Skip to main content
SpringerLink
Log in

Anisotropic Meshes and Stabilization Parameter Design of Linear SUPG Method for 2D Convection-Dominated Convection–Diffusion Equations

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We propose a numerical strategy to generate a sequence of anisotropic meshes and select appropriate stabilization parameters simultaneously for linear SUPG method solving two dimensional convection-dominated convection–diffusion equations. Since the discretization error in a suitable norm can be bounded by the sum of interpolation error and its variants in different norms, we replace them by some terms which contain the Hessian matrix of the true solution, convective field, and the geometric properties such as directed edges and the area of triangles. Based on this observation, the shape, size and equidistribution requirements are used to derive corresponding metric tensor and stabilization parameters. Numerical results are provided to validate the stability and efficiency of the proposed numerical strategy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. Agouzal, A., Yuri V, V.: Minimization of gradient errors of piecewise linear interpolation on simplicial meshes. Comput. Methods Appl. Mech. Eng. 199(33), 2195–2203 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Apel, T., Lube, G.: Anisotropic mesh refinement in stabilized Galerkin methods. Numer. Math. 74(3), 261–282 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Becker, R., Braack, M.: A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38(4), 173–199 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bochev, P., Perego, M., Peterson, K.: Formulation and analysis of a parameter-free stabilized finite element method. SIAM J. Numer. Anal. 53(5), 2363–2388 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bochev, P., Peterson, K.: A parameter-free stabilized finite element method for scalar advection–diffusion problems. Cent. Eur. J. Math. 11(8), 1458–1477 (2013)

    MathSciNet  MATH  Google Scholar 

  6. Brezzi, F., Leopoldo P, F., Alessandro, R.: Further considerations on residual-free bubbles for advective–diffusive equations. Comput. Methods Appl. Mech. Eng. 166(1), 25–33 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brezzi, F., Russo, A.: Choosing bubbles for advection–diffusion problems. Math. Models Methods Appl. Sci. 4(04), 571–587 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brooks, A.N., Hughes, T.J.R.: Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 32(1), 199–259 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cangiani, A., Süli, E.: The residual-free-bubble finite element method on anisotropic partitions. SIAM J. Numer. Anal. 45(4), 1654–1678 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, L., Sun, P., Jinchao, X.: Optimal anisotropic meshes for minimizing interpolation errors in \(l^p\)-norm. Math. Comput. 76(257), 179–204 (2007)

    Article  MathSciNet  Google Scholar 

  11. Codina, R., Oñate, E., Cervera, M.: The intrinsic time for the streamline upwind/Petrov–Galerkin formulation using quadratic elements. Comput. Methods Appl. Mech. Eng. 94(2), 239–262 (1992)

    Article  MATH  Google Scholar 

  12. Dörfler, W., Nochetto, R.H.: Small data oscillation implies the saturation assumption. Numer. Math. 91(1), 1–12 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Farrell, P., Hegarty, A., Miller, J.M., O’Riordan, E., Shishkin, G.I.: Robust Computational Techniques for Boundary Layers. CRC Press, Boca Raton (2000)

    MATH  Google Scholar 

  14. Formaggia, L., Micheletti, S., Perotto, S.: Anisotropic mesh adaptation in computational fluid dynamics: application to the advection–diffusion–reaction and the Stokes problems. Appl. Numer. Math. 51(4), 511–533 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Franca, L.P., Frey, S.L., Hughes, T.J.R.: Stabilized finite element methods: I. Application to the advective–diffusive model. Comput. Methods Appl. Mech. Eng. 95(2), 253–276 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Franca, L.P., Madureira, A.L.: Element diameter free stability parameters for stabilized methods applied to fluids. Comput. Methods Appl. Mech. Eng. 105(3), 395–403 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hachem, E., Jannoun, G., Veysset, J., Coupez, T.: On the stabilized finite element method for steady convection-dominated problems with anisotropic mesh adaptation. Appl. Math. Comput. 232, 581–594 (2014)

    MathSciNet  Google Scholar 

  18. Hecht, F., Ohtsuka, K., Pironneau, O.: Freefem++ Manual. Université Pierre et Marie Curie. http://www.freefem.org/ff++/index.htm

  19. Hecht, F.: Bamg: bidimensional anisotropic mesh generator. INRIA report (1998)

  20. Hu, G., Qiao, Z., Tang, T.: Moving finite element simulations for reaction–diffusion systems. Adv. Appl. Math. Mech. 4(03), 365–381 (2012)

    Article  MathSciNet  Google Scholar 

  21. Huang, W.: Metric tensors for anisotropic mesh generation. J. Comput. Phys. 204(2), 633–665 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hughes, T.J.R.: Recent progress in the development and understanding of supg methods with special reference to the compressible Euler and Navier–Stokes equations. Int. J. Numer. Methods Fluids 7(11), 1261–1275 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hughes, T.J.R., Brooks, A.: A multidimensional upwind scheme with no crosswind diffusion. Finite Elem. Methods Convect. Domin. Flows 34, 19–35 (1979)

    MathSciNet  MATH  Google Scholar 

  24. Hughes, T.J.R., Franca, L.P., Hulbert, G.M.: A new finite element formulation for computational fluid dynamics: Viii. The Galerkin/least-squares method for advective–diffusive equations. Comput. Methods Appl. Mech. Eng. 73(2), 173–189 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  25. John, V.: A numerical study of a posteriori error estimators for convection–diffusion equations. Comput. Methods Appl. Mech. Eng. 190(5), 757–781 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. John, V., Knobloch, P.: A computational comparison of methods diminishing spurious oscillations in finite element solutions of convection–diffusion equations. In: Proceedings of the International Conference Programs and Algorithms of Numerical Mathematics, vol. 13, pp. 122–136 (2006)

  27. John, V., Knobloch, P.: On spurious oscillations at layers diminishing (sold) methods for convection–diffusion equations: part I-a review. Comput. Methods Appl. Mech. Eng. 196(17), 2197–2215 (2007)

    Article  MATH  Google Scholar 

  28. Li, R., Tang, T., Zhang, P.: Moving mesh methods in multiple dimensions based on harmonic maps. J. Comput. Phys. 170(2), 562–588 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  29. Linß, T.: Anisotropic meshes and streamline-diffusion stabilization for convection–diffusion problems. Commun. Numer. Methods Eng. 21(10), 515–525 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Loseille, A., Alauzet, F.: Continuous mesh framework part I: well-posed continuous interpolation error. SIAM J. Numer. Anal. 49(1), 38–60 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Matthies, G., Skrzypacz, P., Tobiska, L.: A unified convergence analysis for local projection stabilisations applied to the oseen problem. ESAIM: Math. Model. Numer. Anal.-Modél. Math. Anal. Numér. 41(4), 713–742 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. Matthies, G., Tobiska, L.: Local projection type stabilization applied to inf-sup stable discretizations of the oseen problem. IMA J. Numer. Anal. 35(1), 239–269 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. Micheletti, S., Perotto, S., Picasso, M.: Stabilized finite elements on anisotropic meshes: a priori error estimates for the advection–diffusion and the Stokes problems. SIAM J. Numer. Anal. 41(3), 1131–1162 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  34. Mittal, S.: On the performance of high aspect ratio elements for incompressible flows. Comput. Methods Appl. Mech. Eng. 188(1), 269–287 (2000)

    Article  MATH  Google Scholar 

  35. Mizukami, A., Hughes, T.J.R.: A petrov-galerkin finite element method for convection-dominated flows: an accurate upwinding technique for satisfying the maximum principle. Comput. Methods Appl. Mech. Eng. 50(2), 181–193 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  36. Nadler, E.: Piecewise linear best \(l_2\) approximation on triangulations. In: Chui, C. K., Schumacher, L. L., ward, J. D. (eds.) Approximation Theory V: Proceedings Fifth International Symposium on Approximation Theory, pp. 499–502. Academic Press, New York (1986)

  37. Nävert, U.: A Finite Element Method for Convection–Diffusion Problems. Chalmers Tekniska Högskola/Göteborgs Universitet, Department of Computer Science (1982)

  38. Nguyen, H., Gunzburger, M., Lili, J., Burkardt, J.: Adaptive anisotropic meshing for steady convection-dominated problems. Comput. Methods Appl. Mech. Eng. 198(37), 2964–2981 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Picasso, M.: An anisotropic error indicator based on Zienkiewicz–Zhu error estimator: application to elliptic and parabolic problems. SIAM J. Sci. Comput. 24(4), 1328–1355 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  40. Principe, J., Codina, R.: On the stabilization parameter in the subgrid scale approximation of scalar convection–diffusion–reaction equations on distorted meshes. Comput. Methods Appl. Mech. Eng. 199(21), 1386–1402 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Roos, H.-G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection–Diffusion–Reaction and Flow Problems, vol. 24. Springer, Berlin (2008)

    MATH  Google Scholar 

  42. Stynes, M.: Steady-state convection–diffusion problems. Acta Numer. 14, 445–508 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  43. Sun, P., Chen, L., Jinchao, X.: Numerical studies of adaptive finite element methods for two dimensional convection-dominated problems. J. Sci. Comput. 43(1), 24–43 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  44. Tang, H.-Z., Tang, T., Zhang, P.: An adaptive mesh redistribution method for nonlinear Hamilton–Jacobi equations in two-and three-dimensions. J. Comput. Phys. 188(2), 543–572 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  45. Tezduyar, T.E., Mittal, S., Ray, S.E., Shih, R.: Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements. Comput. Methods Appl. Mech. Eng. 95(2), 221–242 (1992)

    Article  MATH  Google Scholar 

  46. Tezduyar, T.E., Park, Y.J.: Discontinuity-capturing finite element formulations for nonlinear convection–diffusion–reaction equations. Comput. Methods Appl. Mech. Eng. 59(3), 307–325 (1986)

    Article  MATH  Google Scholar 

  47. Tobiska, L., Verfürth, R.: Robust a posteriori error estimates for stabilized finite element methods. IMA J. Numer. Anal. 35(4), 1652–1671 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  48. Xie, H., Yin, X.: Metric tensors for the interpolation error and its gradient in \(l^p\) norm. J. Comput. Phys. 256, 543–562 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  49. Zienkiewicz, O.C., Zhu, J.Z.: A simple error estimator and adaptive procedure for practical engineerng analysis. Int. J. Numer. Meth. Eng. 24(2), 337–357 (1987)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

Yana Di is supported in part by the National Natural Science Foundation of China (91630208, 11771437). Hehu Xie is supported in part by the National Natural Science Foundation of China (91730302, 11771434, 91330202, 11371026, 11001259, 11031006, 2011CB309703) and Science Challenge Project (TZ2016002). Xiaobo Yin is supported by National Natural Science Foundation of China (11671165, 91630201), Program for Changjiang Scholars and Innovative Research Team in University \(\sharp \) IRT13066, and self-determined research funds of Central China Normal University (CCNU16A02039). The authors would like to thank Professor Lutz Tobiska for his valuable suggestion to this work. We are also thankful to anonymous reviewers for their remarks and suggestions.

Author information

Authors and Affiliations

  1. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Yana Di & Hehu Xie

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Yana Di & Hehu Xie

  3. Hubei Key Laboratory of Mathematical Sciences, School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China

    Xiaobo Yin

Authors
  1. Yana Di
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hehu Xie
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaobo Yin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaobo Yin.

Additional information

Yana Di and Hehu Xie are supported in part by the National Center for Mathematics and Interdisciplinary Science, CAS.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Di, Y., Xie, H. & Yin, X. Anisotropic Meshes and Stabilization Parameter Design of Linear SUPG Method for 2D Convection-Dominated Convection–Diffusion Equations. J Sci Comput 76, 48–68 (2018). https://doi.org/10.1007/s10915-017-0610-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-017-0610-9

Keywords

Mathematic Subject Classification

Search

Navigation