Ordering of Elements for the Volume & Neighbors Algorithm Constructing Elimination Trees for 2D and 3D h- adaptive FEM

Abstract

In this paper we analyze the optimality of the volume and neighbors algorithm constructing elimination trees for three dimensional h-adaptive finite element method codes. The algorithm is a greedy algorithm that constructs the elimination trees based on the bottom up analysis of the computational mesh. We compare the results of the volume and neighbors greedy algorithm with the global dynamic programming optimization performed on a class of elimination trees. The comparison is based on the Directed Acyclic Graph (DAG) constructed for model grids. We construct DAGs for two model grids: a two dimensional grid refined towards point singularitiy and a two dimensional grid refined towards edge singularity. We show that the quasi-optimal trees created by the volume and neighbors algorithm for the model grids are also captured by the dynamic programming procedure. It means that created elimination trees are optimal in the considered class of elimination trees. We show that different element orderings at the input of the volume and neighbors algorithm result in different computational costs of the multi-frontal solver algorithm executed over the resulting elimination trees. Finally we present the ordering of elements that results in optimal (in the considered class) elimination trees. The theoretical results are verified with numerical experiments performed on a three dimensional grids with point, edge and face singularities.