A direct solver with reutilization of previously-computed LU factorizations for h-adaptive finite element grids with point singularities

TL;DR

This paper introduces a direct solver for h-adaptive finite element grids with point singularities that achieves linear computational cost by reusing LU factorizations, significantly improving efficiency over traditional methods.

Contribution

The paper presents a novel LU factorization reutilization technique tailored for h-adaptive meshes with point singularities, reducing computational complexity from quadratic to linear.

Findings

Achieves O(N) computational cost for mesh sequences

Reutilization of LU factorizations reduces overall computation

Numerical results confirm theoretical efficiency gains

Abstract

This paper describes a direct solver algorithm for a sequence of finite element meshes that are h-refined towards one or several point singularities. For such a sequence of grids, the solver delivers linear computational cost O(N) in terms of CPU time and memory with respect to the number of unknowns N. The linear computational cost is achieved by utilizing the recursive structure provided by the sequence of h-adaptive grids with a special construction of the elimination tree that allows for reutilization of previously computed partial LU factorizations over the entire unrefined part of the computational mesh. The reutilization technique reduces the computational cost of the entire sequence of h-refined grids from O(N^2) down to O(N). Theoretical estimates are illustrated with numerical results on two- and three-dimensional model problems exhibiting one or several point singularities.