Based on a fully overlapping domain decomposition technique, a parallel stabilized equal-order finite element method for the steady Stokes equations is presented and studied. In this method, each processor computes a local stabilized finite element solution in its own subdomain by solving a global problem on a global mesh that is locally refined around its subdomain, where the lowest equal-order finite element pairs (continuous piecewise linear, bilinear or trilinear velocity and pressure) are used for the finite element discretization and a pressure-projection-based stabilization method is employed to circumvent the discrete inf–sup condition that is invalid for the used finite element pairs. The parallel stabilized method is unconditionally stable, free of parameter and calculation of derivatives, and is easy to implement based on an existing sequential solver. Optimal error estimates are obtained by the theoretical tool of local a priori error estimates for finite element solutions. Numerical results are also given to verify the theoretical predictions and illustrate the effectiveness of the method.

中文翻译：

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基于最低等阶单元的不可压缩流动平行稳定有限元方法

基于完全重叠域分解技术，提出并研究了稳态Stokes方程的并行稳定等阶有限元方法。在这种方法中，每个处理器通过在其子域周围局部细化的全局网格上解决全局问题来计算其子域中的局部稳定有限元解，其中最低等阶有限元对（连续分段线性、双线性或三线速度和压力）用于有限元离散化，并采用基于压力投影的稳定方法来规避对使用的有限元对无效的离散 inf-sup 条件。平行稳定法无条件稳定，无需参数和导数计算，并且基于现有的顺序求解器很容易实现。最优误差估计是通过有限元解的局部先验误差估计的理论工具获得的。还给出了数值结果以验证理论预测并说明该方法的有效性。