Convergence of Rothe's scheme for the Navier‐Stokes equations with slip conditions in 2D domains

Abstract

Abstract We investigate Rothe's scheme for the Navier‐Stokes equations in two dimensional bounded domains with slip boundary conditions admitting flow across the boundary. The structure of the model enables to reformulate it into the coupled system of the vorticity and velocity problems. The obtained convergence of our first order approximation is the same as for case in domains without boundaries although we admit inhomogeneous boundary data. The method is based on the maximum principle for the vorticity equation which delivers a new bound for the solutions. However our study can be realized only under a geometrical constraint which restricts the shape of the domain.