A direct solver with reutilization of LU factorizations for h-adaptive finite element grids with point singularities
KAUST DepartmentApplied Mathematics and Computational Science Program
Earth Science and Engineering Program
Numerical Porous Media SRI Center (NumPor)
Physical Sciences and Engineering (PSE) Division
Environmental Science and Engineering Program
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AbstractThis 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(N2) down to O(N). Theoretical estimates are illustrated with numerical results on two- and three-dimensional model problems exhibiting one or several point singularities. © 2013 Elsevier Ltd. All rights reserved.
SponsorsMP was supported by Polish National Science Center grants no NN 519 447739 and NN 519 405737. DP was partially funded by the Project of the Spanish Ministry of Sciences and Innovation MTM2010-16511, the Laboratory of Mathematics (UFI 11/52), and the Ibero-American Project CYTED 2011 (P711RT0278). MP and DP were partially supported by the Center for Numerical Porous Media at KAUST.