Unstabilized Hybrid High-order Method for a Class of Degenerate Convex Minimization Problems

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.