Efficient nonlinear iteration schemes based on algebraic splitting for the incompressible Navier-Stokes equations

We propose new, efficient, and simple nonlinear iteration methods for the incompressible Navier-Stokes equations. The methods are constructed by applying Yosida-type algebraic splitting to the linear systems that arise from grad-div stabilized finite element implementations of incremental Picard and Newton iterations. They are efficient because at each nonlinear iteration, the same symmetric positive definite Schur complement system needs to be solved, which allows for CG to be used for inner and outer solvers, simple preconditioning, and the reusing of preconditioners. For the proposed incremental Picard-Yosida and Newton-Yosida iterations, we prove under small data conditions that the methods converge to the solution of the discrete nonlinear problem. Numerical tests are performed which illustrate the effectiveness of the method on a variety of test problems.