A geometric multigrid solver for the incompressible Navier-Stokes equations using discretely divergence-free finite elements in 3D

A geometric multigrid solution technique for the incompressible Navier-Stokes equations in three dimensions is presented, utilizing the concept of discretely divergence-free finite elements without requiring the explicit construction of a basis on each mesh level. For this purpose, functions are constructed in an a priori manner spanning the subspace of discretely divergence-free functions for the Rannacher-Turek finite element pair under consideration. Compared to mixed formulations, this approach yields smaller system matrices with no saddle point structure. This prevents the use of complex Schur complement solution techniques and more general preconditioners can be employed. While constructing a basis for discretely divergence-free finite elements may pose significant challenges and its use prevents a structured assembly routine, a basis is utilized only on the coarsest mesh level of the multigrid algorithm. On finer grids, this information is extrapolated to prescribe boundary conditions efficiently. Here, special attention is required for geometries introducing bifurcations in the flow. In such cases, so called ‘global’ functions with an extended support are defined, which can be used to prescribe the net flux through different branches. Various numerical examples for meshes with different shapes and boundary conditions illustrate the strengths, limitations, and future challenges of this solution concept.