convergence of a homotopy finite element method for computing steady states of Burgers' equation

In this paper, the convergence of a homotopy method (1.1) for solving the steady state problem of Burgers’ equation is considered. When ν is fixed, we prove that the solution of (1.1) converges to the unique steady state solution as e → 0, which is independent of the initial conditions. Numerical examples are presented to confirm this conclusion by using the continuous finite element method. In contrast, when ν = e →, numerically we show that steady state solutions obtained by (1.1) indeed depend on initial conditions.