Summary of Nonlinear Solvers Available In PETSc#
See the paper Composing Scalable Nonlinear Algebraic Solvers for details on the algorithms.
Algorithm 
Associated Type 
Notes 

Newton’s method 
Use 

Newton’s method with trust region 

Single linearization 
Essentially one step of Newtwon without a line search 

QuasiNewton method (BFGS) 

Nonlinear CG 
Requires nearly symmetric Jacobian for good convergence 

Nonlinear GMRES 

Anderson mixing 

Nonlinear Richardson 

Nonlinear GaussSiedel 

Full Approximation Scheme (nonlinear multigrid) 

Nonlinear additive Schwarz 

Nonlinear additive Schwarz preconditioned inexact Newton (ASPIN) methods 

Composite (combine several nonlinear solvers) 

Preconditioned nonlinear solver 
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See 