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Domain decomposition with nonmatching grids: augmented Lagrangian approach

Authors: Patrick Le Tallec and Taoufik Sassi
Journal: Math. Comp. 64 (1995), 1367-1396
MSC: Primary 65N55; Secondary 65M55, 73V20
MathSciNet review: 1308457
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Abstract: We propose and study a domain decomposition method which treats the constraint of displacement continuity at the interfaces by augmented Lagrangian techniques and solves the resulting problem by a parallel version of the Peaceman-Rachford algorithm. We prove that this algorithm is equivalent to the fictitious overlapping method introduced by P.L. Lions. We also prove its linear convergence independently of the discretization step h, even if the finite element grids do not match at the interfaces. A new preconditioner using fictitious overlapping and well adapted to three-dimensional elasticity problems is also introduced and is validated on several numerical examples.

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  • [1] C. Bernardi, Y. Maday, and T. Patera, A new nonconforming approach to domain decomposition: the mortar element method, Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, Pitman, 1990; also report 89027 of Laboratoire d'Analyse Numérique, Univ. Paris 6.
  • [2] 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), 1097-1120. MR 865945 (88h:65188)
  • [3] J.F. Bourgat, R. Glowinski, P. Le Tallec, and M. Vidrascu, Variational formulation and algorithm for trace operator in domain decomposition calculations, Proc. 2nd Internat. Sympos. on Domain Decomposition Methods (Los Angeles, CA, January 1988), SIAM, Philadelphia, PA, 1989. MR 992000 (90b:65198)
  • [4] J.H. Bramble, J.E. Pasciak, and A.H. Schatz, An iterative method for elliptic problems on regions partitioned into substructures, Math. Comp. 46 (1986), 361-369. MR 829613 (88a:65123)
  • [5] -, The construction of preconditioners for elliptic problems by substructuring IV, Math. Comp. 53 (1989), 1-24. MR 970699 (89m:65098)
  • [6] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer-Verlag, New York, 1991. MR 1115205 (92d:65187)
  • [7] P. Clement, Approximation by finite element functions using local regularization, R.A.I.R.O. Anal. Numér. 9 (1974), 77-84. MR 0400739 (53:4569)
  • [8] Y.H. De Roeck, P. Le Tallec, and M. Vidrascu, Domain decomposition methods for large linearly elliptic three dimensional problems, J. Comput. Appl. Math. 34 (1991), 93-117. MR 1095198 (92a:65331)
  • [9] M. Dryja, B. Smith, and O. Widlund, Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions, SIAM J. Numer. Anal. 31 (1994), 1662-1694. MR 1302680 (95m:65211)
  • [10] M. Dryja and O. Widlund, Towards a unified theory of domain decomposition algorithms for elliptic problems, Proc. Third Internat. Sympos. on Domain Decomposition Methods (Houston), SIAM, Philadelphia, PA, 1990. MR 1064335 (91m:65294)
  • [11] C. Farhat and F.X. Roux, Implicit parallel processing in structural mechanics, Computational Mechanics Advance (J. T. Oden, ed.), Vol. 2, North-Holland, Amsterdam, 1994, pp 1-124. MR 1280753 (95c:73078)
  • [12] M. Fortin and R. Glowinski, Augmented Lagrangian methods, North-Holland, Amsterdam, 1983. MR 724072 (85a:49004)
  • [13] D. Gabay, Application of the methods of multipliers to variational inequalities (in [11]).
  • [14] V. Girault and P.A. Raviart, Finite element methods for Navier-Stokes equations. Theory and algorithms, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1986. MR 851383 (88b:65129)
  • [15] E. Givois, Ph.D. Thesis, Univ. Paris Dauphine, Paris, 1992 (In French).
  • [16] R. Glowinski and P. Le Tallec, Augmented Lagrangian and operator splitting methods in nonlinear mechanics, SIAM, Philadelphia, PA, 1989. MR 1060954 (91f:73038)
  • [17] P. Le Tallec, Domain decomposition method in computational mechanics, Computational Mechanics Advance (J. T. Oden, ed.), Vol. 1, North-Holland, Amsterdam, 1994.
  • [18] P. Le Tallec and T. Sassi, Domain decomposition with nonmatching grids: Schur complement approach, Cahiers de mathématiques de la décision, no 9323, CEREMADE, Univ. Paris Dauphine, 1993.
  • [19] P. Le Tallec, T. Sassi, and M. Vidrascu, Three-dimensional domain decomposition methods with nonmatching grids and unstructured coarse solvers, Proc. Seventh Internat. Sympos. on Domain Decomposition Methods, (D. Keyes and J. Xu, eds.), Contemp. Math., vol. 180, Amer. Math. Soc., Providence, RI, 1994, pp. 61-74. MR 1312378 (95j:65167)
  • [20] P.L. Lions, On the Schwarz alternating method III: A variant for nonoverlapping subdomains, In same proceedings as [10].
  • [21] P.L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal. 16 (1979), 964-979. MR 551319 (81g:47070)
  • [22] S.V. Nepomnyaschikh, Mesh theorems on traces, normalizations of function traces and their inversion, Soviet J. Numer. Anal. Math. Modelling 6 (1991), 223-242. MR 1126677 (93h:65148)

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Article copyright: © Copyright 1995 American Mathematical Society

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