Zsaki, Attila and Rixen, Daniel and Paraschivoiu, Marius (2003) A substructure-based iterative inner solver coupled with Uzawa's algorithm for the Stokes problem. International Journal for Numerical Methods in Fluids, 43 (2). pp. 215-230. ISSN 0271-2091
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Official URL: http://dx.doi.org/10.1002/fld.612
A domain decomposition method with Lagrange multipliers for the Stokes problem is developed and analysed. A common approach to solve the Stokes problem, termed the Uzawa algorithm, is to decouple the velocity and the pressure. This approach yields the Schur complement system for the pressure Lagrange multiplier which is solved with an iterative solver. Each outer iteration of the Uzawa procedure involves the inversion of a Laplacian in each spatial direction. The objective of this paper is to effectively solve this inner system (the vector Laplacian system) by applying the finite-element tearing and interconnecting (FETI) method. Previously calculated search directions for the FETI solver are reused in subsequent outer Uzawa iterations. The advantage of the approach proposed in this paper is that pressure is continuous across the entire computational domain. Numerical tests are performed by solving the driven cavity problem. An analysis of the number of outer Uzawa iterations and inner FETI iterations is reported. Results show that the total number of inner iterations is almost numerically scalable since it grows asymptotically with the mesh size and the number of subdomains.
|Divisions:||Concordia University > Faculty of Engineering and Computer Science > Mechanical and Industrial Engineering|
|Authors:||Zsaki, Attila and Rixen, Daniel and Paraschivoiu, Marius|
|Journal or Publication:||International Journal for Numerical Methods in Fluids|
|Keywords:||Uzawa's algorithm • Stokes problem • incompressible flows|
|Deposited By:||ANDREA MURRAY|
|Deposited On:||05 Jul 2010 15:36|
|Last Modified:||08 Dec 2010 23:10|
1. Dryja M, Widlund OB. Domain decomposition algorithms with small overlap. SIAM Journal on Scientific Computing 1994; 15(3): 604-620.
2. Klawonn A, Pavarino LF. Overlapping Schwarz methods for elasticity and Stokes problems. Computer Methods in Applied Mechanics and Engineering 1998; 165(1-4): 233-245.
3. Ronquist E. Domain decomposition methods for the steady Stokes equations. In Eleventh International Conference on Domain Decomposition Methods, Lai C-H, Bjorstad PE, Cross M, Widlund OB (eds). (DDM.org), 1999.
4. Fisher PF. An overlapping Schwarz method for spectral element solution of the incompressible Navier-Stokes equations. Journal of Computational Physics 1997; 133: 84-101.
5. Calgaro C, Laminie J. On the domain decomposition method for the generalized Stokes problem with continuous pressure. Numerical Methods for Partial Differential Equations 2000; 16(1): 84-106.
6. Jing Li. A dual-primal FETI method for incompressible Stokes equations. Technical Report TR2001-816, Courant Institute, New York, July 2001.
7. Le Tallec P, Patra A. Non-overlapping domain decomposition methods for adaptive hp approximations of the Stokes problem with discontinuous pressure field. Computer Methods in Applied Mechanics and Engineering 1997; 145: 361-379.
8. Pavarino L, Widlund O. Balancing Neumann-Neumann methods for incompressible Stokes equations. Technical Report TR2001-813, Courant Institute, New York, March 2001.
9. Gosselet P, Rey Ch, Lene F, Dasset P. A domain decomposition method for quasi-incompressible formulations with discontinuous pressure fields. Revue Européenne des Elements Finis 2002, submitted.
10. Farhat C, Roux F-X. Implicit parallel processing in structural mechanics. Computational Mechanics Advances 1994; 2(1): 1-124.
11. Farhat C, Lesoinne M, LeTallec P, Pierson K, Rixen D. FETI-DP: a dual-primal unified FETI method - part i: a faster alternative to the two-level FETI method. International Journal for Numerical Methods in Engineering 2001; 50(7): 1523-1544.
12. Mandel J. Balancing domain decomposition. Communications in Applied Numerical Methods and Engineering 1993; 9: 233-241.
13. Patra A. Newton-Krylov domain decomposition solvers for adaptive hp approximations of the steady incompressible Navier-Stokes equations with discontinuous pressure fields. In Ninth International Conference on Domain Decomposition Methods, BjPE, Espedal MS, Keyes DE (eds). (DDM.org), 1998.
14. Taylor C, Hood P. A numerical solution of the Navier-Stokes equations using the finite element method. Computers and Fluids 1973; 1: 73-100.
15. Elman H, Golub GH. Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM Journal on Numerical Analysis 1994; 31(6): 1645-1661.
16. Elman H. Multigrid and Krylov subspace methods for discrete Stokes equations. International Journal for Numerical Methods in Fluids 1996; 22: 755-770.
17. Maday Y, Meiron D, Patera AT, Ronquist EM. Analysis of iterative methods for the steady and the unsteady Stokes problem: application to spectral element discretizations. SIAM Journal on Scientific Computing 1993; 14: 310-337.
18. Farhat C, Roux F-X. A method of finite element tearing and interconnecting and its parallel solution algorithm. Computer Methods in Applied Mechanics and Engineering 1991; 32: 1205-1227.
19. Rixen D. Extended preconditioners for FETI method applied to constrained problems. International Journal for Numerical Methods in Engineering 2002; 54(1): 1-26.
20. Saad Y. On the Lanczos method for solving symmetric linear systems with several right-hand sides. Mathematics of Computation 1987; 48: 651-662.
21. Farhat C, Crivelli L, Roux FX. Extending substructures based iteratives solvers to multiple load and repeated analyses. Computer Methods in Applied Mechanics and Engineering 1994; 117: 195-209.
22. Roux F-X. Parallel implementation of a domain decomposition method for non-linear elasticity problems. In Domain-Based Parallelism and Problem Decomposition Methods in Computational Science and Engineering, Keyes D, Saad Y, Truhlar D (eds). SIAM: Philadelphia, PA, 1995; 161-175.
23. Risler F, Rey C. Iterative accelerating algorithms with Krylov subspaces for the solution to large-scale non linear problems. Numerical Algorithms 2000; 23: 1-30.
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