"Some new results for Yosida splitting methods for Navier-Stokes equations"

LEO REBHOLZ

Mathematics Department

Clemson University

## Abstract:

Solving the saddle point systems arising in Navier-Stokes equations is well known to be very difficult. To circumvent the issue, various types of splitting methods have been introduced which reduce the linear algebraic difficulties, but which also introduce error into the solutions. One such method is called the Yosida method, which falls in the class of ‘discretize-then-split’ techniques, and is particularly well-suited for problems where adaptive timestepping is important. In our talk, after we review the basics of the Yosida method, we will show that by applying the Yosida approximation to a system written in terms of a change of variables, asymptotic error is improved by a full order of accuracy. We show the theory both in terms of linear algebra style proofs, as well as in the finite element setting. Several numerical experiments are given which show the effectiveness of the solver, including lift and drag measurements in flow past a cylinder, as well as for turbulent channel flow. This is joint work with Mengying Xiao.