A posteriori error estimates for spectral element solutions to viscoelastic flow problems

Computer Methods in Applied Mechanics and Engineering

Volumn 164, Issue 3-4, 1998, Pages 375-395

  Owens, Robert G ^a  

^a EDINBURGH NAPIER UNIVERSITY   (United Kingdom)

Author keywords

Error; Viscoelastic flow

Indexed keywords

BENCHMARKING; BOUNDARY VALUE PROBLEMS; ERROR ANALYSIS; VISCOUS FLOW;

ERROR ESTIMATION; VISCOELASTIC FLOW;

VISCOELASTICITY;

ERROR; VISCOELASTIC FLOW;

EID: 0344889117     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0045-7825(98)00182-0     Document Type: Article

References (35)

1. [1] I. Babuška and W.C. Rheinboldt, A posteriori error estimates for the finite element method, Int. J. Numer. Methods Engrg. 12 (1978) 1597-1615.
2. [2] I. Babuška and W.C. Rheinboldt, Error estimates for finite element computations, SIAM J. Numer. Anal. 15 (1978) 736-754.
3. [3] L. Demkowicz, J.T. Oden and T. Strouboulis, Adaptive finite element methods for flow problems with moving boundaries. Part I: Variational principles and a posteriori estimates, Comput. Methods Appl. Mech. Engrg. 46 (1984) 217-251.
4. [4] R.E. Bank and A. Weiser, Some a posteriori error estimates for elliptic partial differential equations, Math. Comput. 44 (1985) 283-301.
5. [5] J.T. Oden, L. Demkowicz, T. Strouboulis and P. Devloo, Adaptive methods for problems in solid and fluid mechanics, in: I. Babuška, O.C. Zienkiewicz, J.J. Gago and E.R. de A. Olviera, eds., Accuracy Estimates and Adaptive Refinement in Finite Element Computations (Wiley, Chichester, UK, 1986) 249-280.
6. [6] R.E. Bank and B.D. Welfert, A posteriori error estimators for the Stokes equations: A comparison, Comput. Methods Appl. Mech. Engrg. 82 (1990) 323-340.
7. [7] R.E. Bank and B.D. Welfert, A posteriori error estimates for the Stokes problem, SIAM J. Numer. Anal. 28 (1991) 591-623.
8. [8] R. Verfürth, A posteriori error estimators for the Stokes equations II. Non-conforming discretizations, Numer. Math. 60 (1992) 235-249.
9. [9] M. Ainsworth and J.T. Oden, A posteriori error estimators for the Stokes and Oseen equations, SIAM J. Numer. Anal. 34 (1997) 228-245.
10. [10] M. Ainsworth and J.T. Oden, A unified approach to a posteriori error estimation using element residual methods, Numer. Math. 65 (1993) 23-50.
11. [11] J.T. Oden, W. Wu and M. Ainsworth, An a posteriori error estimate for finite element approximation of the Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 111 (1994) 185-202.
12. [12] R.R. Rao and B.A. Finlayson, On the quality of viscoelastic flow solutions: An adaptive refinement study of a Newtonian and a Maxwell fluid, Int. J. Numer. Methods Fluids 11 (1990) 571-585.
13. [13] R.R. Rao and B.A. Finlayson, Adaptive refinement of a viscoelastic flow problem with the explicitly elliptic momentum equation, J. Non-Newt. Fluid Mech. 38 (1991) 223-246.
14. [14] R.R. Rao and B.A. Finlayson, Adaptive refinement of one-dimensional viscoelastic problem, Comm. Appl. Numer. Methods 8 (1992) 41-49.
15. [15] H. Jin and R.I. Tanner, An a posteriori error estimate and adaptive refinement procedure for viscoelastic fluid flows: The modified upper-convected Maxwell model, Comput. Mech. 13 (1994) 400-413.
16. [16] H. Jin, A subdomain residual method for a posteriori error estimates of finite element solutions of viscoelastic fluid flows, Comput. Methods Appl. Mech. Engrg. 130 (1996) 93-113.
17. [17] I. Mutlu, P. Townsend and M.F. Webster, Adaptive solutions for viscoelastic flows, Comm. Numer. Methods Engrg. 12 (1996) 643-655.
18. [18] F. Yurun and M.J. Crochet, High order finite element methods for steady viscoelastic flows, J. Non-Newt. Fluid Mech. 57 (1995) 283-311.
19. [19] V. Warichet and V. Legat, An adaptive h-p finite element method for viscoelastic flows simulations, in: Proc. 2nd European Computational Fluid Dynamics Conference (1994) 181-187.
20. [20] V. Warichet and V. Legat, Adaptive hp-finite element viscoelastic flow calculations, Comput. Methods Appl. Mech. Engrg. 136 (1996) 93-110.
21. [21] V. Warichet and V. Legat, Adaptive high-order prediction of the drag correction factor for the upper-convected Maxwell fluid, J. Non-Newt. Fluid Mech. 73 (1997) 95-114.
22. [22] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations (Springer-Verlag, Berlin, 1986).
23. [23] Y. Maday and A.T. Patera, Spectral element methods for the incompressible Navier-Stokes equations, in: A.K. Noor and J.T. Oden, eds., State of the Art Surveys in Computational Mechanics (American Society of Mechanical Engineers, New York, 1989) 71-143.
24. [24] F. Brezzi, On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers, RAIRO Anal. Numer. 8 (R.2) (1974) 129-151.
25. [25] M. Fortin and R. Pierre, On the convergence of the mixed method of Crochet and Marchal for viscoelastic flows, Comput. Methods Appl. Mech. Engrg. 73 (1989) 341-350.
26. [26] M.I. Gerritsma and T.N. Phillips, Compatible spectral approximations for the velocity-pressure-stress formulation of the Stokes problem, to appear in SIAM J. Sci. Comput. (1998).
27. [27] M.I. Gerritsma and T.N. Phillips, Discontinuous spectral element approximations for the velocity-pressure-stress formulation of the Stokes problem, to appear in Int. J. Numer. Meth. Engrg. (1998).
28. [28] C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods in Fluid Dynamics (Springer-Verlag, Berlin, 1988).
29. [29] R.G. Owens and T.N. Phillips, Steady viscoelastic flow past a sphere using spectral elements, Int. J. Numer. Methods Engng. 39 (1996) 1517-1534.
30. [30] Y. Maday, C. Mavriplis and A.T. Patera, Nonconforming mortar element methods: applications to spectral discretisations, in: T.F. Chan, R. Glowinski, J. Periaux and O.B. Widlund, eds., Proc. Second Int. Symposium on Domain Decomposition Methods (SIAM, Philadelphia, 1989).
31. [31] B. Caswell, Report on the IXth International Workshop on Numerical Methods in non-Newtonian Flows, J. Non-Newt. Fluid Mech. 62 (1996) 99-110.
32. [32] R.G. Owens and T.N. Phillips, Decoupled spectral element methods for steady viscoelastic flow past a sphere, in: Proc. ICOSAHOM.95, Houston J. Math. (1996) 287-294.
33. [33] W.J. Gordon and C.A. Hall, Transfinite element methods: blending-function interpolation over arbitrary curved element domains, Numer. Math. 21 (1973) 109-129.
34. [34] A. Fortin and M. Fortin, A preconditioned generalized minimal residual algorithm for the numerical solution of viscoelastic fluid flows, J. Non-Newt. Fluid Mech. 36 (1990) 277-288.
35. [35] Y. Saad and M.H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 7 (1986) 856-869.