We analyze and test a simple-to-implement two-step iteration for the incompressible Navier-Stokes equations that consists of first applying the Picard iteration and then applying the Newton iteration to the Picard output. We prove that this composition of Picard and Newton converges quadratically, and our analysis (which covers both the unique solution and non-unique solution cases) also suggests that this solver has a larger convergence basin than usual Newton because of the improved stability properties of Picard-Newton over Newton. Numerical tests show that Picard-Newton dramatically outperforms both the Picard and Newton iterations, especially as the Reynolds number increases. We also consider enhancing the Picard step with Anderson acceleration (AA), and find that the AAPicard-Newton iteration has even better convergence properties on several benchmark test problems.

翻译：我们分析并测试了一种针对不可压缩Navier-Stokes方程的简单易实现的两步迭代方法：首先应用Picard迭代，然后对Picard迭代输出执行Newton迭代。我们证明这种Picard-Newton组合方法具有二次收敛性，且我们的分析（涵盖唯一解与非唯一解两种情况）表明，由于Picard-Newton相比传统Newton具有更优的稳定性特征，该求解器具有更大的收敛域。数值实验显示，Picard-Newton在性能上显著优于单独使用Picard或Newton迭代，尤其随着雷诺数增大优势更为明显。我们还研究了通过Anderson加速(AA)增强Picard步骤的方案，发现在多个基准测试问题中，AA-Picard-Newton迭代展现出更优的收敛特性。