $\|\sigma - \sigma_h\|_{L^{4/3}(\Omega)}^2$ (dotted line) in \ref{['sec:expLaplaceSquare']} for $4$-Laplace with $k$ from \ref{['fig:legend']} on uniform (left) and adaptive (right) meshes
• Figure 4: Convergence history plot of $E(u) - \mathrm{LEB}$ (solid line) and discrete duality gap $E_h(u_h) - E^*(\sigma_h)$ (dashed line) in \ref{['sec:expLaplaceSquare']} for $4$-Laplace with $k$ from \ref{['fig:legend']} on uniform (left) and adaptive (right) meshes
• Figure 5: Adaptive mesh of L-shape domain for $4$-Laplace in \ref{['sec:expLaplaceLshape']} with (a) 431 triangles (1055 dof) for $k = 0$ (left) and (b) 481 triangles (10710 dof) for $k = 4$ (right)

Theorems & Definitions (29)

• Theorem 1.1: a priori
• Theorem 1.2: a posteriori
• Lemma 2.1
• Proof 1
• Remark 2.2: monotonicity
• Theorem 2.3
• Proof 2
• Lemma 3.1
• Proof 3
• Theorem 3.2