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

Authors:
Maciej Paszynski;David Pardo;Victor M. Calo
Affiliations:
AGH University of Science and Technology, Faculty of Computer Science, Electronics and Telecommunication, Department of Computer Science, al. A. Mickiewicza 30, 30-059 Krakow, Poland;Department of Applied Mathematics, Statistics and Operational Research, University of the Basque Country UPV/EHU, Leioa, Spain and IKERBASQUE (Basque Foundation for Science), Bilbao, Spain;Center for Numerical Porous Media, Applied Mathematics & Computational Science and Earth Science & Engineering, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia
Venue:
Computers & Mathematics with Applications
Year:
2013

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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 (or Cholesky) 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.