The Prolonged Adaptive Multigrid method for finite element Navier-Stokes equations

Computer Methods in Applied Mechanics and Engineering

Volumn 138, Issue 1-4, 1996, Pages 227-271

  Wille, S O ^a  

^a OSLO METROPOLITAN UNIVERSITY   (Norway)

Author keywords

[No Author keywords available]

Indexed keywords

FINITE ELEMENT METHOD; INTEGRATION; ITERATIVE METHODS; MATRIX ALGEBRA; REYNOLDS NUMBER; TREES (MATHEMATICS);

PROLONGED ADAPTIVE MULTIGRID METHOD;

NAVIER STOKES EQUATIONS;

FINITE ELEMENT ANALYSIS; FLUID MECHANICS; GRIDS; NAVIER-STOKES EQUATIONS;

EID: 0030512327     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0045-7825(96)01100-0     Document Type: Article

References (52)

1. T. Utnes, Finite element modeling of quasi-three dimensional nearly horizontal flow, Int. J. Numer. Methods Fluids 12 (1991) 559-576.
2. R.R. Tharera, J.R. Stewart, O. Hassan, K. Morgan and J. Peraire, A point implicit unstructured grid solver for the Euler and Navier-Stokes equations, AIAA 26th Aerospace Sciences Meeting. Reno, Nevada, 1988.
3. T.J.R. Hughes, L.P. Franca and G.M. Hulbert, A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations, Comput. Methods Appl. Mech. Engrg. 73 (1989) 173-189.
4. S.Ø. Wille, Numerical simulations of steady flow inside a three dimensional aortic bifurcation model, J. Biomed. Engrg. 6 (1984) 49-55.
5. S.Ø. Wille and L. Walløe, Pulsatile pressure and flow in arterial stenosis simulated in a mathematical model, J. Biomed. Engrg. 3 (1981) 17-24.
6. S.Ø. Wille, Pulsatile pressure and flow in arterial aneurysm simulated in a mathematical model, J. Biomed. Engrg. 3 (1981) 153-158.
7. M. Jayaram and A. Jameson, An efficient vector and parallel multigrid method for viscous transonic flow, Proceedings of the Int. Symp. High Performance Computing, Montpellier, France (1989) 225-237.
8. S.Ø, Wille, A structured Tri-Tree search method for generation of optimal unstructured finite element grids in two and three dimensions, Int. J. Numer. Methods Fluids 14 (1992) 861-881.
9. G.S. Barussi, WG. Habashi and J.G. Guevremont. A second order finite element for the solution of the transonic Euler and Navier-Stokes equations, Int. J. Numer. Methods Fluids 20 (1995) 671-693.
10. D.L. Hill and E.A. Baskharone, A multiblock Navier-Stokes algorithm using equal-order quadratic finite elements. Int. J. Numer. Methods Fluids 20 (1995) 169-185.
11. Y. Zang and R.L. Street, A composite multigrid method for calculating unsteady incompressible flows in geometrically complex domains, Int. J. Numer. Methods Fluids 20 (1995) 341-361.
12. A.A. Lorber and G.F. Carey, A vector-parallel scheme for Navier-Stokes computations at multi-gigaflop performance rates, Int. J. Numer. Methods Fluids 21 (1995) 445-466.
13. Y. Kallinderis, Adaptive hybrid prismatic/tetrahedral grids, Int. J. Numer. Methods Fluids 20 (1992) 1023-1037
14. R. Verfurth, A posteriori error estmators for the Stokes equations, Numer. Math. 55 (1989) 309-325.
15. R.E. Bank and B.D Welfert, A posteriori error estimates for the Stokes equations: a comparison, Comput. Methods Appl. Mech. Engrg. 82 (1990) 323-340.
16. I. Gustafsson, A class of first order factorization methods, BIT 8 (1978) 142-156.
17. E. Barragy and G.F. Carey, A partitioning scheme and iterative solution for sparse bordered systems, Comput. Methods Appl. Mech. Engrg. 70 (1988) 321-327.
18. H.P. Langtangen, T. Rusten, A. Tveito and S.Ø. Wille, An element by element preconditioner for iterative equation solvers, Proceedings of the VI International Conference on Finite Elements in Water Resources, Lisbon, Portugal, 1986.
19. S.Ø Wille, An element by element preconditioner for refined finite element grids, Proceedings of the First International Conference on Numerical Grid Generation in Computational Fluid Dynamics, Landshut, Germany, 1986.
20. S.Ø. Wille, Local Gaussian preconditioning of symmetric and non symmetric finite element equation solvers, Proceedings of the First International Conference on Industrial and Appl. Mathematics, ICIAM 87. Paris, France, 1987.
21. G.F. Carey, K.C. Wang and W.D. Joubert, Performance of iterative methods for Newtonian and generalized Newtonian flows, Int. J. Numer. Methods Fluids 9 (1989) 127-150.
22. J.A. Meijerink and H.A van der Vorst, An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Math. Comput. 31 (1977) 148-162.
23. D. Howard, W.M. Connolley and J.S. Rollett, Unsymmetric conjugate gradient methods and sparse direct methods in finite element flow simulation, Int. J. Numer. Methods Fluids 10 (1990) 925-945.
24. O.G. Johnson, C.A. Michelli and G. Paul, Polynomial preconditioners for conjugate gradient calculations, SIAM J. Numer. Anal. 20 (1983) 362-376.
25. J. Peraire, M. Vadati, K. Morgan and O.C. Zienkiewicz, Adaptive remeshing for compressible flow computations, J. Comput. Phys. 71 (1987) 449-466.
26. R. Lohner, K. Morgan and O.C. Zienkiewicz, An adaptive finite element procedure for compressible high speed flows, Comput. Methods Appl. Mech. Engrg. 51 (1985) 441-464.
27. R. Lohner, Generation of three-dimensional unstructured grids by the advancing-front method, AIAA 26th Aerospace Science Meeting, Reno, AIAA Paper 88-1515. January, 1988.
28. H.K. Ruud and S.Ø. Wille, An advancing front algorithm for three dimensional mesh generation, Proc. NUMETA 90. Numer. Methods Engineering: Theory and Applications, January, 1990.
29. N.P. Weatherill, A method for generating irregular computational grids in multiply connected planar domains, Int. J. Numer. Methods Fluids 8 (1988) 181-197.
30. T.J. Baker, Three dimensional mesh generation by triangulation of arbitrary point sets, AIAA 8th Computational Fluid Dynamics, AIAA Paper 87-1124-CP, Hawaii, June 1987.
31. W.J. Schroeder and M.S. Shepard, A combined Octree/Delauney method for fully automatic 3-D mesh generation, Int. J. Numer Methods Engrg. 29 (1990) 37-55.
32. T.J. Baker, Unstructured mesh generation by a generalized Delauney algorithm, Presented at AGARD, Loen, 1989.
33. O.C. Zienkiewicz and J.Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. Part 2: Error Estimates and Adaptivity, Int. J. Numer. Methods Engrg. 33 (1992) 1365-1382.
34. D.M. Young and K.C. Yea. Generalized conjugate-gradient acceleration of non-symmetrizable iterative methods, Lin. Alg. Applic. 34 (1980) 159-194.
35. P.K.W. Vinsome, Orthomin, an iterative method for solving sparse sets of simultaneous linear equations, in: Society of Petroleum Engineers of AIME. Proceedings of the Fourth Symposium on Reservoir Simulation, Society of Petroleum Engineers of AIME (1976) 147-159.
36. P. Sonneveld, CGS, a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Stat. Comput. 10 (1987) 36-52.
37. H.A. van der Vorst and P. Sonneveld, A more smoothly converging variant of CG-S. Report 90-50, Faculty of Technical Mathematics and Informatics, Delft University of Technology. The Netherlands. ISSN 0922-5641 (1990).
38. C. Taylor and P. Hood, A numerical solution of the Navier-Stokes equations using the finite element technique, Comput. Fluids 1 (1973) 73-100.
39. S.Ø. Wille, An adaptive unstructured Tri-Tree iterative solver for mixed finite element formulation of the Stokes equations. Int. J. Numer. Methods Fluids (1995) in press.
40. S.Ø. Wille, A nonlinear adaptive Tri-Tree multigrid solver for mixed finite element formulation of the Navier-Stokes equations, Int. J. Numer. Methods in Fluids (1996) in press.
41. R.E. Bank and B.D. Welfert and H. Yserentant, A class of iterative methods for solving saddle point problems, Numer. Math. 56 (1990) 645-666.
42. S.Ø. Wille, A preconditioned alternating inner-outer iterative solution method for mixed finite element formulation of the Navier-Stokes equations. Int. J. Numer. Methods Fluids 18 (1994) 1135-1151.
43. O. Dahl and S.Ø. Wille, An ILU preconditioner with coupled node fill In for iterative solution of the mixed finite element formulation of the 2D and 3D Navier-Stokes equations. Int. J. Numer. Methods Fluids 15 (1992) 525-544.
44. A. George and J.W. Liu, Computer Solutions of Large Sparse Positive Definite Systems (Prentice-Hall, Englewood Cliffs, NJ, 1981).
45. S.Ø. Wille, A local solution adapted Tri-Tree multigrid generator and equation solver for mixed finite element formulation of the Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 131 (1996) 109-132.
46. S.Ø. Wille, Adaptive linearization and grid iterations with the Tri-Tree multigrid refinement - Recoarsement algorithm for the Navier-Stokes equations, Int. J. Numer. Methods Fluids, in press.
47. S.Ø. Wille, A Tri-Tree recoarsement algorithm for the finite element formulation of the Navier-Stokes Equations, Comput. Methods Appl. Mech. Engrg. 135 (1996) 129-142.
48. S.Ø. Wille, A local predictive convection - diffusion refinement indicator for the Tri-Tree adapted finite element multigrid algorithm of the Navier-Stokes equations. Submitted to Comput. Methods Appl. Mech. Engrg. 134 (1996) 181-196.
49. S.Ø. Wille, A nonlinear adaptive full Tri-Tree multigrid method for the mixed finite element formulation of the Navier-Stokes equations, Int. J. Numer. Methods Fluids 22 (1996) 1041-1059.
50. C. Vincent and R. Boyer, A preconditioned conjugate gradient Uzawa - type method for the solution of the Stokes problem by mixed Q1 - P0 stabilized finite elements. Int. J. Numer. Methods Fluids 14 (1992) 289-298.
51. W. Hackbush, Multi-Grid Methods and Applications, Springer series in Computational Mathematics (Springer-Verlag. Berlin, Heidelber, New York and Tokyo, 1985).
52. P. Wesseling, An Introduction to Multigrid Methods (John Wiley, 1992).