Workshop on Numerical Methods and Analysis in CFD - Abstract
Bause, Markus
The numerical simulation of incompressible viscous flow continues to remain a challenging task, in particular, if three space dimensions are involved. Using space-time finite element methods (STFEM) allows the natural construction of higher order methods; cf. [5, 1, 2, 3]. They offer the potential to achieve accurate results on computationally feasible grids with a minimum of numerical costs. Geometric multigrid methods (GMG) are known as the most efficient iterative methods for the solution of large linear systems arising from the discretization of partial differential equations. To exploit their potential, they need to be adapted to STFEM; cf. [5, 2]. Time-dependent domains put a further facet of complexity on the numerical simulation of the flow problems; cf. [3] .
Here, families of STFEM with discrete solutions of increasing regularity in time are studied for the Navier--Stokes equations. Firstly, the potential of a Newton--GMRES solver with GMG preconditioning is illustrated for discontinuous Galerkin time-stepping schemes and the 3D DFG flow benchmark. Then, the approximation of the pressure trajectory in families of continuous in time Galerkin schemes is reviewed carefully. The lack of an initial pressure value yields a source of trouble for the definition of an equal-order in time pressure trajectory. A post-processing of the pressure trajectory by collocation techniques is introduced and optimal order error estimates are proved for the pressure approximation. Finally, extensions of SFTEM to flow problems on evolving domains and/or Galerkin time discretizations of higher regularity are discussed.
This is a joint work with M. Anselmann (Helmut Schmidt University), G. Matthies (University of Dresden) and F. Schieweck (University of Magedeburg).
References:
[1] M. Anselmann, M. Bause, Higher order Galerkin-collocation time discretization with Nitsche's method for the Navier--Stokes equations, Math. Comput. Simul., 189 (2021), pp. 141--162.
[2] M. Anselmann, M. Bause, A geometric multigrid method for space-time finite element discretizations of the Navier--Stokes equations and its application to 3D flow simulation, ACM Trans. Math. Softw., submitted (2021), pp. 1--27; arXiv:2107.10561.
[3] M. Anselmann, M. Bause, CutFEM and ghost stabilization techniques for higher order space-time discretizations of the Navier--Stokes equations, Int. J. Numer. Methods Fluids, accepted (2022),
doi: 10.1002/fld.5074, pp. 1--31; arXiv:2103.16249.
[4] M. Anselmann, M. Bause, G. Mathies, S. Schieweck, Optimal order pressure approximation for the Stokes problem by a variational method in time with post-processing, in progress, (2022).
[5] S. Hussain, F. Schieweck, S. Turek, Efficient Newton-multigrid solution techniques for higher order space-time Galerkin discretizations of incompressible flow, Appl. Numer. Math., 83 (2014), pp. 51--71.