Numerical Solution of Overdetermined PDEs

Navier-Stokes are exceptionally useful because they describe the physics of many things of academic and scientific interest. They may be used to model the weather, water flow in a pipe, the air’s flow around a wing, motion of stars inside a galaxy. The Navier-Stokes equations simplify to give the (stationary) Stokes equations: −µ∆u +∇p = f − ∇ · u = 0 where u is the velocity field, p is the pressure and µ is viscosity. By taking divergence of the first equation we obtain −∆p = − ∇ · f The new equation is called a differential consequence or integrability condition of the initial system. We call the new system completed system. Putting y = (u, p) we can write the whole system a