Analysis of an Adaptive Safeguarded Newton-Anderson Algorithm with Applications to Fluid Problems

The purpose of this paper is to develop a practical strategy to accelerate Newton's method in the vicinity of singular points. We do this by developing an adaptive safeguarding scheme, which we call gamma-safeguarding, that one can apply to Anderson accelerated Newton's method when solving problems near singular points. The key features of adaptive gamma-safeguarding are that it converges locally for singular problems, and it can detect nonsingular problems, in which case the Newton-Anderson iterates are scaled towards a standard Newton step. This leads to faster local convergence compared to both Newton's method and Newton-Anderson without safeguarding, at no additional computational cost. We demonstrate three strategies one can use when implementing Newton-Anderson and gamma-safeguarded Newton-Anderson to solve parameter-dependent problems near singular points. For our benchmark problems, we take two parameter-dependent incompressible flow systems: flow in a channel and Rayleigh-Benard convection.