We consider the Navier-Stokes equations for the incompressible flow in channels with forward and backward steps. The paper consists of two main parts. In the first part we investigate a posteriori error estimates for the Stokes and Navier-Stokes equations on two-dimensional polygonal domains. We apply the a posteriori estimates to solve an incompressible flow problem in a domain with corners that cause singularities in the solution. Second part of the paper stands on the result on the asymptotics
of the solution in the vicinity of nonconvex internal angles. Using now a priori error estimates we suggest an alternative approach to the adaptive mesh refinement near the corners. This approach gives very precise results in a cheap way. We give numerical results and show the pros and cons of both approaches.