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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 composition 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.

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The nonlocal continuum dielectric model is an important extension of the classical Poisson dielectric model, but it is very expensive to be solved in general. In this paper, we prove that the solution of one commonly used nonlocal continuum dielectric model of water can be split as a sum of two functions, and these two functions are simply the solutions of one Poisson equation and one Poisson-like equation. With this new solution splitting formula, we develop a fast finite element algorithm and a program package in Python based on the DOLFIN program library such that a nonlocal dielectric model can be solved numerically in an amount of computation that merely doubles that of solving a classic Poisson dielectric model. Using the new solution splitting formula, we also derive the analytical solutions of two nonlocal model problems. We then solve these two nonlocal model problems numerically by our program package and validate the numerical solutions through a comparison with the analytical solutions. Finally, our study of free energy calculation by a nonlocal Born ion model demonstrates that the nonlocal dielectric model is a much better predictor of the solvation free energy of ions than the local Poisson dielectric model.

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