Algebraic Multigrid Methods for the Numerical Solution of the Incompressible Navier-Stokes Equations
The Navier-Stokes equations from an important set of nonlinear partial differential equations for the description of even very complex phenomena of fluid flow. The present work provides concepts for the application of algebraic multigrid (AMG) methods for their numerical solution, resp. the numerical solution of their linearized form, the Oseen equations. The "classical" methods in this area perform an iterative decoupling of the system to equations for pressure and velocity, respectively, and apply AMG to the resulting (scalar) problems. Our main focus is different from that, it is the development of AMG methods for the solution of the whole system. For that we show how the coarse levels can be constructed (where stability is an important issue) and which smoothers could be used. To prove the efficiency of our methods experimentally, we apply them to finite element discretizations of various problems (model problems and also more involved industrial settings) and compare them with classical approaches.