一类退化凸极小化问题的无稳定化混合高阶方法
Unstabilized Hybrid High-Order method for a class of degenerate convex minimization problems
AI总结：
本文针对一类退化凸极小化问题，提出无稳定化的混合高阶方法，利用分片 Raviart-Thomas 或 BDM 有限元重构梯度，获得唯一协调应力逼近，并给出先验与后验误差估计及可计算下能量界。
摘要：
变分法中的松弛化促使对一类退化凸极小化问题进行数值分析，这类问题具有非严格凸的能量密度，并带有一定的凸性控制及双边 $p$-增长条件。极小化元在原变量上可能不唯一，但会导出唯一的应力 $σ\in H(\operatorname{div},Ω;\mathbb{M})$。典型例子包括 p-Laplacian、拓扑优化中的一个最优设计问题以及凸化双井问题。采用混合高阶方法（HHO）进行逼近时，利用分片 Raviart-Thomas 或 BDM 有限元对梯度进行重构，在单纯形正则三角剖分上无需稳定化。将该 HHO 方法应用于这类退化凸极小化问题，可以得到唯一的 $H(\operatorname{div})$ 协调应力逼近 $σ_h$。主要结果包括应力误差 $σ-σ_h$ 在 Lebesgue 范数下的先验与后验误差估计，以及一个可计算的下能量界。数值基准测试表明，更高的多项式次数可获得更高的收敛阶，并包含自适应网格细化，首次实现了保证下能量界的超线性收敛速率。
英文摘要：
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.
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