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

TL;DR

This paper introduces a novel unstabilized hybrid high-order method for degenerate convex minimization problems, achieving accurate stress approximations and superlinear convergence in adaptive mesh refinement.

Contribution

It develops an unstabilized hybrid high-order approach that provides unique stress solutions and error estimates for a class of degenerate convex minimization problems, including p-Laplacian and topology optimization.

Findings

Higher convergence rates with increased polynomial degree

Effective adaptive mesh refinement with superlinear convergence

Reliable lower energy bounds for solution quality

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 $\sigma \in H(\operatorname{div},\Omega;\mathbb{M})$. 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 $\sigma_h$. The main results are a~priori and a posteriori error estimates for the stress error $\sigma-\sigma_h$ 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.