Efficient parallel solvers for the biharmonic equation

Parallel Computing

Volumn 30, Issue 1, 2004, Pages 35-55

  Mihajlović, Milan D ^a   Silvester, David J ^a  

^a UNIVERSITY OF MANCHESTER   (United Kingdom)

Author keywords

Biharmonic equation; Finite elements; Krylov solvers; Multigrid; Preconditioning; Sparse direct linear solvers

Indexed keywords

ALGORITHMS; APPROXIMATION THEORY; BENDING MOMENTS; BOUNDARY ELEMENT METHOD; COMPUTATIONAL METHODS; COMPUTER ARCHITECTURE; CONVERGENCE OF NUMERICAL METHODS; DATA STORAGE EQUIPMENT; EIGENVALUES AND EIGENFUNCTIONS; FINITE ELEMENT METHOD; FLUID MECHANICS; ITERATIVE METHODS; LAPLACE TRANSFORMS; PROBLEM SOLVING; VECTORS;

BIHARMONIC EQUATIONS; KRYLOV SOLVERS; MULTIGRIDS; PRECONDITIONING; SPARSE DIRECT LINEAR SOLVERS;

LINEAR ALGEBRA;

EID: 0347601911     PISSN: 01678191     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.parco.2003.07.002     Document Type: Article

References (25)

1. Babuška I., Osborn J., Pitkäranta J. Analysis of mixed methods using mesh dependent norms. Math. Comp. 35:1980;1039-1062.
2. Battermann A., Sachs E.W. Block preconditioners for KKT systems in PDE-governed optimal control problems. Hoppe R.H.W., Hoffmann K.-H., Schulz V. Proceedings of Workshop on Fast Solutions of Discretised Optimisation Problems. 2001;1-18 Birkhäuser-Verlag, Berlin.
3. Bjørstad P.E., Tjøstheim B.P. Efficient algorithms for solving a fourth order equation with the spectral Galerkin method. SIAM J. Sci. Comp. 18:1997;621-632.
4. Bjørstad P.E., Tjøstheim B.P. High precision solutions of two fourth order eigenvalue problems. Computing. 63:1999;97-107.
5. Brezzi F., Fortin M. Mixed and Hybrid Finite Element Methods. 1991;Springer-Verlag, New York.
6. Brown B.M., Jimack P.K., Mihajlović M.D. A new direct solver for a class of mixed finite element problems. Appl. Numer. Math. 38(1-2):2001;1-20.
7. Brown B.M., Davies E.B., Jimack P.K., Mihajlović M.D. A numerical investigation of the solution of a class of fourth-order eigenvalue problems. Proc. Roy. Soc. (London) A. 456:2000;1505-1521.
8. Dongarra J.J., Duff I.S., Sorensen D.C., van der Vorst H.A. Numerical Linear Algebra for High-Performance Computers. 1998;SIAM, Philadelphia.
9. Glowinski R., Pironneau O. Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem. SIAM Rev. 21:1979;167-212.
10. Gould N.I.M., Hribar M.E., Nocedal J. On the solution of equality constrained quadratic programming problems arising in optimisation. SIAM J. Sci. Comp. 23(4):2001;1375-1394.
11. HSL 2000, A collection of Fortran codes for large scale scientific computation. Available from 〈http://www.numerical.rl.ac.uk〉.
12. Keller C., Gould N.I.M., Wathen A.J. Constraint preconditioning for indefinite linear systems. SIAM J. Matrix Anal. Appl. 21(4):2000;1300-1317.
13. Kozlov V.A., Kondratev V.A., Mazya V.G. On sign variation and the absence of "strong" zeros of solutions of elliptic equations. Mat. USSR Izv. 34:1990;337-353.
14. MG Net. Available from 〈http://www.mgnet.org/mgnet-codes-bunner.html〉
15. Moffat K. Viscous and restrictive eddies near a sharp corner. J. Fluid. Mech. 18:1964;1-18.
16. http://www.mpi-forum.org/docs/mpi-20-html/mpi2-report.html.
17. Numerical Algorithms Group: NAG manual, Fortran Library Mark 20, 2002.
18. Peisker P. A multilevel algorithm for the biharmonic problem. Numer. Math. 46:1985;623-634.
19. Perugia I., Simoncini V. Block-diagonal and indefinite symmetric preconditioners for mixed finite element formulations. Numer. Linear Alg. Appl. 7(7-8):2000;585-616.
20. Scott J.A. A parallel frontal solver for finite element applications. Int. J. Numer. Meth. Eng. 50:2001;1131-1144.
21. D.J. Silvester, M.D. Mihajlović, A black-box multigrid preconditioner for the biharmonic equation, BIT, submitted for publication. (Available from 〈http://www.maths.ac.uk/̃nareports/narep362.ps.gz〉).
22. Sleijpen G.L.G., Fokkema D.R. BiCGSTAB. (ℓ) for linear equations involving unsymmetric matrices with complex spectrum Electron. Trans. Numer. Anal. 1:1993;11-32.
23. Stüben K. A review of algebraic multigrid. J. Comput. Appl. Math. 128:2001;281-309.
24. Wathen A.J. Realistic eigenvalue bounds for the Galerkin Mass Matrix. IMA J. Numer. Anal. 7:1987;449-457.
25. P. Wesseling, MGD1 - A robust and efficient multigrid method, in: Lecture Notes in Mathematics, vol. 960, 1982, pp. 614-630.