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A Convergence Study of the 3D Dynamic Diffusion Method

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Computational Science and Its Applications – ICCSA 2021 (ICCSA 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12949))

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Abstract

In this work we present a convergence study of the multiscale Dynamic Diffusion (DD) method applied to the three-dimensional steady-state transport equation. We consider diffusion-convection and diffusion-convection-reaction problems, varying the diffusion coefficient in order to obtain an increasingly less diffusive problem. For both cases, the convergence order estimates are evaluated in the energy norm and the \(L^2(\varOmega )\) and \(H^1(\varOmega )\) Sobolev spaces norms. In order to investigate the meshes effects on the convergence, the numerical experiments were carried out on two different sets of meshes: one with structured meshes and the other with unstructured ones. The numerical results show optimal convergence rates in all norms for the dominant convection case.

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References

  1. Friedman, A., Littman, W.: Industrial mathematics: a course in solving real-world problems. Society for Industrial and Applied Mathematics, pp. 27–45 (1994)

    Google Scholar 

  2. Kuttler, C.: Reaction-Diffusion equations with applications. In: Internet siminar, Technische Unversität München, pp. 3–11 (2011)

    Google Scholar 

  3. Jüngel, A.: Diffusive and nondiffusive population models. In: Naldi, G., Pareschi, L., Toscani, G. (eds) Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences. Modeling and Simulation in Science, Engineering and Technology, pp 397–425. Birkhäuser Boston (2010) https://doi.org/10.1007/978-0-8176-4946-3_15

  4. Colombo, R.M., Garavello, M., Lécureux-Mercier, M.: A class of nonlocal models for pedestrian traffic. Math. Models Methods Appl. Sci. 22(04), 1150023 (2012). https://doi.org/10.1142/S0218202511500230

    Article  MathSciNet  MATH  Google Scholar 

  5. Lopes, R.J.G., Quinta-Ferreira, R.M.: Numerical assessment of diffusion-convection-reaction model for the catalytic abatement of phenolic wastewaters in packed-bed reactors under trickling flow conditions. Comput. Chem. Eng. 35(12), 2706–2715 (2011)

    Article  Google Scholar 

  6. Yin, S., Towler, B.F., Dusseault, M.B., Rothenburg, L.: Fully coupled THMC modeling of wellbore stability with thermal and solute convection considered. Transp. Porous Media 84(3), 773–798 (2010). https://doi.org/10.1007/s11242-010-9540-9

    Article  MathSciNet  Google Scholar 

  7. Quarteroni, A.: Numerical Models for Differential Problems, vol. 16. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-49316-9_21

  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(1–3), 199–259 (1982)

    Article  MathSciNet  Google Scholar 

  9. 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)

    Google Scholar 

  10. Franca, L., Valentin, F.: On an improved unusual stabilized finite element method for the advective-reactive-diffusive equation. Comput. Methods Appl. Mech. Eng. 190(13), 1785–1800 (2000). https://doi.org/10.1016/S0045-7825(00)00190-0

    Article  MathSciNet  MATH  Google Scholar 

  11. 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 

  12. Knobloch, P., Lube, G.: Local projection stabilization for advection-diffusion-reaction problems: one-level vs. two-level approach. Appl. Numer. Math. 59(12), 2891–2907 (2009)

    Google Scholar 

  13. 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 

  14. 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 

  15. Johnson, C., Nävert, U., Pitkäranta, J.: Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Eng. 45(1), 285–312 (1984). https://doi.org/10.1016/0045-7825(84)90158-0

    Article  MathSciNet  MATH  Google Scholar 

  16. Allendes, A., Barrenechea, G., Rankin, R.: Fully computable error estimation of a nonlinear, positivity-preserving discretization of the convection-diffusion-reaction equation. SIAM J. Sci. Comput. 39(5), A1903–A1927 (2017). https://doi.org/10.1137/16M1092763

  17. 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)

    Google Scholar 

  18. 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 

  19. 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 

  20. Hughes, T.J., 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 

  21. 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 

  22. 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 

  23. 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 

  24. 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 

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

    Article  MathSciNet  Google Scholar 

  26. 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 

  27. Azevedo, R.Z.S., Santos, I.P.: Multiscale finite element formulation for the 3D diffusion-convection equation. In: Gervasi, O., et al. (eds.) ICCSA 2020. LNCS, vol. 12251, pp. 455–469. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-58808-3_33

    Chapter  Google Scholar 

  28. Santos, I.P., Malta, S.M., Valli, A.M., Catabriga, L., Almeida, R.C.: Convergence analysis of a new dynamic diffusion method (2021, submitted)

    Google Scholar 

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

  30. 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)

    Google Scholar 

  31. Becker, E.B., Carey, G.F., Oden, J.T.: Finite Elements - An Introduction, vol. 1. Prentice-Hall, New Jersey (1981)

    MATH  Google Scholar 

  32. Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method - Volume 1: The Basis, 5th edn. Butterworth-Heinemann, Massachusetts (2000)

    Google Scholar 

  33. Geuzaine, C., Remacle, J.F.: Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Num. Methods Eng. 79(11), 1309–1331 (2009). https://gmsh.info

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Acknowledgments

This work was supported by the Espírito Santo State Research Support Foundation (FAPES), under Grant Term 181/2017, and by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.

Author information

Authors and Affiliations

  1. Laboratório de Otimização e Modelagem Computacional, Programa de Pós-Graduação em Informática, Universidade Federal do Espírito Santo, Vitória, ES, Brazil

    Ramoni Z. S. Azevedo, Lucia Catabriga & Isaac P. Santos

  2. Departamento de Informática, Universidade Federal do Espírito Santo, Vitória, ES, Brazil

    Lucia Catabriga

  3. Departamento de Matemática Aplicada, Universidade Federal do Espírito Santo, São Mateus, ES, Brazil

    Isaac P. Santos

Authors
  1. Ramoni Z. S. Azevedo
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  2. Lucia Catabriga
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  3. Isaac P. Santos
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Corresponding authors

Correspondence to Ramoni Z. S. Azevedo , Lucia Catabriga 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. Covenant University, Ota, Nigeria

    Sanjay Misra

  4. University of Cagliari, Cagliari, Italy

    Chiara Garau

  5. University of Cagliari, Cagliari, Italy

    Ivan Blečić

  6. Monash University, Clayton, VIC, Australia

    David Taniar

  7. 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

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Azevedo, R.Z.S., Catabriga, L., Santos, I.P. (2021). A Convergence Study of the 3D Dynamic Diffusion Method. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2021. ICCSA 2021. Lecture Notes in Computer Science(), vol 12949. Springer, Cham. https://doi.org/10.1007/978-3-030-86653-2_5

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  • DOI: https://doi.org/10.1007/978-3-030-86653-2_5

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