On equilibration and sparse factorization of matrices arising in finite element solutions of partial differential equations

Numerical Methods for Partial Differential Equations

Volumn 16, Issue 1, 2000, Pages 11-29

  De Almeida, Valmor F ^a   Chapman, Andrew M ^b   Derby, Jeffrey J ^a  

^a University of Minnesota   (United States)

^b NEC Systems Inc   (United States)

Author keywords

Incomplete LUfactorization; Incompressible fluids; Matrix equilibration; Navier Stokes equations; Pivoting; Preconditioning; Scaling; Sparse finite element matrices

Indexed keywords

EID: 0004829329     PISSN: 0749159X     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1098-2426(200001)16:1<11::AID-NUM2>3.0.CO;2-X     Document Type: Article

References (33)

1. A. M. Chapman, Y. Saad, and L. Wigton, High order ILU preconditioners for CFD problems, Int J Numer Methods Fluids, to appear.
2. W. Hackbush and G. Wittum (editors), Incomplete decomposition (ILU) - algorithms, theory, and applications, Proc GAMM-Seminar, January 24-26, 1992. Vieweg, Germany, 1993.
3. A. van der Sluis, Condition, equilibration and pivoting in linear algebraic systems, Numer Math 15 (1970), 74-86.
4. F. L. Bauer, Optimally scaled matrices, Numer Math 5 (1963), 73-87.
5. C. McCarthy and G. Strang, Optimal conditioning of matrices, SIAM J Numer Anal 10 (1973), 370-388.
6. A. van der Sluis, Condition numbers and equilibration of matrices, Numer Math 14 (1969), 14-23.
7. R. D. Skeel, Scaling for numerical stability in Gaussian elimination, J ACM 26 (1979), 494-526.
8. G. Poole and L. Neal, Gaussian elimination: when is scaling beneficial, Linear Algebra Appl 162 (1992), 309-324.
9. G. Dahlquist and Å. Björck, "Numerical methods," Series in Automatic Computation, Prentice-Hall, Englewood Cliffs, NJ, 1974.
10. G. H. Golub and C. F. Van Loan, "Matrix computations," Johns Hopkins series in the mathematical sciences, Johns Hopkins Univ, Baltimore, 1989.
11. C. T. Kelley, "Iterative methods for linear and nonlinear equations," Frontiers in applied mathematics, Soc Indust Appl Math, Philadelphia, 1995.
12. W. Lin and K. E. Benson, The science and engineering of large-diameter Czochralski silicon crystal growth, Ann Rev Mater Sci 17 (1987), 273-298.
13. J. C. Rojo and J. J. Derby, On the formation of rotational spoke patterns on the melt surface during czochralski growth of molten bismuth silicon oxide, J Crystal Growth 198 (1999), 154-160.
14. S. Chandrasekhar, Hydrodynamic and hydromagnetic stability, Dover New York, 1981.
15. J. D. Bierlein, H. Vanherzeele, Potassium titanyl phosphate: properties and new applications, J Opt Soc Am B 6 (1989), 622-633.
16. B. Vartak, Y. I. Kwon, A. Yeckel, and J. J. Derby, Fluid flow and mass transfer during the solution growth of potassium titanyl phosphate, J Crystal Growth, to appear.
17. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Vol 15, Computational mathematics, Springer-Verlag, New York, 1991.
18. M. D. Gunzburger, "Finite element methods for viscous incompressible flows," Computer science and scientific computing, Academic, Boston, 1989.
19. V. F. de Almeida and J. J. Derby, Construction of solution curves for large 2-D problems of steady-state flows of incompressible fluids, SIAM J Sci Comput, to appear.
20. V. Girault and P. A. Raviart, "Finite element methods for Navier-Stokes equations," Springer series in computational mathematics, Springer-Verlag, New York, 1986.
21. E. L. Allgower and K. Georg, Numerical continuation methods, Vol 15, Springer series in computational mathematics, Springer-Verlag, New York, 1990.
22. H. B. Keller, Numerical methods for two-point boundary-value problems, Dover, New York, 1992.
23. Y. Saad, "Iterative methods for sparse linear systems," PWS Series in computer science, PWS, New York, 1996.
24. B. F. Smith, P. E. Bjøorstad, and W. D. Gropp, Domain decomposition parallel multilevel methods for elliptic partial differential equations, Cambridge Univ, New York, 1996.
25. I. S. Duff, A. M. Erisman, and J. K. Reid, "Direct methods for sparse matrices," Monographs on numerical analysis, Clarendon, Oxford, England, 1989.
26. J. J. Dongarra, I. S. Duff, D. C. Sorensen, and H. A. van der Vorst, Solving linear systems on vector and shared memory computers. Soc Indust Appl Math, Philadelphia, 1991.
27. I. S. Duff and J. A. Scott, MA42 - a new frontal code for solving sparse unsymmetric systems, Research Report RAL 93-064, Central Computing Department, Atlas Centre, Rutherford Appleton Laboratory, Oxon OX11 0QX, U.K., 1993.
28. V. F. de Almeida A. M. Chapman, and J. J. Derby, On equilibration and sparse factorization of matrices arising in finite element solutions of partial differential equations, Minnesota Supercomputer Institute Research Report 98/165, Univ Minnesota, Dept Chem Eng Materials Sci, Minneapolis, MN 55455-0132, 1998.
29. S. W. Sloan, A FORTRAN program for profile and wavefront reduction, Int J Num Meth Eng 28 (1989), 2651-2679.
30. N. J. Higham, The accuracy of solutions to triangular systems, SIAM J Numer Anal 26 (1989), 1252-1265.
31. P. N. Brown and Y. Saad, Hybrid Krylov methods for nonlinear systems of equations, SIAM J Sci Stat Comput 3 (1990), 450-481.
32. V. F. de Almeida J. J. Derby, Practical mixed finite elements for 3-D flows of incompressible fluids. Minnesota Supercomputer Institute Research Report 99, Univ Minnesota, Dept Chem Eng Materials Sci, Minneapolis, MN 55455-0132, to appear.
33. A. J. Yeckel and J. J. Derby, Parallel computation of incompressible flows in materials processing: Numerical experiments in diagonal preconditioning, Parallel Comp 23 (1997), 1379-1400.