离散Poincaré和Friedrichs不等式中的常数及离散拟插值
Constants in Discrete Poincaré and Friedrichs Inequalities and Discrete Quasi-Interpolation
AI总结：
本文研究了离散Poincaré不等式中的显式常数及离散拟插值的稳定性，通过适应性三角剖分和最小角条件，给出了不等式中的常数界。
摘要：
本文提供了在n维空间上简单形K上的离散Poincaré不等式，其显式常数C(n)仅依赖于维度n=2,3。该不等式界定了积分均值为零的函数在K上的分片导数范数及所有内部边上的跳跃积分范数，由Lebesgue范数乘以C(n)乘以K的直径。第二部分证明了富集算子的稳定性，从而保证了离散拟插值在证明离散Friedrichs不等式和离散可靠性估计时的稳定性及逼近性，并给出了常数Λ₁和Λ₃在适应性公理中的界，确保了批量参数的实用选择和最优收敛率。
英文摘要：
This paper provides a discrete Poincaré inequality in $n$ space dimensions on a simplex $K$ with explicit constants. This inequality bounds the norm of the piecewise derivative of functions with integral mean zero on $K$ and all integrals of jumps zero along all interior sides by its Lebesgue norm by $C(n)\operatorname{diam}(K)$. The explicit constant $C(n)$ depends only on the dimension $n=2,3$ in case of an adaptive triangulation with the newest vertex bisection. The second part of this paper proves the stability of an enrichment operator, which leads to the stability and approximation of a (discrete) quasi-interpolator applied in the proofs of the discrete Friedrichs inequality and discrete reliability estimate with explicit bounds on the constants in terms of the minimal angle $ω_0$ in the triangulation. The analysis allows the bound of two constants $Λ_1$ and $Λ_3$ in the axioms of adaptivity for the practical choice of the bulk parameter with guaranteed optimal convergence rates.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
