Skip to main content
SpringerLink
Log in

Multiscale Finite Element Formulation for the 3D Diffusion-Convection Equation

  • Conference paper
  • First Online:
Computational Science and Its Applications – ICCSA 2020 (ICCSA 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12251))

Included in the following conference series:

Abstract

In this work we present a numerical study of the Dynamic Diffusion (DD) method for solving diffusion-convection problems with dominant convection on three dimensional domains. The DD method is a two-scale nonlinear model for convection-dominated transport problems, obtained by incorporating to the multiscale formulation a nonlinear dissipative operator acting isotropically in both scales of the discretization. The standard finite element spaces are enriched with bubble functions in order to add stability properties to the numerical model. The DD method for two dimensional problems results in good solutions compared to other known methods. We analyze the impact of this methodology on three dimensional domains comparing its numerical results with those obtained using the Consistent Approximate Upwind (CAU) method.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Dehghan, M.: On the numerical solution of the one-dimensional convection-diffusion equation. Math. Probl. Eng. 2005(1), 61–74 (2005)

    Article  Google Scholar 

  2. Donea, J., Huerta, A.: Finite Element Method for Flow Problems. Wiley, Hoboken (2003)

    Book  Google Scholar 

  3. Bause, M.: Stabilized finite element methods with shock-capturing for nonlinear convection-diffusion-reaction models. In: Kreiss, G., Lötstedt, P., Målqvist, A., Neytcheva, M. (eds.) ENUMATH 2009, Numerical Mathematics and Advanced Applications 2009, pp. 125–133. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-11795-4_12

  4. Barrenechea, G.R., John, V., Knobloch, P.: A local projection stabilization finite element method with nonlinear crosswind diffusion for convection-diffusion-reaction equations. ESAIM: Math. Model. Numer. Anal. 47(5), 1335–1366 (2013)

    Article  MathSciNet  Google Scholar 

  5. Roos, H., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion-Reaction and Flow Problems, 2nd edn. Springer Science & Business Media, Berlin (2008). https://doi.org/10.1007/978-3-540-34467-4

    Book  MATH  Google Scholar 

  6. 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(1–3), 199–259 (1982)

    Article  MathSciNet  Google Scholar 

  7. Johnson, C., Nävert, U., Pitkäranta, J.: Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Eng. 45(1), 285–312 (1984)

    Article  MathSciNet  Google Scholar 

  8. 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  Google Scholar 

  9. Burman, E., Hansbo, P.: Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems. Comput. Methods Appl. Mech. Eng. 193(15), 1437–1453 (2004)

    Article  MathSciNet  Google Scholar 

  10. Barrenechea, G.R., Burman, E., Karakatsani, F.: Blending low-order stabilised finite element methods: a positivity-preserving local projection method for the convection-diffusion equation. Comput. Methods Appl. Mech. Eng. 317, 1169–1193 (2017)

    Article  MathSciNet  Google Scholar 

  11. Valli, A.M.P., Almeida, R.C., Santos, I.P., Catabriga, L., Malta, S.M.C., Coutinho, A.L.G.A.: A parameter-free dynamic diffusion method for advection-diffusion-reaction problems. Comput. Math. Appl. 75(1), 307–321 (2018)

    Article  MathSciNet  Google Scholar 

  12. 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  Google Scholar 

  13. John, V., Knobloch, P.: On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations: part II - analysis for P1 and Q1 finite elements. Comput. Methods Appl. Mech. Eng. 197(21–24), 1997–2014 (2008)

    Article  Google Scholar 

  14. Hughes, T.J.R.: Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput. Methods Appl. Mech. Eng. 127(1–4), 387–401 (1995)

    Article  MathSciNet  Google Scholar 

  15. Hughes, T.J.R., Stewart, J.R.: A space-time formulation for multiscale phenomena. J. Comput. Appl. Math. 74(1–2), 217–229 (1996)

    Article  MathSciNet  Google Scholar 

  16. Brezzi, F., Franca, L.P., Hughes, T.J.R., Russo, A.: b = \(\smallint \)g. Comput. Methods Appl. Mech. Eng. 145, 329–339 (1997)

    Google Scholar 

  17. Hughes, T.J.R., Feijóo, G.R., Mazzei, L., Quincy, J.-B.: The variational multiscale method - a paradigm for computational mechanics. Comput. Methods Appl. Mech. Eng. 166(1–2), 3–24 (1998)

    Article  MathSciNet  Google Scholar 

  18. Hughes, T.J.R., Scovazzi, G., Franca, L.P.: Multiscale and stabilized methods. In: Encyclopedia of Computational Mechanics, 2nd edn, pp. 1–64 (2018)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Franca, L.P., Nesliturk, A., Stynes, M.: On the stability of residual-free bubbles for convection-diffusion problems and their approximation by a two-level finite element method. Comput. Methods Appl. Mech. Eng. 166(1–2), 35–49 (1998)

    Article  MathSciNet  Google Scholar 

  21. Guermond, J.-L.: Stabilization of Galerkin approximations of transport equation by subgrid modeling. Math. Model. Numer. Anal. 33, 1293–1316 (1999)

    Article  MathSciNet  Google Scholar 

  22. Guermond, J.-L.: Subgrid stabilization of Galerkin approximations of linear monotone operators. IMA J. Numer. Anal. 21, 165–197 (2001)

    Article  MathSciNet  Google Scholar 

  23. Santos, I.P., Almeida, R.C.: A nonlinear subgrid method for advection-diffusion problems. Comput. Methods Appl. Mech. Eng. 196(45–48), 4771–4778 (2007)

    Article  MathSciNet  Google Scholar 

  24. Santos, I.P., Almeida, R.C., Malta, S.M.C.: Numerical analysis of the nonlinear subgrid scale method. Comput. Appl. Math. 31(3), 473–503 (2012)

    Article  MathSciNet  Google Scholar 

  25. Arruda, N. C. B., Almeida, R. C., Dutra do Carmo, E. G.: Dynamic diffusion formulations for advection dominated transport problems. In: Dvorkin, E., Goldschmit, M., Storti, M. (eds.) Mecánica Computacional. vol. 29, pp. 2011–2025. Asociación Argentina de Mecánica Computacional, Buenos Aires (2010)

    Google Scholar 

  26. Bento, S.S., de Lima, L.M., Sedano, R.Z., Catabriga, L., Santos, I.P.: A nonlinear multiscale viscosity method to solve compressible flow problems. In: Gervasi, O., et al. (eds.) ICCSA 2016. LNCS, vol. 9786, pp. 3–17. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-42085-1_1

    Chapter  Google Scholar 

  27. Bento, S. S., Sedano, R. Z., Lima, L. M., Catabriga, L., Santos, I. P.: Comparative studies to solve compressible problems by multiscale finite element methods. Proc. Ser. Braz. Soc. Comput. Appl. Math. 5(1) (2017). https://doi.org/10.5540/03.2017.005.01.0317

  28. Baptista, R., Bento, S.S., Santos, I.P., Lima, L.M., Valli, A.M.P., Catabriga, L.: A multiscale finite element formulation for the incompressible Navier-Stokes equations. In: Gervasi, O., et al. (eds.) ICCSA 2018. LNCS, vol. 10961, pp. 253–267. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-95165-2_18

    Chapter  Google Scholar 

  29. Galeão, A.C., Dutra do Carmo, E.G.: A consistent approximate upwind Petrov-Galerkin method for convection-dominated problems. Comput. Methods Appl. Mech. Eng. 68(1), 83–95 (1988)

    Article  MathSciNet  Google Scholar 

  30. Raviart, P.A., Thomas, J.M.: Introduction à l’analyse numérique des équations aux dérivées partielles. Masson, Paris (1992)

    Google Scholar 

  31. Hughes, T. J. R.: The Finite Element Method: Linear Static and Dymanic Finite Element Analysis. Prentice-Hall International (1987)

    Google Scholar 

  32. Galeão, A.C., Almeida, R.C., Malta, S.M.C., Loula, A.F.D.: Finite element analysis of convection dominated reaction-diffusion problems. Appl. Numer. Math. 48(2), 205–222 (2004)

    Article  MathSciNet  Google Scholar 

  33. Carmo, E.G.D., Alvarez, G.B.: A new stabilized finite element formulation for scalar convection-diffusion problems: the streamline and approximate upwind/Petrov-Galerkin method. Comput. Methods Appl. Mech. Eng. 192(31–32), 3379–3396 (2003)

    Article  MathSciNet  Google Scholar 

  34. Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements, vol. 159, 1st edn. Springer Science & Business Media, New York (2004). https://doi.org/10.1007/978-1-4757-4355-5

    Book  MATH  Google Scholar 

  35. Saad, Y., Schultz, H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM. J. Sci. Stat. Comput. 7(3), 856–869 (1986)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the support through the Espírito Santo State Research Support Foundations (FAPES), under Grant Term 181/2017, and the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.

Author information

Authors and Affiliations

  1. Optimization and Computational Modeling Lab, Federal University of Espírito Santo, Vitória, ES, Brazil

    Ramoni Z. S. Azevedo & Isaac P. Santos

  2. Department of Applied Mathematics, Federal University of Espírito Santo, São Mateus, ES, Brazil

    Isaac P. Santos

Authors
  1. Ramoni Z. S. Azevedo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Isaac P. Santos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Ramoni Z. S. Azevedo or Isaac P. Santos .

Editor information

Editors and Affiliations

  1. University of Perugia, Perugia, Italy

    Osvaldo Gervasi

  2. University of Basilicata, Potenza, Potenza, Italy

    Beniamino Murgante

  3. Chair- Center of ICT/ICE, Covenant University, Ota, Nigeria

    Sanjay Misra

  4. University of Cagliari, Cagliari, Italy

    Chiara Garau

  5. University of Cagliari, Cagliari, Italy

    Ivan Blečić

  6. Clayton School of Information Technology, Monash University, Clayton, VIC, Australia

    David Taniar

  7. Department of Information Science, Kyushu Sangyo University, Fukuoka, Japan

    Bernady O. Apduhan

  8. University of Minho, Braga, Portugal

    Ana Maria A. C. Rocha

  9. Polytechnic University of Bari, Bari, Italy

    Eufemia Tarantino

  10. Polytechnic University of Bari, Bari, Italy

    Carmelo Maria Torre

  11. Department of Neurology, University of Massachusetts Medical School, Worcester, MA, USA

    Yeliz Karaca

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Azevedo, R.Z.S., Santos, I.P. (2020). Multiscale Finite Element Formulation for the 3D Diffusion-Convection Equation. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2020. ICCSA 2020. Lecture Notes in Computer Science(), vol 12251. Springer, Cham. https://doi.org/10.1007/978-3-030-58808-3_33

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-58808-3_33

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-58807-6

  • Online ISBN: 978-3-030-58808-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation