A two-level newton, finite element algorithm for approximating electrically conducting incompressible fluid flows

Abstract

We consider the approximation of stationary, electrically conducting, incompressible fluid flow problems at small magnetic Reynolds number. The finite element discretization of these systems leads to a very large system of nonlinear equations. We consider a solution algorithm which involves solving a much smaller number of nonlinear equations on a coarse mesh, then one large linear system on a fine mesh. Under a uniqueness condition, this one-step, two-level Newton-FEM procedure is shown to produce an optimally accurate solution. This result extends both the two-level method of Xu [1,2] from elliptic boundary value problems to MHD problems, and the energy norm error analysis of Peterson [3] (see also [4]) of MHD problems at a small magnetic Reynolds number to L2 error estimates and multilevel discretization and solution procedures.