Nonlinear Preconditioning for Implicit Solution of Discretized PDEs

Nonlinear preconditioning refers to transforming a nonlinear algebraic system to a form for which Newton-type algorithms have improved success through quicker advance to the domain of quadratic convergence. We place these methods, which go back at least as far as the Additive Schwarz Preconditioned Inexact Newton (ASPIN, 2002) in the context of a proliferation distinguished by being left- or right-sided, multiplicative or additive, and partitioned by subdomain, field type, or other criteria. We present the Nonlinear Elimination Preconditioned Inexact Newton (NEPIN, 2022), which is based on a heuristic “bad/good” heuristic splitting of equations and corresponding degrees of freedom. We augment basic forms of nonlinear preconditioning with the feature of practical interest: an adaptive switchover to ordinary Newton as the domain of convergence is approached. Various nonlinearly stiff algebraic and model PDE problems are considered for insight.

Lulu Liu is an associate professor at the School of Mathematics and Statistics, Nanjing University of Science and Technology. In 2015, she obtained doctoral degree from King Abdullah University of Science and Technology in Saudi Arabia. Mainly engaged in the design of nonlinear preconditioned parallel algorithms and their application research in oil reservoir simulation, fluid mechanics, combustion, and other fields. The results have been published in authoritative journals with international influence such as SINUM, SISC, JCP, JACS, etc. Hosted general and youth programs of the National Natural Science Foundation of China, as well as youth programs of the Jiangsu Provincial Natural Science Foundation. In 2022, she won the Youth Award of the Science and Technology Award of the Jiangsu Industrial and Applied Mathematics Society.