Discretely Divergence-Free Q2-P1 Finite Elements for Incompressible Flow Problems

Discretely divergence-free finite elements are characterized by vector-valued shape functions that satisfy the discrete incompressibility condition in an a priori manner. When applied to the incompressible Navier-Stokes equations, this approach yields a problem defined solely for the velocity field, thereby avoiding the typical saddle-point structure of mixed formulations. Consequently, iterative solvers do not require adaptation to a zero block and highly efficient solution strategies become available. In this book, Christoph Lohmann exploits this concept to develop a geometric multigrid solver for three-dimensional flow problems. The proposed method achieves a mesh-independent convergence behavior under standard assumptions, while its performance is validated in several linear and nonlinear test cases. Numerical examples demonstrate the necessity of using so called `global' finite element functions to accurately predict the flow behavior through geometries with multiple connected branches. These functions can be implicitly incorporated into the proposed multigrid framework using suitable filtering techniques.