Speaker
Description
In this talk, a geometric multigrid solution technique for the incompressible Navier-Stokes equations in three dimensions is presented, utilizing discretely divergence-free finite elements. For this purpose, a set of shape functions is constructed in an a priori manner to span the subspace of discretely divergence-free functions for the Rannacher-Turek finite element pair under consideration. Compared to primal formulations, this approach yields smaller system matrices without a saddle point structure. This prevents the use of complex Schur complement solution techniques and more general preconditioners can be employed. Furthermore, the prolongation operator is designed to preserve globally linear functions while maintaining a high computational efficiency.
Constructing a basis for the subspace of discretely divergence-free finite elements in three dimensions poses significant challenges, making it a bottleneck of this approach. Therefore, the explicit basis construction is utilized only on the coarsest mesh level of the multigrid algorithm for the construction of direct solvers. On finer grids, this information is extrapolated to enforce boundary conditions efficiently. Here, special attention is required for meshes introducing bifurcations in the flow. In such cases, 'global' shape functions are incorporated, which describe the net flux through different branches. This framework also facilitates enforcing total fluxes, which is notoriously difficult in primal approaches.
Various numerical examples for meshes with different shapes and boundary conditions illustrate the strengths, limitations, and future challenges of this solution concept for three-dimensional flow simulations.
