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A direct solver for finite element matrices requiring \(O(N \log N)\) memory places. (English) Zbl 1340.65038
Brandts, J. (ed.) et al., Proceedings of the international conference ‘Applications of mathematics’, Prague, Czech Republic, May 15–17, 2013. In honor of the 70th birthday of Karel Segeth. Prague: Academy of Sciences of the Czech Republic, Institute of Mathematics (ISBN 978-80-85823-61-5). 225-239 (2013).
The paper presents a direct solver for sparse matrices arising from discretization of partial differential equations by means of the finite element method. In particular, a numbering of unknowns is constructed such that the fill-in of matrix factors is minimized. The a priori construction of this numbering is devised from the hierarchical structure of the finite elements, and a sequence of static condensation of unknowns follows. The method is demonstrated on a triangular domain, which allows for a straightforward sequence of subdivisions into triangular elements. Apart of the construction of the numbering, the factorization and back-substitution resembles modern sparse direct solvers. Details of the algorithm are given in the paper. Performance of the new approach is evaluated in Matlab, including comparison to the built-in sparse-direct solver, and it leads to very convincing results.
For the entire collection see [Zbl 1277.00032].
MSC:
65F05 Direct numerical methods for linear systems and matrix inversion
65F50 Computational methods for sparse matrices
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Software:
Matlab
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