Unstabilized Hybrid High-Order method for a class of degenerate convex minimization problems
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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∈ H(div,Ω;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(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.
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