I am trying to solve the Navier Stokes equations using the finite element method. I plan on using the pressure correction method to deal with the pressure and an implicit time stepping scheme for diffusion and an explicit time stepping scheme for convection. My question is: How does one deal with the non-linear term using the finite element method? I have investigated a couple approaches. For example, I understand for steady problems it is common to use a Newton iteration. Since I am interested in unsteady problems I don't think this method is suitable. I have also seen an approach where part of the non-linear term is frozen at the last time step so that at each time step the problem appears as a linear advection problem, i.e.$u_{i}\frac{\partial{u_{i}}}{\partial{x_{j}}} \approx{} u_{i}^{n-1}\frac{\partial{u_{i}^{n}}}{\partial{x_{j}}}$In other words the velocity at the previous time step ($u_{i}^{n-1}$) is treated as constant and no shape function expansion is used in approximating this term. Is this method any good? What methods should I use to deal with the non-linear term in the unsteady Navier Stokes equations where accuracy, ease of implementation, stability, and robustness to different Reynolds numbers are all considerations?Thanks
How to deal with nonlinear term in Navier Stokes equations (finite element code)
finite element;fluid dynamics;navier stokes
You can absolutely use the Newton method to linearize the system of equations that results after you have discretized in time. It's a pretty common approach. It might be overkill in lots of cases, but it's not inappropriate. Freezing like you suggest and not iterating at all within a timestep can lead to some pretty awful solutions unless your timestep is miniscule. But you don't want to do that since you are interested in implicit approaches. Write the split equations, and then discretize in time to give a non-linear PDE in space. Then apply Newton's method. Then discretize in space with FEM. 