A gradient method approach to optimization-based multidisciplinary simulations and nonoverlapping domain decomposition algorithms

SIAM Journal on Numerical Analysis

Volumn 37, Issue 5, 2000, Pages 1513-1541

  Du, Qiang ^a,b   Gunzburger, Max D ^b  

^a HONG KONG UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Hong Kong)

^b IOWA STATE UNIVERSITY   (United States)

Author keywords

Convergence; Domain decomposition; Finite element; Multidisciplinary simulations and optimization; Nonoverlapping subdomains; Optimization; Parallel computation

Indexed keywords

EID: 0011446532     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142998343087     Document Type: Article

References (44)

1. P. E. BJØRSTAD AND O. B. WIDLUND, Iterative methods for the solution of elliptic problems on regions partitioned into substructures, SIAM J. Numer. Anal., 23 (1986), pp. 1097-1120.
2. J.-F. BOURGAT, R. GLOWINSKI, P. LE TALLEC, AND M. VIDRASCU, Variational formulation and algorithm for trace operator in domain decomposition calculations, in Proceedings of the Second International Symposium on Domain Decomposition Methods, T. Chan, R. Glowinski, J. Periaux, and O. B. Widlund, eds., SIAM, Philadelphia, 1989, pp. 3-16.
3. J. BRAMBLE J. PASCIAK, AND A. SCHATZ, The construction of preconditioners for elliptic problems by substructuring, I-IV, Math. Comp., 47 (1986), pp. 103-134; 49 (1987), pp. 1- 16; 51 (1988), pp. 415-430; 53 (1989), pp. 1-24.
4. J. BRAMBLE J. PASCIAK, AND A. SCHATZ, The construction of preconditioners for elliptic problems by substructuring, I-IV, Math. Comp., 47 (1986), pp. 103-134; 49 (1987), pp. 1-16; 51 (1988), pp. 415-430; 53 (1989), pp. 1-24.
5. J. BRAMBLE J. PASCIAK, AND A. SCHATZ, The construction of preconditioners for elliptic problems by substructuring, I-IV, Math. Comp., 47 (1986), pp. 103-134; 49 (1987), pp. 1- 16; 51 (1988), pp. 415-430; 53 (1989), pp. 1-24.
6. J. BRAMBLE J. PASCIAK, AND A. SCHATZ, The construction of preconditioners for elliptic problems by substructuring, I-IV, Math. Comp., 47 (1986), pp. 103-134; 49 (1987), pp. 1- 16; 51 (1988), pp. 415-430; 53 (1989), pp. 1-24.
7. 2 projection, Math. Comp., 56 (1991), pp. 463-476.
8. X. CAI, M. DRYJA, AND M. SARKIS, Overlapping Non-Matching Grid Mortar Element Methods for Elliptic Problems, Tech. Report, Univ. of Colorado, Boulder, CO, 1997.
9. T. CHAN R. GLOWINSKI, J. PERIAUX, AND O. WIDLUND, EDS., Proceedings of the Second International Symposium Domain Decomposition Methods, SIAM, Philadelphia, 1989.
10. T. CHAN, R. GLOWINSKI, J. PERIAUX, AND O. WIDLUND, EDS., Proceedings of the Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, 1990.
11. T. CHAN, D. KEYES, G. MEURANT, J. SCROGGS, AND R. VOIGT, Proceedings of the Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, 1992.
12. T. CHAN AND T. MATHEW, Domain decomposition algorithms, Acta Numerica, (1994), pp. 61-143.
13. Z. CHEN AND J. ZOU, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math., 79 (1998), pp. 175-202.
14. P. CIARLET, Finite Element Methods for Elliptic Problems, North-Holland, Amsterdam, 1978.
15. P. CIARLET, Introduction to Linear Algebra and Optimization, Cambridge University Press, Cambridge, UK, 1989.
16. M. DAUGE, Elliptic Boundary Value Problems on Corner Domains, Lecture Notes in Math. 1341, Springer-Verlag, Berlin, 1988.
17. Q. DENG, An analysis for a nonoverlapping domain decomposition iterative procedure, SIAM J. Sci. Comput., 18 (1997), pp. 1517-1525.
18. Q. DINH, R. GLOWINSKI, AND J. PERIAUX, Solving elliptic problems by domain decomposition methods with application, in Elliptic Problem Solvers, II, G. Birkoff and A. Schoenstadt, eds., Academic Press, New York, 1984, pp. 395-426.
19. M. DRYJA, B. F. SMITH, AND O. B. WIDLUND, Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions, SIAM J. Numer. Anal., 31 (1994), pp. 1662-1694.
20. Q. DU, Optimization Based Nonoverlapping Domain Decomposition Methods and Their Convergence, preprint, 1997.
21. Q. Du AND X. WU, Numerical Solution of the Three Dimensional Ginzburg-Landau Models of Superconductivity via Artificial Boundary, preprint, 1997.
22. V. GIRAULT AND P.-A. RAVIART, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986.
23. R. GLOWINSKI, G. H. GOLUB, G. A. MEURANT, AND J. PERIAUX, Proceedings of the First International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, 1988.
24. R. GLOWINSKI, Y. A. KUZNETSOV, G. MEURANT, J. PERIAUX, AND O. B. WIDLUND, Proceedings of the Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, 1991.
25. R. GLOWINSKI AND P. LE TALLEC, Augmented Lagrangian interpretation of the nonoverlapping Schwarz alternating method, in Proceedings of the Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, T. F. Chan, R. Glowinski, J. Periaux, and O. B. Widlund, eds., SIAM, Philadelphia, 1990, pp. 224-231.
26. R. GLOWINSKI, J. PERIAUX, Z. SHI, AND O. B. WIDLUND, Domain Decomposition Methods in Scientific and Engineering Computing, John Wiley, New York, 1996.
27. R. GLOWINSKI, J. PERIAUX, AND G. TERRASSON, On the coupling of viscous and inviscid models for compressible fluid flows via domain decomposition, in Proceedings of the Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, T. F. Chan, R. Glowinski, J. Periaux, and O. B. Widlund, eds., SIAM, Philadelphia, 1990, pp. 64-97.
28. P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.
29. M. GUNZBURGER AND S. L. HOU, Treating inhomogeneous essential boundary conditions in finite element methods and the calculation of boundary stresses, SIAM J. Numer. Anal., 29 (1991), pp. 390-424.
30. M. GUNZBURGER, M. HEINKENSCHLOSS, AND H. LEE, Solution of elliptic PDEs by an optimization-based domain decomposition method, Appl. Math. Comput., to appear.
31. M. GUNZBURGER, H. LEE, AND J. PETERSON, An optimization based domain decomposition method for partial differential equations, Comput. Math. Appl., 37 (1999), pp. 77-93.
32. M. D. GUNZBURGER AND H. LEE, An optimization based domain decomposition method for the Navier-Stokes equations, SIAM J. Numer. Anal., to appear.
33. M. GUNZBURGER AND R. NICOLAIDES, On sub structuring algorithms and solution techniques for the numerical approximation of partial differential equations, Appl. Numer. Math., 2 (1986), pp. 243-256.
34. D. E. KEYES AND W. D. GROPP, A comparison of domain decomposition techniques for elliptic partial differential equations and their parallel implementation, SIAM. J. Sci. Statist. Comput., 8 (1987), pp. S166-S202.
35. D. KEYES AND J. XU, Domain Decomposition Methods in Scientific and Engineering Computing, AMS, Providence, RI, 1994.
36. P. L. LIONS, On the Schwarz alternating method I, in Proceedings of the First International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski, G. H. Golub, G. A. Meurant, and J. Periaux, eds., SIAM, Philadelphia, 1988, pp. 1-42.
37. P.L. LIONS, On the Schwarz alternating method III, in Proceedings of the Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, T. Chan, R. Glowinski, J. Periaux, and O. B. Widlund, eds., SIAM, Philadelphia, 1990, pp. 202-223.
38. D. FUNARO, A. QUARTERONI, AND P. ZANOLLI, An iterative procedure with interface relaxation for domain decomposition methods, SIAM J. Numer. Anal., 25 (1988), pp. 1213-1236.
39. L. MARINI AND A. QUARTERONI, A relaxation procedure for domain decomposition methods using finite elements, Numer. Math., 55 (1989), pp. 575-598.
40. A. MATSOKIN AND S. NEPOMNYASHCHIKH, On the convergence of the alternating subdomain Schwartz method without intersections, Sov. Numer. Anal. Math. Mod., 4 (1989), pp. 471-477.
41. H. MCFADDIN AND J. RICE, Collaborating PDE solvers, Appl. Numer. Math., 10 (1992), pp. 279-295.
42. M. MU, Solving composite problems with interface relaxation, SIAM J. Sci. Comput., 20 (1999), pp 1394-1416.
43. R. SCOTT AND S. ZHANG, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp., 54 (1990), pp. 483-493.
44. J. XU, Iterative methods by space decomposition and subspace correction, SIAM Rev., 34 (1992), pp. 518-613.