A finite element approximation for the steady solution of a second-grade fluid model

The aim of this work is to present a finite element method for the approximation of the steady solution of an incompressible second-grade fluid model in two dimensions. The equations for second-grade fluids form a system of nonlinear partial differential equations of mixed elliptic–hyperbolic type (in the steady state). Using a fixed-point argument, associated with the decomposition of the system into a transport equation and a Stokes system, existence and uniqueness of the approximate solution are proved and error estimates are obtained. This technique allows the construction of a decoupled fixed-point algorithm converging to the discrete solution of the original problem.