Volume and neighbors algorithm for finding elimination trees for three dimensional [formula omitted]-adaptive grids

Abstract

This paper presents an algorithm called “volume & neighbors” for finding elimination trees for multifrontal solver algorithm applied to three dimensional h-adaptive finite element method computations. The algorithm is described in a pseudo-code and explained on exemplary h-refined grids. The algorithm is implemented in three dimensional h-adaptive finite element method code and tested on a sequence of representative grids, namely uniform grid, grid with point singularity, grid with edge singularity and grid with face singularity. The number of floating point operations for the multifrontal solver algorithm working with the elimination trees generated by the volume & neighbors algorithm is compared with the number of floating-point operations resulting from execution of the state-of-the-art multifrontal direct solver MUMPS with state-of-the-art algorithms for constructing elimination trees like nested-dissection from METIS library and approximate minimum degree AMD algorithm as well as PORD algorithm. In all the cases the volume & neighbors algorithm outperforms other state-of-the art algorithms. The only exception is the case of the uniform grid, where the algorithm results in a similar number of FLOPs than nested dissection algorithm.