Abstracts - Dipartimento di Elettronica Applicata

Meta 2010 & FEM 2010 – Rome, 13-15 December 2010
A Comparison of Direct Methods for the Solution
of Finite Element Systems on Shared Memory
Computers
Giuseppe Borzì
University of Messina, Department of Civil Engineering
Messina, Italy – E-mail: gborzi@ieee.org
The numerical solution of electromagnetic problems by means of the finite element
method involves the construction of a linear algebraic system and its solution. When
the order of the system is very high, the solution is achieved by means of iterative
solvers such as those of the Lanczos family [1], Multigrid [2] or Algebraic Multigrid
[3]. The convergence characteristics of iterative solvers are not well understood,
except for a few special cases, like Symmetric Positive Definite matrices for Lanczos
based solvers. Moreover, for complex linear systems the convergence theory is even
less understood than for real linear systems. So, for small and medium size linear
systems, direct solvers for sparse matrices can be more suitable. Unlike iterative
solvers, direct ones can be used as ‘black boxes’, that is to say, the user does not need
to give parameters such as the
convergence tolerance, or preconditioning parameters. Most of these parameters are
chosen heuristically, and an unwise choice may lead wrong results or a breakdown of
the iteration. The downside of direct methods is their higher memory and CPU usage
when compared with iterative solvers, but with the commercial availability of multi
core/threaded CPUs with huge memory this is no more a serious drawback. In this
paper, some public available direct solvers for sparse matrices, such as those included
in the suitesparse package [4], superlu and superlu_mt [5-6] and spooles [7] are
compared on some test matrices resulting from the finite element discretization of
electromagnetic problems.
References
[1] G. H. Golub and C. F. van Loan, Matrix Computations, 3rd Ed., The John Hopkins University
Press, Baltimore, USA, 1996
[2] W. L. Briggs, A Multigrid Tutorial, SIAM Books, Philadelphia, USA, 1987
[3] K. Stueben, “Algebraic multigrid (AMG): An introduction with applications,” GMD -
Forschungszentrum Informationstechnik GmbH, Tech. Rep. 53, Sankt Augustin, Germany, Mar.
1999
[4] T. A. Davis, Direct Methods for Sparse Linear Systems, SIAM Books, Philadelphia, USA, 2006
[5] J. W. Demmel, S. C. Eisenstat, J. R. Gilbert, X. S. Li, and J W. H. Liu, “A supernodal approach to
sparse partial pivoting,” SIAM J. Matrix Analysis and Applications, 20, 720-755, 1999
[6] J. W. Demmel, J. R. Gilbert, and X. S. Li, “An Asynchronous Parallel Supernodal Algorithm for
Sparse Gaussian Elimination,” SIAM J. Matrix Analysis and Applications, 20, 915-952, 1999
[7] C. Ashcraft, and R. Grimes, “SPOOLES: An object-oriented sparse matrix library,” Proceedings of
the Ninth SIAM Conference on Parallel Processing, San Antonio, Texas, USA, March 22-24,
1999.
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