Deepness analysis: Bringing optimal fronts to triangular finite element method
A scheme is presented for analizing finite-element triangulations. The method takes a random triangulated planar graph and gives a multifrontal way to solve the corresponding physical problem, so that the maximum bandwidth of each front is guaranteed to be optimal. The interesting characteristic of this scheme is that it introduces large-grained parallelism dictated by the domain structure.
A way to extend these results to unsymmetric systems is given. Some experimental results are also presented.
Unable to display preview. Download preview PDF.
- 1.J.J. Dongarra, I.S. Duff, D.C. Sorensen and H.A. Van der Vorst. Solving Linear Systems on Vector and Shared Memory Computers. 1991.Google Scholar
- 2.I.S. Duff. Parallel implementation of multifrontal schemes. Parallel Computing, 3:193–204, 1986.Google Scholar
- 3.I.S. Duff, A.S. Erisman and J.K. Reid. Direct Methods for Sparse Matrices. Oxford Science Publications, 1986.Google Scholar
- 4.I.S. Duff and J.K. Reid. The multifrontal solution of indefinite sparse symmetric linear systems. ACM Trans. Math. Softw., 9:302–325, 1983.Google Scholar
- 5.I.S. Duff and J.K. Reid. The multifrontal solution of unsymmetric sets of linear equations. SIAM J. Sci. Stat. Comput., 5:633–641, 1984.Google Scholar
- 6.I.S. Duff and J.A. Scott. MA42 — A new frontal code for solving sparse unsymmetric systems. Rutherford Appleton Laboratory, Report RAL-93-064, September 93.Google Scholar
- 7.D.J. Evans. The use of preconditioning in iterative methods for solving linear equations with symmetric positive definite matrices. J. Inst. Maths. Applics., 4:295–314, 1967.Google Scholar
- 8.J. Galtier Deepness analysis: bringinig optimal fronts to triangular finite element analysis. ACAPS memo 73, ACAPS Lab., McGill University, Montreal, Canada.Google Scholar
- 9.A. George. Nested dissection of a regular finite element mesh. SIAM J. Numer. Anal. Vol.10 No2, April 1973Google Scholar
- 10.A.J. Hoffman, M.S. Martin and D.J. Rose. Complexity bounds for regular finite difference and finite element grids. SIAM J. Numer. Anal., Vol.10 No2, April 1973Google Scholar
- 11.P. Hood. Frontal Solution Program for Unsymmetric Matrices. Int. J. num. Meth. Engng., 10:379–399, 1976.Google Scholar
- 12.B. Irons. A Frontal Solution Program for Finite Element Analysis. Int. J. num. Meth. Engng., 2:5–32, 1970.Google Scholar
- 13.D.A.H. Jacobs. Preconditioned conjugate gradient methods for solving systems of algebraic equations. Note No. RD/L/Nl93/80, CERL, Leatherhead, 1980.Google Scholar
- 14.P. King. Automatic reordering scheme for simultaneous equations derived from network systems. Int. J. num. Meth. Engng., 2:523–533, 1970.Google Scholar
- 15.R.J. Lipton, D.J. Rose and R.E. Tarjan. Generalized nested dissection. SIAM J. Numer. Anal., Vol. 16, No 2:346–358, 1979.Google Scholar
- 16.J.K. Reid. Frontal methods for solving finite-element systems of linear equations. Sparse Matrices and Their Uses, I.S. Duff (editor), Academic Press Inc. (London) LTD, 1981.Google Scholar
- 17.D.J. Rose. Triangulated graphs and the elimination process. J. of Math. Anal. and Appl., 32:597–609, 1970.Google Scholar
- 18.S.W. Sloan and M.F. Randolph. Automatic Element Reordering for Finite Element Analysis with frontal solution schemes. Int. J. num. Meth. Engng., 19:1153–1181, 1983.Google Scholar
- 19.W.F. Tinney and J.W. Walker. Direct solutions of sparse network equations by optimally ordered triangular factorization. Proc. of the IEEE, Vol.55, No11, November 1967.Google Scholar
- 20.M. Yannakakis. Computing the minimum fill-in is NP-complete. SIAM J. Alg. Disc. Meth., Vol.2, No1, March 1981.Google Scholar