Download PDF - Stabilized Galerkin finite element methods for convection dominated and incompressible flow problemsAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Stabilized Galerkin finite element methods for convection dominated and incompressible flow problems

Authors 1

Affiliations

  1. Mathematics Department, Magdeburg University of Technology, PF 4120, D-39016 Magdeburg, Germany

Abstract

In this paper, we analyze a class of stabilized finite element formulations used in computation of (i) second order elliptic boundary value problems (diffusion-convection-reaction model) and (ii) the Navier-Stokes problem (incompressible flow model). These stabilization techniques prevent numerical instabilities that might be generated by dominant convection/reaction terms in (i), (ii) or by inappropriate combinations of velocity/pressure interpolation functions in (ii). Stability and convergence results on non-uniform meshes are given in the whole range from diffusion to convection/reaction dominated situations. In particular, we recover results for the streamline upwind and Galerkin/least-squares methods. Numerical results are presented for low order interpolation functions.

Bibliography

  1. [BF] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer, New York 1991.
  2. [DW] J. Douglas, Jr., and J. Wang, An absolutely stabilized finite element method for the Stokes problem, Math. Comp. 52 (186) (1989), 495-508.
  3. [FD] L. P. Franca and E. G. Dutra do Carmo, The Galerkin-gradient-least-squares method, Comput. Methods Appl. Mech. Engrg. 74 (1989), 41-54.
  4. [FFH] L. P. Franca, S. L. Frey and T. J. R. Hughes, Stabilized finite element methods: I. Applications to the advective-diffusive model, ibid., to appear.
  5. [FS] L. P. Franca and R. Stenberg, Error analysis of some Galerkin-least-squares methods for the elasticity equations, SIAM J. Numer. Anal. (1991), to appear.
  6. [GR] V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer, 1986.
  7. [HS] P. Hansbo and A. Szepessy, A velocity-pressure streamline diffusion finite element method for the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 84 (1990), 175-192.
  8. [Hu] T. J. R. Hughes, Recent progress in the development and understanding of SUPG methods with special reference to the compressible Euler and Navier-Stokes equations, Internat. J. Numer. Methods Fluids 7 (1987), 1261-1275.
  9. [Ja] C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Studentlitteratur, Sweden, 1987.
  10. [Jb] C. Johnson, Adaptive finite element methods for diffusive and convective problems, Comput. Methods Appl. Mech. Engrg. 82 (1990), 301-322.
  11. [KR] R. Kornhuber and R. Roitzsch, On adaptive grid refinement in the presence of internal and boundary layers, IMPACT of Computing in Science and Engineering 2 (1990), 40-72.
  12. [LA] G. Lube and A. Auge, Regularized mixed finite element approximations of incompressible flow problems. II. Navier-Stokes flow, preprint, TU Magdeburg, 1991.
  13. [P] R. Pierre, Simple C0 approximation for the computation of incompressible flow, Comput. Methods Appl. Mech. Engrg. 68 (1988), 205-227.
  14. [T] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conf. Ser. in Appl. Math. 41, SIAM, 1983.
  15. [Tea] T. E. Tezduyar, R. Shih, S. Mittal and S. E. Ray, Incompressible flow computations with stabilized bilinear and linear equal-order interpolation velocity-pressure elements, preprint UMSI 90/165, Univ. of Minnesota, 1990.
  16. [Teb] T. E. Tezduyar, Stabilized finite element formulations for incompressible flow computations, von Karman Institute for Fluid Dynamics, Lecture Series 1991-01.
  17. [Wa] L. B. Wahlbin, Local behavior in finite element methods, in: Handbook of Numerical Analysis, P. G. Ciarlet and J. L. Lions (eds.), Vol. II, North-Holland, 1990.
  18. [We] D. Weiß, Numerische Simulation von Temperaturfeldern bei Schweißvorgängen, Forschungsbericht, TU Magdeburg, 1991.
Banach Center Publications
1994/29/1
Export to PBN, Opens in new tab
Pages:
85-104
Main language of publication
English
Published
1994
Exact and natural sciences