COMPOSING SCALABLE NONLINEAR ALGEBRAIC SOLVERS

Abstract. Most efficient linear solvers use composable algorithmic components, with the most common model being the combination of a Krylov accelerator and one or more preconditioners. A similar set of concepts may be used for nonlinear algebraic systems, where nonlinear composition of different nonlinear solvers may significantly improve the time to solution. We describe the basic concepts of nonlinear composite combination and preconditioning and present a number of solvers applicable to nonlinear partial differential equations. We have developed a software framework in order to easily explore the possible combinations of solvers. We show that the performance gains from using composed solvers can be substantial compared with gains from standard Newton-Krylov methods. 1. Introduction. Large-scale algebraic solvers for nonlinear partial differential equations (PDEs) are an essential component of modern simulations. Newton-Krylov methods [20] have well-earned dominance. They are generally robust and may be built from preexisting linear solvers and preconditioners, including fast multilevel preconditioners such as multigrid [64, 58, 7, 5, 9] and domain decomposition methods [54, 53, 46, 56]. Newton’s methods start from a global linearization of the nonlinear operator. The linearization lead