Efficiency of local Vanka smoother geometric multigrid preconditioning for space‐time finite element methods to the Navier–Stokes equations

Abstract

Numerical simulation of incompressible viscous flow continues to remain a challenging task, in particular if three space dimensions are involved. Space-time finite element methods feature the natural construction of higher order discretization schemes. They offer the potential to achieve accurate results on computationally feasible grids. Using a temporal test basis supported on the subintervals and linearizing the resulting algebraic problems by Newton's method yield linear systems of equations with block matrices built of (k + 1) × (k + 1) saddle point systems, where k denotes the polynomial order of the variational time discretization. We demonstrate numerically the efficiency of preconditioning GMRES iterations for solving these linear systems by a V -cycle geometric multigrid approach based on a local Vanka smoother. The studies are done for the two-and three-dimensional benchmark problem of flow around a cylinder. The robustness of the solver with respect to the piecewise polynomial order k in time is analyzed and confirmed numerically.

Benchmark computations of dynamic poroelasticity

“…The pair $$\mathbb{Q}_r^d/\mathbb{P}_{r-1}^{\operatorname{disc}}$$ with a discontinuous approximation of p in the broken polynomial space $$Q_h$$ has proved excellent accuracy and stability properties for higher order approximations of mixed (or saddle point) systems like the Navier–Stokes equations and the applicability of geometric multigrid preconditioner for the algebraic systems; cf. for example, [19, 20].…”