Solving the (Navier-)Stokes equations with space and time adaptivity using deal.II
This work addresses computational fluid dynamics problems for researchers and engineers, but it is incremental as it applies existing methods to new implementations.
The authors tackled solving Stokes and Navier-Stokes equations by leveraging the deal.II library's multigrid and adaptive-mesh capabilities, resulting in efficient iterative solvers for stationary, transient, and incompressible cases.
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.