Unstabilized Hybrid High-Order method for a class of degenerate convex minimization problems
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| Format: | Preprint |
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2020
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| author | Carstensen, C. Tran, N. T. |
| author_facet | Carstensen, C. Tran, N. T. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2011_15059 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Unstabilized Hybrid High-Order method for a class of degenerate convex minimization problems Carstensen, C. Tran, N. T. Numerical Analysis 65N12 (Primary), 65N30, 65Y20 (Secondary) 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. |
| title | Unstabilized Hybrid High-Order method for a class of degenerate convex minimization problems |
| topic | Numerical Analysis 65N12 (Primary), 65N30, 65Y20 (Secondary) |
| url | https://arxiv.org/abs/2011.15059 |