Nils Margenberg

Moataz Dawor, Nils Margenberg, Markus Bause

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. Due to the high dimensionality, the solution of the arising algebraic systems do not become feasible without efficient solvers, preconditioners, and software architectures. A stochastic Galerkin discretization with an embedded slabwise finite element approximation of the space and time variables is proposed and analyzed numerically. For solving the linear systems, GMRES iterations are block-preconditioned by a geometric multigrid (GMG) technique using a local Vanka smoother for the space-time subsystems. Monte-Carlo methods are also used for solving random parabolic problems and studied here for the purpose of comparison. The Monte-Carlo approach is built on the space-time finite element formulation together with the GMRES-GMG solver technology. All algorithms have been implemented in a unified matrix-free framework based on the deal.II software library. Comparative numerical evaluations illustrate the performance properties of both approaches, including convergence of the discretizations and statistics of the algebraic solver. Superiority of the stochastic Galerkin approach is observed.

Nils Margenberg, Carolin Mehlmann

Fully implicit tensor-product space-time discretizations of time-dependent $(p,δ)$-Navier-Stokes models yield, on each time step, large nonlinear monolithic saddle-point systems. In the shear-thinning regime $1<p<2$, especially as $p\downarrow 1$ and $δ\downarrow 0$, the decisive difficulty is the constitutive tangent: its ill-conditioning impairs Newton globalization and the preconditioning of the arising linear systems. We therefore develop a scalable monolithic modified Newton framework for tensor-product space-time finite elements in which the exact constitutive tangent in the Jacobian action is replaced by a better-conditioned surrogate. Picard and exact Newton serve as reference linearizations within the same algebraic framework. Scalability is achieved through matrix-free operator evaluation, a monolithic multigrid V-cycle preconditioner, order-preserving reduced Gauss-Radau time quadrature, and an inexact space-time Vanka smoother with single-time-point coefficient freezing in local patch matrices. We prove coercivity of the linearized viscous-Nitsche term in the uniformly elliptic regime $ν_\infty>0$ and consistency of the reduced time quadrature. Numerical tests demonstrate robustness with respect to model parameters, nonlinear and linear iteration counts, and scalable parallel performance.

Peter Munch, Marc Fehling, Martin Kronbichler et al.

In this article, we solve the Stokes and Navier-Stokes equations with the deal$.$II finite-element library. In particular, we use its multigrid, adaptive-mesh, and matrix-free infrastructures to design efficient linear and nonlinear iterative solvers, respectively. We solve the stationary Stokes equations on hp-adaptive meshes with a hp-multigrid approach, the transient Stokes equations with space-time finite elements and space-time multigrid, and, finally, the stabilized incompressible Navier-Stokes equations on locally refined meshes with a monolithic multigrid solver. The selected examples underline the flexibility and modularity of the multigrid infrastructure of deal$.$II.