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

TL;DR

This paper demonstrates the use of deal.II's adaptive and multigrid infrastructure to efficiently solve various forms of the Navier-Stokes equations with space and time adaptivity.

Contribution

It introduces a modular approach leveraging deal.II's infrastructure to solve Navier-Stokes equations efficiently with adaptive meshes and multigrid methods.

Findings

Efficient solutions for stationary and transient Stokes equations using hp-multigrid.

Successful application of space-time finite elements for transient problems.

Effective monolithic multigrid solver for incompressible Navier-Stokes equations.

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.