Conference Agenda

In this talk, we analyse a higher-order unfitted finite element method for the incompressible Stokes equations, yielding a pointwise divergence-free solution in the entire domain. The approach is based on the isoparametric Scott--Vogelius velocity-pressure pair on a background mesh, together with a stabilized mixed Nitsche/Lagrange multiplier formulation for imposing Dirichlet boundary conditions. We use a higher-order Lagrange multiplier space to ensure stability and to mitigate the loss of pressure robustness typically associated with the weak enforcement of Dirichlet boundary conditions. The key results are a new inf-sup stability result for the isoparametric Scott--Vogelius pair on unfitted meshes and a combined inf-sup stability result for the bilinear forms associated with the pressure and the Lagrange multiplier. We show stability and convergence properties of the method, including geometry approximation errors introduced by the isoparametric approximation. This demonstrates optimal-order velocity convergence of the velocity in the $H^1$- and $L^2$-norms. Furthermore, we establish optimal $H^1$-convergence and nearly optimal $L^2$-convergence of a post-processed pressure. Numerical examples illustrate the theoretical findings.

Space-time finite element methods (STFEMs) have demonstrated their potential for the accurate and efficient approximation of incompressibe viscous flow on computationally feasible grids. Stochastic Galerkin methods offer unexplored potential for the numerical simulation of parabolic problems with random variables, in particular if they are combined with variational discretizations of the space and time variables. In both cases, application readiness relies on algebraic solvers, preconditioners and software architectures that are tailored to the algebraic system structure induced by the tensor product discrete spaces. For preconditioning the Navier-Stokes system, we present and analyze hp multigrid techniques with coarse grid correction in space and time polynomial orders and for the spatial mesh [N. Margenberg, M. Bause, A monolithic hp space-time multigrid preconditioned Newton-Krylov solver for space-time FEM applied to the incompressible Navier-Stokes equations, SIAM J. Sci. Comput., submitted (2026), pp.1-26; arXiv:2602.13841]. For preconditioning Stochastic Galerkin approximations of parabolic problems, we propose and analyze block-preconditioner, using geometric multigrid techniques with local Vanka smoother for the space--time subsystems [M. Dawor, N. Margenberg, M. Bause, Stochastic Galerkin and Monte--Carlo methods for parabolic problems: Numerical performance of variational matrix-free approximations, PAMM, submitted (2026), pp. 1-26]. The implementations use a unified matrix-free framework based on the deal.II library.

This talk presents a parallel-in-time formulation for the unsteady Reynolds-Averaged Navier-Stokes equations applied to incompressible periodic ducted flows under turbulent conditions. The method is based on a multiple-shooting approach, in which the time domain is decomposed into temporal subdomains, enabling the forward solution to be computed in parallel. The implementation relies on the open-source CFD framework OpenFOAM, using a finite-volume URANS solver based on the PIMPLE algorithm.

A key feature of the approach is that the flow solver is treated largely as a black box. Consequently, no fundamental modification of the underlying solver structure is required, making the method non-intrusive and potentially transferable to other CFD solvers. At the same time, the efficiency of the framework depends on the coordination of repeated solver calls, the exchange of data between shooting nodes, and the availability of suitable linearized solution procedures. We present numerical studies addressing the influence of shooting nodes, initial conditions, and parallel performance, and provide insight into the use of this framework for primal and adjoint computations in optimal shape design problems.

In this talk, we show progress on a mechanism to systematically evaluate the performance of numerical schemes and software packages on their abilities to create correct contacts and bounces. The benchmark we propose consists in the free fall of a deformable elastic ball inside a viscous incompressible fluid coupled through no-slip boundary condition. The expectation is that this ball bounces off the ground. Given the singular circumstances, which pose mathematical-numerical challenges, we decided for a setting with minimal complexity to ease immitations. Hence it is within the two dimensional space, but with a deformable elastic solid. This is actually critically for a rebound as non-deformable elastic solids would not bounce inside fluids. For this fluid-structure-contact interaction benchmark five independent groups with independent numerical discretizations, solution schemes, and codes have contributed and compared their numerical findings. In this talk, we explain two methods in more detail, namely fully Eulerian fluid-structure using a ghost-penalty cut finite element approach and fluid-structure interaction in arbitrary Lagrangian-Eulerian coordinates. Several quantities of interest like the motion of the center of mass, the contact time, the energy loss (due to dissipation), and some more have been collected on several refinement scales in space and time by each group in order to substaniate the benchmark setting. Finally, we finish with a summary of successes obtained so far and ongoing challenges in designing such a fluid-structure-contact interaction benchmark.