Discretization of the Navier-Stokes equationswith slip boundary condition II

Abstract

Multilevel methods are a common tool for solving nonlinear systems arising from discretizations of elliptic boundary value problems (see, e.g., [1,2]). Multilevel methods consist of solving the nonlinear problem on a coarse mesh and then performing one or two Newton correction steps on each subsequent mesh thus solving only one or two larger linear systems. In a previous paper by the author [3], a two-level method was proposed for the stationary Navier-Stokes equations with slip boundary condition. Uniqueness of solution to the nonlinear problem (Step 1 of the two-level method) is guaranteed provided that the data of the problem (i.e., the Reynolds number, and the forcing term) is bounded. In practice, however, the aforementioned bound on the data is rarely satisfied. Consequently, a priori error estimates should not rely upon this bound. Such estimates will be established in this report. For the two-level method, the scalings of the meshwidths that guarantee optimal accuracy in the H 1-norm, are equally favorable to those in the uniqueness case.