Multilevel Homotopic Adaptive Finite Element Methods for Convection Dominated Problems

  • Long Chen
  • Pengtao Sun
  • Jinchao Xu
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 40)

Summary

A multilevel homotopic adaptive finite element method is presented in this paper for convection dominated problems. By the homotopic method with respect to the diffusion parameter, the grids are iteratively adapted to better approximate the solution. Some new theoretic results and practical techniques for the grid adaptation are presented. Numerical experiments show that a standard finite element scheme based on this properly adapted grid works in a robust and efficient manner.

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References

  1. I. Babuska and T. Strouboulis. The finite element method and its reliability. Numerical Mathematics and Scientific Computation. Oxford Science Publications, 2001.Google Scholar
  2. R. E. Bank, A. H. Sherman, and A. Weiser. Refinement algorithms and data structures for regular local mesh refinement. In R. S. et al., editor, Scientific Computing, pages 3–17. IMACS/North-Holland Publishing Company, Amsterdam, 1983.Google Scholar
  3. R. E. Bank and R. K. Smith. Mesh smoothing using a posteriori error estimates. SIAM Journal on Numerical Analysis, 34:979–997, 1997.MathSciNetCrossRefGoogle Scholar
  4. R. E. Bank and J. Xu. Asymptotically exact a posteriori error estimators, Part I: Grids with superconvergence. SIAM J. on Numerical Analysis, 41(6):2294–2312, 2003a.MathSciNetCrossRefGoogle Scholar
  5. R. E. Bank and J. Xu. Asymptotically exact a posteriori error estimators, Part II: General unstructured grids. SIAM J. on Numerical Analysis, 41(6):2313–2332, 2003b.MathSciNetCrossRefGoogle Scholar
  6. C. Carstensen and S. Bartels. Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. I. low order conforming, nonconforming, and mixed FEM. Math. Comp., 71(239):945–969, 2002.MathSciNetCrossRefGoogle Scholar
  7. C. Chen and Y. Huang. High accuracy theory of finite element methods. Hunan, Science Press, Hunan, China (in Chinese), 1995.Google Scholar
  8. L. Chen, P. Sun, and J. Xu. Optimal anisotropic simplicial meshes for minimizing interpolation errors in Lp-norm. Submitted to Math. Comp., 2003.Google Scholar
  9. L. Chen and J. Xu. Optimal Delaunay triangulation. J. of Comp. Math., 22(2):299–308, 2004a.MathSciNetGoogle Scholar
  10. L. Chen and J. Xu. Stability and accuracy of adapted finite element methods for singularly perturbed problems. Technical report, 2004b.Google Scholar
  11. E. D'Azevedo and R. Simpson. On optimal interpolation triangle incidences. SIAM Journal on Scientific and Statistical Computing, 6:1063–1075, 1989.MathSciNetCrossRefGoogle Scholar
  12. A. S. Dvinsky. Adaptive grid generation from harmonic maps on riemannian manifolds. J. Comput. Phys., 95(2):450–476, 1991.MATHMathSciNetCrossRefGoogle Scholar
  13. W. Habashi, M. Fortin, J. Dompierre, M. Vallet, D. A. A. Yahia, Y. Bourgault, M. Robichaud, A. Tam, and S. Boivin. Anisotropic mesh optimization for structured and unstructured meshes. In 28th Computational Fluid Dynamics Lecture Series 1997-02. von Karman Institute for Fluid Dynamics, 1997.Google Scholar
  14. W. Huang. Variational mesh adaptation: isotropy and equidistribution. J. Comput. Phys., 174:903–924, 2001.MATHMathSciNetCrossRefGoogle Scholar
  15. W. Huang and R. D. Russell. Moving mesh strategy based on a gradient flow equation for two-dimensional problems. SIAMJournal on Scientific Computing, 20(3):998–1015, 1999.MathSciNetCrossRefGoogle Scholar
  16. W. Huang and W. Sun. Variational mesh adaptation II: Error estimates and monitor functions. J. Comput. Phys., 2003.Google Scholar
  17. M. Jones, L. Freitag, and P. Plassmann. An efficient parallel algorithm for mesh smoothing. In 4th Int. Meshing Roundtables, pages 47–58. Sandia Labs, 1995.Google Scholar
  18. R. Kornhuber and R. Roitzsch. On adaptive grid refinement in the presence of internal or boundary layers. IMPACT Comput. Sci. Engrg., 2:40–72, 1990.CrossRefGoogle Scholar
  19. C. Lawson. Software for C1 surface interpolation. In J. Rice, editor, Mathematical Software III, pages 161–194. Academic Press, 1977.Google Scholar
  20. B. Li and Z. Zhang. Analysis of a class of superconvergence patch recovery techniques for linear and bilinear finite elements. Numerical Methods for Partial Differential Equations, pages 151–167, 1999.Google Scholar
  21. V. Liseikin. Grid Generation Methods. Springer Verlag, Berlin, 1999.Google Scholar
  22. J. Miller, E. O'Riordan, and G. Shishkin. Fitted Numerical Methods For Singular Perturbation Problems. World Scientific, 1996.Google Scholar
  23. M. Rivara. Mesh refinement processes based on the generalized bisection of simplexes. SIAM J. Numer. Anal., 21:604–613, 1984.MATHMathSciNetCrossRefGoogle Scholar
  24. H. Roos. Optimal convergence of basic schemes for elliptic boundary value problems with strong parabolic layers. J. Math. Anal. Appl., 267:194208, 2002.MathSciNetCrossRefGoogle Scholar
  25. H. Roos, M. Stynes, and L. Tobiska. Numerical Methods for Singularly Perturbed Differential Equations, volume 24 of Springer Series in Computational Mathematics. Springer Verlag, 1996.Google Scholar
  26. G. Shishkin. Grid approximation of singularly perturbed elliptic and parabolic equations (in Russian). PhD thesis, Second doctorial thesis, Keldysh Institute, Moscow, 1990.Google Scholar
  27. L. Wahlbin. Superconvergence in Galkerkin finite element methods. Springer Verlag, Berlin, 1995.Google Scholar
  28. J. Xu and Z. Zhang. Analysis of recovery type a posteriori error estimators for mildly structured grids. Math. Comp., 73(247):1139–1152, 2004. URL http://www.ams.org/mcom/2004-73-247/S0025-5718-03-01600-4/home.html.MathSciNetCrossRefGoogle Scholar
  29. Z. Zhang. Ultraconvergence of the patch recovery technique II. Mathematics Of Computation, 69(229):141–158, 1999.CrossRefGoogle Scholar
  30. O. Zienkiewicz and J.Z. Zhu. The superconvergence patch recovery and a posteriori error estimates. Part 1: The recovery techniques. Int. J. Number. Methods Engrg., 33:1331–1364, 1992a.MathSciNetCrossRefGoogle Scholar
  31. O. Zienkiewicz and J.Z. Zhu. The superconvergence patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity. Int. J. Number. Methods Engrg., 33:1365–1382, 1992b.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Long Chen
    • 1
  • Pengtao Sun
    • 1
  • Jinchao Xu
    • 1
  1. 1.Department of MathematicsThe Pennsylvania State University

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