Efficient and effective algebraic splitting‐based solvers for nonlinear saddle point problems

• Published in: Mathematical Methods in the Applied Sciences
• Main Authors: Liu, Jia, Rebholz, Leo G., Xiao, Mengying
• Other Authors: National Science Foundation
• Format: Article in Journal/Newspaper
• Language: English
• Published: Wiley 2023
• Subjects: Bingham; Saddle Point; IPY
• Online Access: http://dx.doi.org/10.1002/mma.9665; https://onlinelibrary.wiley.com/doi/am-pdf/10.1002/mma.9665; https://onlinelibrary.wiley.com/doi/pdf/10.1002/mma.9665

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.