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Daily Overview |
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Num3: Numerical Mathematics and Scientific Computing Location: A703 Session Chair: Stefan Frei | |
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Matrix-free multigrid techniques for variational discretizations: Navier-Stokes equations and random parabolic problems 1: Helmut Schmidt University, Germany; 2: Otto von Guericke University of Magdeburg, Germany 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. | |
