Lagrangian formulation of domain decomposition methods: A unified theory

Applied Mathematical Modelling

Volumn 30, Issue 7, 2006, Pages 593-615

  Magoulès, Frédéric ^a   Roux, François Xavier ^b  

^a UNIVERSITÉ HENRI POINCARÉ NANCY 1   (France)

^b ONERA   (France)

Author keywords

Augmented Lagrangian; Domain decomposition; Interface conditions; Lagrange multipliers; Saddle point problem

Indexed keywords

MATHEMATICAL MODELS; PROBLEM SOLVING;

AUGMENTED LAGRANGIAN; DOMAIN DECOMPOSITION; INTERFACE CONDITIONS; SADDLE POINT PROBLEM;

LAGRANGE MULTIPLIERS;

LAGRANGE MULTIPLIERS; MATHEMATICAL MODELS; PROBLEM SOLVING;

EID: 33645781091     PISSN: 0307904X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apm.2005.06.016     Document Type: Article

References (44)

1. Smith B., Bjorstad P., and Gropp W. Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations (1996), Cambridge University Press
2. Quarteroni A., and Valli A. Domain Decomposition Methods for Partial Differential Equations (1999), Oxford University Press, Oxford
3. A. Toselli, O.B. Widlund, Domain Decomposition Methods: Algorithms and Theory, Springer Series in Computational Mathematics, vol. 34, 2004.
4. C.-H. Lai, P.E. Bjørstad, M. Cross, O. Widlund, Eleventh International Conference on Domain Decomposition Methods, in: Proceedings of the 11th International Conference on Domain Decomposition Methods in Greenwich, England, 20-24 July 1998.
5. Dryja M., and Widlund O.B. Towards a unified theory of domain decomposition algorithms for elliptic problems. Proceedings of the Third International Symposium on Domain Decomposition Methods for Partial Differential, held in Houston, Texas, 20-22 March 1989 (1990), SIAM
6. Collino F., Ghanemi S., and Joly P. Domain decomposition method for harmonic wave propagation: a general presentation. Comput. Methods Appl. Mech. Eng. 184 171-211 (2000)
7. Farhat C., and Le Tallec P. Vistas in domain decomposition and parallel processing in computational mechanics. Comput. Methods Appl. Mech. Eng. 184 2-4 (2000)
8. Lai C.-H., Ierotheou C.S., Palansuriya C.J., Espedal M.S., and Tai X.-C. Accuracy of a domain decomposition method for the recovering of discontinuous heat sources in metal sheet cutting. Comput. Visual. Sci. 2 (1999) 149-152
9. Keyes D.E. Domain decomposition methods in the mainstream of computational science. Proceedings of the 14th International Conference on Domain Decomposition Methods (2003), UNAM Press, Mexico City 79-93
10. Lions P.-L. In: Glowinski R., Golub G.H., Meurant G.A., and Périaux J. (Eds). On the Schwarz Alternating Method I (1988), SIAM, Philadelphia, PA 1-42
11. Lions P.-L. In: Chan T., Glowinski R., Périaux J., and Widlund O. (Eds). On the Schwarz Alternating Method II (1989), SIAM, Philadelphia, PA 47-70
12. Lions P.-L. In: Chan T., Glowinski R., Périaux J., and Widlund O. (Eds). On the Schwarz Alternating Method. III: A Variant for Nonoverlapping Sub-domains (1989), SIAM, Philadelphia, PA
13. Le Tallec P. In: Tinsley J. (Ed). Domain Decomposition Methods in Computational Mechanics vol. 1(2) (1994), North-Holland, Oden 121-220
14. Farhat C., and Roux F.X. Implicit parallel processing in structural mechanics. Comput. Mech. Adv. 2 (1994) 1-124
15. Carvalho L.M., Giraud L., and Le Tallec P. Algebraic two-level preconditioners for the Schur complement method. SIAM J. Sci. Comput. 22 6 (2001) 1987-2005
16. Blank L. Preconditioning via a Schur complement method: an application in state estimation. SIAM J. Sci. Comput. 25 3 (2003) 942-960
17. Mandel J., and Dohrmann C.R. A Convergence of a balancing domain decomposition by constraints and energy minimization. Numer. Linear Algebra Appl. 10 (2003) 639-659
18. Farhat C. A Lagrange multiplier based on divide and conquer finite element algorithm. J. Comput. System. Eng. 2 149-156 (1991)
19. Farhat C., and Roux F.-X. A method of finite element tearing and interconnecting and its parallel solution algorithm. Int. J. Numer. Methods Eng. 32 (1991) 1205-1227
20. Farhat C., and Roux F.X. An unconventional domain decomposition method for an efficient parallel solution of large-scale finite element systems. SIAM J. Sci. Statist. Comput. 13 1 (1992) 379-396
21. Klawonn A., and Widlund O.B. A domain decomposition method with Lagrange multipliers and inexact solvers for linear elasticity. SIAM J. Sci. Comput. 22 4 (2000) 1199-1219
22. Farhat C., Macedo A., Lesoinne M., Roux F.-X., Magoulès F., and de la Bourdonnaye A. Two-level domain decomposition methods with Lagrange multipliers for the fast iterative solution of acoustic scattering problems. Comput. Methods Appl. Mech. Eng. 184 2 (2000) 213-240
23. Farhat C., and Roux F.X. The dual Schur complement method with well-posed local Neumann problems. Contemp. Math. 157 (1994) 193-201
24. F. Nataf, F. Rogier, E. de Sturler, Optimal Interface Conditions for Domain Decomposition Methods, CMAP (Ecole Polytechnique), 1994.
25. Benamou J.-D., and Després B. A domain decomposition method for the Helmholtz equation and related optimal control problems. J. Comput. Physics 136 (1997) 68-82
26. Chevalier P., and Nataf F. Symmetrized method with optimized second-order conditions for the Helmholtz equation. Contemp. Math. 218 (1998) 400-407
27. L.C. McInnes, R. Susan-Resiga, D.E. Keyes, H.M. Atassi, Additive Schwarz methods with nonreflecting boundary conditions for the parallel computation of helmholtz problems, in: J. Mandel, et al. (Ed.), Proceedings of the 10th International Conference on Domain Decomposition Methods, 1998, pp. 349-357.
28. M. Gander, L. Halpern, F. Nataf, Optimal convergence for overlapping and non-overlapping Schwarz waveform relaxation, in: Proceedings of the 11th International Symposium on Domain Decomposition Methods, 1999.
29. Gander M., Magoulès F., and Nataf F. Optimized Schwarz methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput. 24 1 (2002) 38-60
30. Magoulès F., Roux F.-X., and Salmon S. Optimal discrete transmission conditions for a non-overlapping domain decomposition method for the Helmholtz equation. SIAM J. Sci. Comput. 25 5 (2004) 1497-1515
31. Magoulès F., Ivànyi P., and Topping B.H.V. Non-overlapping Schwarz methods with optimized transmission conditions for the Helmholtz equation. Comput. Methods Appl. Mech. Eng. 193 45-47 (2004) 4797-4818
32. Magoulès F., Ivànyi P., and Topping B.H.V. Convergence analysis of Schwarz methods without overlap for the Helmholtz equation. Comput. Struct. 82 (2004) 1835-1847
33. Ciarlet P.G. The Finite Element Method for Elliptic Problems (1979), North Holland
34. Brezzi F., and Fortin M. Mixed and Hybrid Finite Element Methods (1991), Springer-Verlag
35. Hughes T.J.R. The Finite Element Method (1987), Prentice Hall International Inc.
36. G. Anagnostou, Y. Maday, C. Mavriplis, A.T. Patera. On the mortar element method: generalizations and implementation, Domain decomposition methods for partial differential equations, in: Proceedings of the 3rd International Symposium, Houston/TX, USA, 1990, pp. 157-173.
37. Bernardi C., Maday Y., and Patera A.T. A new nonconforming approach to domain decomposition: the mortar element method. In: Brezis H., and Lions J.L. (Eds). Nonlinear Partial Differential Equations and their Applications (1992), Pitman, Boston
38. Ben Belgacem F., Buffa A., and Maday Y. The mortar method for the Maxwell's equations in 3D. C.R. Acad. Sci., Paris, Ser. I, Math. 329 10 (1999) 903-908
39. Ben Belgacem F., Chilton L.K., and Seshaiyer P. The hp Mortar finite element method for the mixed elasticity and stokes problems. Comput. Math. Appl. 46 (2003) 35-55
40. Chilton L.K., and Seshaiyer P. The hp mortar domain decomposition method for problems in fluid mechanics. Int. J. Numer. Methods Fluids 40 12 (2002) 1561-1570
41. Stefanica D. A numerical study of FETI algorithms for mortar finite element methods. SIAM J. Sci. Comput. 23 4 (2001) 1135-1160
42. Stefanica D. Parallel FETI algorithms for mortars. Appl. Numer. Math. (2003)
43. Saad Y. Iterative Methods for Linear Systems (1996), PWS Publishing, Boston
44. Rapetti F., and Toselli A. A FETI preconditioner for two dimensional edge element approximations of Maxwell's equations on nonmatching grids. SIAM J. Sci. Comput. 23 1 (2001) 92-108