Abstract Based on a fully overlapping domain decomposition approach, local and parallel stabilized finite element algorithms are proposed and investigated for the steady incompressible Navier–Stokes equations, where the inf − sup unstable lowest equal-order P 1 − P 1 finite element pairs are used and the stabilized term is based on two local Gauss integrations defined by the difference between a consistent and under-integrated matrix of pressure interpolants. In these algorithms, each processor computes a local stabilized solution in its own subdomain using a global grid that is locally refined around its own subdomain, making the algorithms have low communication cost and easy to implement based on a sequential solver. Using the technical tool of the local a priori estimate for the stabilized solution, error bounds of the proposed algorithms are derived. Theoretical and numerical results show that, the algorithms can yield an approximate solution with an accuracy comparable to that of the standard stabilized finite element solution with a substantial decrease in CPU time.

中文翻译：

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基于稳定 Navier-Stokes 方程的最低等阶元的局部和并行稳定有限元算法

摘要 基于完全重叠域分解方法，提出并研究了稳态不可压缩 Navier-Stokes 方程的局部和并行稳定有限元算法，其中使用了 inf − sup 不稳定的最低等阶 P 1 − P 1 有限元对稳定项基于两个局部高斯积分，由压力插值的一致矩阵和欠积分矩阵之间的差异定义。在这些算法中，每个处理器使用围绕其自身子域局部细化的全局网格计算其子域中的局部稳定解，使得算法具有低通信成本且易于基于顺序求解器实现。使用稳定解的局部先验估计的技术工具，推导出所提出算法的误差界限。理论和数值结果表明，该算法可以产生一个近似解，其精度可与标准稳定有限元解相媲美，同时大大减少了 CPU 时间。