三维曲域上 Stokes 问题的压力鲁棒有限元方法

中文摘要 AI 辅助

本文针对曲域上的三维 Stokes 问题，发展了一种无散度、inf-sup 稳定、最优收敛且压力鲁棒的有限元方法。几何形状通过等参四面体网格近似，而速度空间由 Alfeld 分裂上的 Scott--Vogelius 空间经 Piola 变换获得。离散 inf-sup 条件利用合适的面气泡函数证明。曲三角界面上的求积规则精度不足会导致一致性误差，可能引起次优收敛。解决方法是引入合适的一致性修正，无需稳定化项。此外，从局部交换插值构造了曲域上的交换算子，并证明其全局相容。这些算子用于近似载荷并获得压力鲁棒的离散化。提供了数值算例以验证理论结果。

英文摘要

This paper develops a divergence-free, inf-sup stable, optimally convergent and pressure-robust finite element method for the three-dimensional Stokes problem on curved domains. The geometry is approximated by an isoparametric tetrahedral mesh, while the velocity space is obtained from Scott--Vogelius spaces on an Alfeld split by the Piola transform. The discrete inf-sup condition is proved using suitable face bubble functions. The insufficient accuracy of quadrature rules on curved triangular interfaces leads to a consistency error that may cause suboptimal convergence. The remedy is to introduce a suitable consistency correction without stabilization terms. Moreover, commuting operators on curved domains are constructed from local commuting interpolants and shown to be globally conforming. The operators are used to approximate the load and obtain a pressure-robust discretization. Numerical examples are provided to validate the theoretical results.

发表机构

• LMAM and School of Mathematical Sciences, Peking University（北京大学数学科学学院）
• Chongqing Research Institute of Big Data, Peking University（北京大学重庆大数据研究院）
• Institut für Mathematik, Friedrich-Schiller-Universität Jena（耶拿弗里德里希·席勒大学数学研究所）

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