Marc Fehling | Searcharxiv

arXiv2026

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

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.

math.NA↗

arXiv2022

Algorithms for Parallel Generic $hp$-adaptive Finite Element Software

The $hp$-adaptive finite element method (FEM) - where one independently chooses the mesh size ($h$) and polynomial degree ($p$) to be used on each cell - has long been known to have better theoretical convergence properties than either $h$- or $p$-adaptive methods alone. However, it is not widely used, owing at least in parts to the difficulty of the underlying algorithms and the lack of widely usable implementations. This is particularly true when used with continuous finite elements. Herein, we discuss algorithms that are necessary for a comprehensive and generic implementation of $hp$-adaptive finite element methods on distributed-memory, parallel machines. In particular, we will present a multi-stage algorithm for the unique enumeration of degrees of freedom (DoFs) suitable for continuous finite element spaces, describe considerations for weighted load balancing, and discuss the transfer of variable size data between processes. We illustrate the performance of our algorithms with numerical examples, and demonstrate that they scale reasonably up to at least 16,384 Message Passing Interface (MPI) processes. We provide a reference implementation of our algorithms as part of the open-source library deal.II.

math.NA↗