Researches on Dynamic Load Balancing Algorithms and hp-Adaptivity in 3-D Parallel Adaptive Finite Element Computations

TL;DR

This paper advances parallel adaptive finite element computations by developing new algorithms for mesh partitioning, Hamiltonian path construction, and hp-adaptive strategies, validated through implementation in the PHG toolbox.

Contribution

It introduces efficient algorithms for Hamiltonian paths, high-dimensional Hilbert ordering, and improved hp-adaptive strategies within the PHG framework.

Findings

Proved existence of Hamiltonian paths and cycles in tetrahedral meshes.

Developed linear complexity algorithms for Hamiltonian path construction.

Demonstrated exponential convergence and superior accuracy of the new hp-adaptive strategy.

Abstract

This work is related to PHG (Parallel Hierarchical Grid). PHG is a toolbox for developing parallel adaptive finite element programs, which is under active development at the State Key Laboratory of Scientific and Engineering Computing. The main results of this work are as follows. 1) For the tetrahedral meshes used in PHG, under reasonable assumptions, we proved the existence of through-vertex Hamiltonian paths between arbitrary two vertices, as well as the existence of through-vertex Hamiltonian cycles, and designed an efficient algorithm with linear complexity for constructing through-vertex Hamiltonian paths. The resulting algorithm has been implemented in PHG, and is used for ordering elements in the coarsest mesh for the refinement tree mesh partitioning algorithm. 2) We designed encoding and decoding algorithms for high dimensional Hilbert order. Hilbert order has good locality, and it has wide applications in various fields in computer science, such as memory management, database, and dynamic load balancing. 3) We implemented refinement tree and space filling curve based mesh partitioning algorithms in PHG, and designed the dynamic load balancing module of PHG. The refinement tree based partitioning algorithm was originally proposed by Mitchell, the one implemented in PHG was improved in several aspects. The space filling curve based mesh partitioning function in PHG can use either Hilbert or Morton space filling curve. 4) We studied existing hp-adaptive strategies in the literature, and proposed a new strategy. Numerical experiments show that our new strategy achieves exponential convergence, and is superior, in both precision of the solutions and computation time, to the strategy compared. This part of the work also serves to validate the hp-adaptivity module of PHG.