Based on full domain partition, three parallel iterative finite-element algorithms are proposed and analyzed for the Navier–Stokes equations with nonlinear slip boundary conditions. Since the nonlinear slip boundary conditions include the subdifferential property, the variational formulation of these equations is variational inequalities of the second kind. In these parallel algorithms, each subproblem is defined on a global composite mesh that is fine with size h on its subdomain and coarse with size H (H ≫ h) far away from the subdomain, and then we can solve it in parallel with other subproblems by using an existing sequential solver without extensive recoding. All of the subproblems are nonlinear and are independently solved by three kinds of iterative methods. Compared with the corresponding serial iterative finite-element algorithms, the parallel algorithms proposed in this paper can yield an approximate solution with a comparable accuracy and a substantial decrease in computational time. Contributions of this paper are as follows: (1) new parallel algorithms based on full domain partition are proposed for the Navier–Stokes equations with nonlinear slip boundary conditions; (2) nonlinear iterative methods are studied in the parallel algorithms; (3) new theoretical results about the stability, convergence and error estimates of the developed algorithms are obtained; (4) some numerical results are given to illustrate the promise of the developed algorithms.

中文翻译：

---

带有非线性滑移边界条件的Navier–Stokes方程的并行迭代有限元算法

基于全域划分，提出了三种并行的迭代有限元算法，并对带有非线性滑移边界条件的Navier-Stokes方程进行了分析。由于非线性滑移边界条件包括亚微分性质，因此这些方程式的变分公式是第二类的变分不等式。在这些并行算法中，每个子问题是在全球复合材料网状物，其与粒径微细定义ħ它的子域，与粗尺寸ħ（ħ  »  ħ）远离子域，然后我们可以使用现有的顺序求解器与其他子问题并行解决它，而无需进行大量重新编码。所有子问题都是非线性的，并且可以通过三种迭代方法独立解决。与相应的串行迭代有限元算法相比，本文提出的并行算法可以产生近似的解决方案，并且具有相当的精度，并且大大减少了计算时间。本文的主要工作如下：（1）针对具有非线性滑移边界条件的Navier-Stokes方程，提出了一种基于全域划分的并行算法。（2）在并行算法中研究了非线性迭代方法；（3）关于稳定性的新理论结果，获得了所开发算法的收敛性和误差估计；（4）给出了一些数值结果，说明了改进算法的前景。