SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1348-1373, January 2021.
The relaxation in the calculus of variation motivates the numerical analysis of a class of degenerate convex minimization problems with nonstrictly convex energy densities with some convexity control and two-sided $p$-growth. The minimizers may be nonunique in the primal variable but lead to a unique stress $\sigma \in H({div},\Omega;\mathbb{M})$. 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 Brezzi--Douglas--Marini 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 $\sigma_h$. 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.

中文翻译：

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一类退化凸最小化问题的不稳定稳定高阶方法

SIAM数值分析学报，第59卷，第3期，第1348-1373页，2021年1月。
变分演算的弛豫促使了对一类退化的凸最小化问题的数值分析，该问题具有非严格的凸能量密度，具有一定的凸控制和两侧的$ p $-增长。极小值在原始变量中可能是不唯一的，但会导致唯一的应力$ \ sigma \ in H（{div}，\ Omega; \ mathbb {M}）$。示例包括$ p $ -Laplacian，拓扑优化中的最佳设计问题和凸双阱问题。通过混合高阶方法（HHO）进行的逼近利用分段Raviart-Thomas或Brezzi-Douglas-Marini有限元的梯度重建方法，而无需将规则三角剖分的稳定化为单纯形。将此HHO方法应用于退化凸极小化问题类别，可以使唯一的$ H（{div}）$符合应力近似值$ \ sigma_h $。主要结果是对Lebesgue规范中的应力误差$ \ sigma- \ sigma_h $的先验和后验误差估计，以及可计算的较低能界。数值基准对于较高的多项式度显示较高的收敛速度，并且包括自适应网格细化，其中第一个超线性收敛速度可确保较低的能量范围。