Analysis of the Picard-Newton iteration for the Navier-Stokes equations: global stability and quadratic convergence

Abstract

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.

Figures (11)

• Figure 1: Convergence plots of the analytical test with various initial values $u_0 =(c,0)^T, c =0,1,5,10,20$.
• Figure 2: Shown above are streamlines of velocity solutions found for the 2D driven cavity problems with $Re$=1000, 5000, 10000, and 12000 (from top left to bottom right).
• Figure 3: Shown above are convergence plots for the Picard-Newton iteration for varying $Re$, for meshes using $h=\frac{1}{32}$, $h=\frac{1}{64}$, $h=\frac{1}{128}$ and $h=\frac{1}{196}$ before barycenter refinement.
• Figure 4: Shown above are convergence plots for the Picard (top left), Newton (top right) and Line Search Newton (bottom center), all using varying Re, but all using a barycenter refined $h=\frac{1}{64}$ mesh.
• Figure 5: Shown above is the method used to split a rectangular box into 6 tetrahedra.

Theorems & Definitions (11)

• Lemma 3.1
• Remark 3.2
• proof
• Theorem 3.1
• Remark 3.3
• proof : Proof of Theorem \ref{['convthm']}
• Theorem 3.4: Local convergence to nonsingular solutions for any problem data
• proof
• Remark 3.5
• Remark 4.1