Nonlinear Preconditioning for Implicit Solution of Dis-cretized 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, multiplica-tive or additive, and partitioned by field, subdomain, 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. NEPIN is shown to be fairly insensitive to mesh res-olution and “bad” subproblem identification based on the local Mach number or the local nonlinear residual for transonic flow over a wing.

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