Block-diagonal and indefinite symmetric preconditioners for mixed finite element formulations

Numerical Linear Algebra with Applications

Volumn 7, Issue 7-8, 2000, Pages 585-616

  Perugia, I ^a,c   Simoncini, V ^b  

^a UNIVERSITY OF PAVIA   (Italy)

^b CNR   (Italy)

^c UNIVERSITY OF MINNESOTA   (United States)

Author keywords

Finite elements; Iterative methods; Mixed methods; Preconditioning

Indexed keywords

EID: 0034368563     PISSN: 10705325     EISSN: None     Source Type: Journal    
DOI: 10.1002/1099-1506(200010/12)7:7/8<585::aid-nla214>3.0.co;2-f     Document Type: Article

References (52)

1. Wathen AJ, Silvester D. Fast iterative solution of stabilized Stokes systems - Part I: Using general block preconditioned. SIAM Journal on Numerical Analysis 1993; 30:630-649.
2. Rusten T, Winther R. A preconditioned iterative method for saddle point problems. SIAM Journal on Matrix Analysis and Applications 1992; 13:887-902.
3. Arnold DN, Falk RS, Winther R. Preconditioning in H(div) and applications. Mathematics of Computation 1997; 66:957-984.
4. Rusten T, Winther R. Substructure preconditioners for elliptic saddle point problems. Mathematics of Computation 1993; 60:23-48.
5. Rusten T, Vassilevski PS, Winther R. Interior penalty preconditioners for mixed finite element approximations of elliptic problems. Mathematics of Computation 1996; 65:447-466.
6. Ewing RE, Lazarov RD, Lu P, Vassilevski PS. Preconditioning indefinite systems arising from mixed finite element discretization of second-order elliptic problems. In Notes in Mathematics, Springer, 1990; 1457: 28-43.
7. Elman H. Multigrid and Krylov subspace methods for the discrete Stokes equations. International Journal for Numerical Methods in Fluids 1996; 22:755-770.
8. Silvester D, Wathen AJ. Fast iterative solution of stabilized Stokes systems - Part II: Using general block preconditioners. SIAM Journal on Numerical Analysis 1994; 31:1352-1367.
9. Bramble JH, Pasciak JE. A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems. Mathematics of Computation 1988; 50:1-17.
10. Fischer B, Ramage A, Silvester D, Wathen A. Minimum residual methods for augmented systems. BIT 1998; 38:527-543.
11. Elman HC, Silvester DJ, and Wathen AJ. Iterative methods for problems in computational fluid dynamics. In Iterative Methods in Scientific Computing, Chan CT and Golub G (eds). Springer-Verlag, 1997.
12. Klawonn A. An optimal preconditioner for a class of saddle point problems with a penalty term. SIAM Journal on Scientific Computing 1998; 19:540-552.
13. Klawonn A. Block-triangular preconditioners for saddle-point problems with a penalty term. SIAM Journal on Scientific Computing 1998; 19:172-184.
14. Bossavit A. 'Mixed' systems of algebraic equations in computational electromagnetism. COMPEL 1998; 17:59-63.
15. Perugia I, Simoncini V, Arioli M. Linear algebra methods in a mixed approximation of magneto static problems. SIAM Journal on Scientific Computing 2000; 21(3): 1085-1101.
16. Gill PE, Murray W, Ponceléon DB, Saunders MA. Preconditioners for indefinite systems arising in optimization, SIAM Journal on Matrix Analysis and Applications 1992; 13:292-311.
17. Lukšan L, Vlček J. Indefinitely preconditioned inexact Newton method for large sparse equality constrained non-linear programming problems. Numerical Linear Algebra with Applications 1998; 5:219-247.
18. Keller C, Gould NIM, Wathen AJ. Constraint preconditioning for indefinite linear systems. Technical Report NA 99/01, Oxford University Computing Laboratory, UK, 1999.
19. Golub GH, Wathen AJ. An iteration for indefinite systems and its application to the Navier-Stokes equations. SIAM Journal on Scientific Computing 1998; 19:530-539.
20. Bramble JH. Multigrid Methods, Pitman Research Notes in Mathematics Series. Longman, Harlow, UK, 1993.
21. Morano E, Mavriplis DJ, Venkatakrishnan V. Coarsening strategies for unstructured multigrid techniques with application to anisotropic problems. SIAM Journal on Scientific Computing 1998; 20:393-415.
22. Arnold DN, Falk RS, Winther R. Multigrid in H(div) and H(curl). Numerische Mathematik 2000; 85:197-218.
23. Chan T, Go S, Zikatanov L. Lecture Notes on Multilevel Methods for Elliptic Problems on Unstructured Grids. UCLA Computational and Applied Math Reports, March 1997.
24. Rikabi J, Bryant CF, Freeman EM. An error-based approach to complementary formulations of static field solutions. International Journal for Numerical Methods in Engineering 1988; 26:1963-1987.
25. Bossavit A. A rationale for edge-elements in 3D fields computation. IEEE Transactions on Magnetics 1998; 24:74-79.
26. Dular P, Remacle JF, Henrotte F, Genon A, Legros W. Magnetostatic and magnetodynamic mixed formulations compared with conventional formulations. IEEE Transactions on Magnetics 1997; 33:1302-1305.
27. Ren Z, Ida N. Derivation of various dual formulations in magnetostatics via error based energy approach. IEEE Transactions on Magnetics 1999; 35:1167-1170.
28. Perugia I. A field-based mixed formulation for the 2-D magnetostatic problem. SIAM Journal on Numerical Analysis 1997; 34:2382-2391.
29. Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. Springer, New York, 1991.
30. 3. Numerische Mathematik 1980; 35:315-341.
31. 3, Numerische Mathematik 1982; 39:97-112.
32. Quarteroni A, Valli A. Numerical Approximation of Partial Differential Equations. Springer, Berlin, 1994.
33. Raviart PA, Thomas JM. A mixed finite element method for second order elliptic problems, Mathematical Aspects of the Finite Element Method, Lecture Notes in Mathematics, Springer, New York, 1977; 606:10-12.
34. Fernandes P, Guardi G. Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions. Mathematical Models and Methods in the Applied Sciences 1997; 7:957-991.
35. Alotto P, Perugia I. A fieldO-based finite element method for magnetostatics derived from an error minimisation approach. International Journal for Numerical Methods in Engineering 2000; 49;573-598.
36. Perugia I, Simoncini V. Preconditioners for a mixed finite element method in Magnetostatics. In 1999 International Conference on Preconditioning Techniques for Large Sparse Matrix Problems in Industrial Applications, Minneapolis, MN, 1999; 119-123.
37. Stewart GW, Sun JG. Matrix Perturbation Theory. Academic Press, 1990.
38. Greenbaum A. Iterative Methods for Solving Linear Systems, SIAM, PA, 1997.
39. Wathen A, Fischer B, Silvester D. The convergence rate of the minimal residual method for the Stokes problem. Numerische Mathematik 1995; 71:121-134.
40. Saad Y. Iterative Methods for Sparse Linear Systems. PWS Publishing, 1996.
41. Freund RW, Nachtigal NM. Software for simplified Lanczos and QMR algorithms. Applied Numerical Mathematics 1995; 19:319-341.
42. Parlett BN, Taylor DR, Liu ZA. A look-ahead Lanczos algorithm for unsymmetric matrices. Mathematics of Computation 1985; 44:105-124.
43. Freund R, Gutknecht M, Nachtigal N. An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices. SIAM Journal on Scientific Computing 1993; 14:137-158.
44. Rozloznik M, Simoncini V. Short-term recurrences for indefinite preconditioning of saddle point problems. Technical Report 1181, Istituto di Aualisi Numerica del C.N.R., Pavia, July 2000.
45. AEA Technology, Harwell Subroutine Library, Harwell Laboratory, Oxfordshire, UK, December 1995.
46. Matstoms P. Sparse QR factorization with applications to linear least squares problems. PhD thesis, Department of Mathematics, Linköping, Sweden, August 1994.
47. Benzi M, Tuma M. A comparison of some preconditioning techniques for general sparse matrices. In Iterative Methods in Linear Algebra, II, IMACS Series in Computational and Applied Mathematics, Margenov S and Vassilevski P (eds). 1996; 3:191-203.
48. Björck Å. Numerical Methods for Least Squares Problems. SIAM, Philadelphia, PA, USA, 1996.
49. Wathen AJ. An analysis of some element-by-element techniques. Computational Methods Applied to Mechanics and Engineering 1989; 77:271 -287.
50. Axelsson O, Barker VA. Finite Element Solution of Boundary Value Problems - Theory and Computation, Computer Science and Applied Mathematics Series. Academic Press, London, 1984.
51. Wathen AJ. Realistic eigenvalue bounds for the Galerkin mass matrix. IMA Journal of Numerical Analysis 1987; 7:449-457.
52. George PL. Automatic mesh generation and finite element computation, in Handbook of Numerical Analysis, vol. IV, Ciarlet PG, Lions JL (eds). Elsevier Science, 1996; 69-190.