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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 converges more reliably for higher Reynolds numbers and worse initial conditions than Picard and Newton iterations. 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.
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Efficient and effective algebraic splitting‐based solvers for nonlinear saddle point problems
The incremental Picard Yosida (IPY) method has recently been developed as an iteration for nonlinear saddle point problems that is as effective as Picard but more efficient. By combining ideas from algebraic splitting of linear saddle point solvers with incremental Picard‐type iterations and grad‐div stabilization, IPY improves on the standard Picard method by allowing for easier linear solves at each iteration—but without creating more total nonlinear iterations compared to Picard. This paper extends the IPY methodology by studying it together with Anderson acceleration (AA). We prove that IPY for Navier–Stokes and regularized Bingham fits the recently developed analysis framework for AA, which implies that AA improves the linear convergence rate of IPY by scaling the rate with the gain of the AA optimization problem. Numerical tests illustrate a significant improvement in convergence behavior of IPY methods from AA, for both Navier–Stokes and regularized Bingham.
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- Award ID(s):
- 2011490
- PAR ID:
- 10482321
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Mathematical Methods in the Applied Sciences
- Volume:
- 47
- Issue:
- 1
- ISSN:
- 0170-4214
- Format(s):
- Medium: X Size: p. 451-474
- Size(s):
- p. 451-474
- Sponsoring Org:
- National Science Foundation
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