论文标题
一类简并最小化问题的未稳定的混合高级方法
Unstabilized Hybrid High-Order method for a class of degenerate convex minimization problems
论文作者
论文摘要
变异的计算中的放松激发了对一类简并凸的最小化问题的数值分析，这些问题具有一定的凸面控制和两侧$ p $增长的非刻痕凸电能密度。在原始变量中，最小化器可能不是唯一的，但导致h（\ peripatorName {div}，ω; \ mathbb {m}）$中的唯一压力$σ\。示例包括P-Laplacian，拓扑优化方面的最佳设计问题以及凸的双孔问题。混合高级方法（HHO）的近似值利用了分段raviart-thomas或BDM有限元的梯度重建，而无需在常规三角测量中稳定在简单中。将此HHO方法应用于简并最小化问题的类别，允许唯一的$ h（\ permatatorName {div}）$符合压力近似$σ_h$。主要结果是lebesgue规范中应力误差$σ-σ_h$的先验和后验错误估计和可计算的较低能量结合。数值基准显示出更高的多项式度的收敛速率，并具有自适应网状固定率，并具有保证较低能量界限的第一个超级线性收敛速率。
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.
