Ninth Annual Graduate Student Miniconference in Computational Mathematics
Mengying Xiao
Clemson University https://mthsc.clemson.edu/directory/view_person.py?person_id=437 

Abstract 
Incremental PicardYosida iteration for the efficient numerical solution of steady NavierStokes equations
We present new, efficient, nonlinear iteration methods for the incompressible NavierStokes equations. The methods are constructed by applying Yosidatype algebraic splitting to the linear systems that arise from graddiv 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 needs to be solved, which allows for CG to be used for inner and outer solvers, simple preconditioning, and reusing of preconditioners. For the incremental PicardYosida and NewtonYosida iterations, we prove under small data conditions that the methods converge to the solution of the discrete nonlinear problem. Numerical tests are performed which verify excellent convergence properties of the methods on a variety of test problem. 