论文标题
廉价的多项式射击平衡的通量A后验估计值均一代分析
Inexpensive polynomial-degree-robust equilibrated flux a posteriori estimates for isogeometric analysis
论文作者
论文摘要
我们考虑了泊松模型问题的等几何离散，重点是高多项式学位和强大的层次结构改进。我们通过平衡的通量来得出后验误差估计，即位于$ \ boldsymbol {h}（{\ rm div}）$空间中的矢量值映射的分段多项式估计，以适当近似于所需的分解约束。我们的估计值在领先期限，局部有效，并且相对于多项式程度稳健。对于使用层次B-Splines产生的自适应网状精炼中产生的悬挂淋巴结的数量，它们也很健壮。设计了两个统一的分区，一种具有对应于映射的花纹的较大支撑，一个具有较大的支撑，具有对应于映射的分段多线有限元元素帽子基函数的小支持。平衡仅在小支架上执行，避免了较高的大量支撑物甚至全球系统解决方案的平衡计算价格。因此，派生的估计值也尽可能便宜。开发了用于此类设置的抽象框架，其应用于特定情况，仅请求验证一些清晰确定的假设。数值实验说明了理论发展。
We consider isogeometric discretizations of the Poisson model problem, focusing on high polynomial degrees and strong hierarchical refinements. We derive a posteriori error estimates by equilibrated fluxes, i.e., vector-valued mapped piecewise polynomials lying in the $\boldsymbol{H}({\rm div})$ space which appropriately approximate the desired divergence constraint. Our estimates are constant-free in the leading term, locally efficient, and robust with respect to the polynomial degree. They are also robust with respect to the number of hanging nodes arising in adaptive mesh refinement employing hierarchical B-splines. Two partitions of unity are designed, one with larger supports corresponding to the mapped splines, and one with small supports corresponding to mapped piecewise multilinear finite element hat basis functions. The equilibration is only performed on the small supports, avoiding the higher computational price of equilibration on the large supports or even the solution of a global system. Thus, the derived estimates are also as inexpensive as possible. An abstract framework for such a setting is developed, whose application to a specific situation only requests a verification of a few clearly identified assumptions. Numerical experiments illustrate the theoretical developments.
