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Solving the (Navier-)Stokes equations with space and time adaptivity using deal.II

Peter Munch, Marc Fehling, Martin Kronbichler, Nils Margenberg, Laura Prieto Saavedra

Abstract

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.

Solving the (Navier-)Stokes equations with space and time adaptivity using deal.II

Abstract

In this article, we solve the Stokes and Navier-Stokes equations with the dealII 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 dealII.

Paper Structure

This paper contains 6 sections, 10 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Multigrid in deal.II
  3. Multigrid for hp-adaptive stationary Stokes
  4. Space-time multigrid for transient Stokes
  5. Multigrid for stabilized Navier--Stokes on locally refined meshes
  6. Conclusions & Outlook

Figures (5)

  • Figure 1: deal.II supports geometric, polynomial, and non-nested transfer, enabling multigrid variants with the same name. Furthermore, it supports multivectors. Time transfer is not supported, however, can be added in application codes, using inheritance.
  • Figure 2: Transfer types: (geometric) local smoothing, (geometric) global coarsening, and (global) polynomial coarsening. Nesting polynomial and geometric multigrid gives a (hybrid) hp-multigrid.
  • Figure 3: Overview of the meshes used in experiments 1, 3, and 4. The color plots show the pressure and the vector plots the velocity.
  • Figure 4: Flow through Y-pipe (Exp. 1): robustness metrics after solving the Stokes system with successive hp-adaptive refinements on 256 MPI processes. Left: comparison of the number of outer solver iterations, using AMG or hp-multigrid with plane or ASM-enhanced point Jacobi for smoothing. Right: estimation of maximum eigenvalue of the preconditioned $A$-block.
  • Figure 6: Flow around a sphere (Exp. 4): time per linear iteration on each level for the $\bm{Q}_2Q_2$ case with $l=2$. It showcases the average, minimum, and maximum time among all processes.