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Description
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.
Introduction.- Discretely divergence-free Q2-P1 finite elements.- Application to Stokes problem.- Multigrid solution strategy.- Numerical experiments.- Conclusions.- Rannacher-Turek finite elements.- Weakly imposed boundary conditions.- Generating system.
Christoph Lohmann is a postdoctoral researcher in the Department of Mathematics at TU Dortmund University. His research interests include finite element methods satisfying discrete maximum principles, parallel-in-time methods, and efficient solvers for incompressible flow problems.
