发表机构
University of Hamburg; University of Milano-Bicocca（汉堡大学; 米兰比可卡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种新的有限元离散方法，用于具有对称粘性应力的二维不可压缩Stokes方程，通过Clough-Tocher宏单元分割和局部泡函数精确强制应力对称性，实现压力鲁棒且无需额外富集，并验证了最优收敛阶。
AI 中文摘要
我们针对具有对称粘性应力的二维不可压缩Stokes方程的混合形式引入了一种新的有限元离散方法。该方法基于一种保持质量、应力屈服的形式（Gopalakrishnan, Lederer, Schöberl；A mass conserving mixed stress formulation for the Stokes equations；IMA J. Numer. Anal., Vol 40, 2020），并采用对称应力，其关键新颖之处在于应力张量的对称性被作为离散空间的固有属性精确强制实现。该空间构建在Clough--Tocher宏单元分割上，由矩阵值函数组成，这些函数在宏单元界面上的法向-切向分量是连续的。由于精确对称性是通过子单元上的局部多项式泡函数强制实现的，而这些泡函数可以通过静态凝聚消除，因此全局耦合自由度的数量与原始保持质量、应力屈服形式一致。由此产生的压力鲁棒方法无需任何额外富集即可保持稳定。我们为所有变量建立了最优收敛阶，并进行了数值实验，实验结果证实了理论结果。
英文摘要
We introduce a new finite element discretization for a mixed formulation of the two-dimensional incompressible Stokes equations with symmetric viscous stresses. The method is based on a mass-conserving stress-yielding formulation (Gopalakrishnan, Lederer, Schöberl; A mass conserving mixed stress formulation for the Stokes equations; IMA J. Numer. Anal., Vol 40, 2020) with symmetric stresses, with the key novelty that the symmetry of the stress tensor is enforced exactly as an intrinsic property of the discrete space. This space is constructed on the Clough--Tocher macroelement split and consists of matrix-valued functions whose normal-tangential components across macroelement interfaces are continuous. Since the exact symmetry is enforced via local polynomial bubble functions on the subelements, which can be eliminated via static condensation, the number of globally coupled degrees of freedom coincide with those of the original mass-conserving stress-yielding formulation. The resulting pressure-robust method is stable without requiring any additional enrichment. We establish optimal convergence rates for all the variables and present numerical experiments that confirm the theoretical results.
