Error bounds for semi-Galerkin approximations of nonhomogeneous   incompressible fluids
  We consider the spectral semi-Galerkin method applied to the nonhomogeneous Navier-Stokes equations. Under certain conditions it is known that the approximate solutions constructed through this method converge to a global strong solution of these equations. Here, we derive an optimal uniform in time error estimate in the $H^1$ norm for the velocity. We also derive an error estimate for the density in some spaces $L^r$. 
P. Braz e Silva, M. A. Rojas-Medar