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

Abstract

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.

arXiv comment27 pages, 10 figures

References in corpus (9)

• MFEM: a modular finite element methods library
• The deal.II finite element library: design, features, and insights
• Flexible and scalable particle-in-cell methods for massively parallel computations
• Hybrid multigrid methods for high-order discontinuous Galerkin discretizations
• Hierarchical multigrid approaches for the finite cell method on uniform and multi-level hp-refined grids
• Parallelization of the multi-level hp-adaptive finite cell method
• Performance Portable Solid Mechanics via Matrix-Free $p$-Multigrid
• Efficient distributed matrix-free multigrid methods on locally refined meshes for FEM computations
• Implementation of $hp$-adaptive discontinuous finite element methods in Dune-Fem

Cited by in corpus (1)

• Anisotropic hp space-time adaptivity and goal-oriented error control for convection-dominated problems