Navier-Stokes equations for incompressible fluids
Incompressible fluid dynamics is a challenging topic, basically for the “saddle point” nature of the problem. Under certain assumptions, the velocity field that solves this problem is the minimum of an energy constrained by incompressibility. This makes this problem significantly different from elliptic problems (corresponding to free minimization). Pressure is the Lagrange multiplier associated with the incompressibility constraint. For this reason, the numerical solution may be in general expensive to compute and large efforts have been devoted to develop efficient solution schemes.