Abstract
This paper presents a two-scale finite element formulation for a variant of the nonlinear Dynamic Diffusion (DD) method, applied to advection-diffusion-reaction problems. The approach, named here new-DD method, introduces locally and dynamically an extra stability through a nonlinear operator acting in all scales of the discretization, and it is designed to be bounded. We use bubble functions to approximate the subgrid scale space, which are locally condensed on the resolved scales. The proposed methodology is solved by an iterative procedure that uses the bubble-enriched Galerkin solution as the correspondent initial approximation, which is automatically recovered wherever stabilization is not required. Since the artificial diffusion introduced by the new-DD method relies on a problem-depend parameter, we investigate alternative choices for this parameter to keep the accuracy of the method. We numerically evaluate stability and accuracy properties of the method for problems with regular solutions and with layers, ranging from advection-dominated to reaction-dominated transport problems.
Supported by organizations CNPq, FAPERJ and FAPES.
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References
Arruda, N., Almeida, R., do Carmo, E.D.: Dynamic diffusion formulation for advection dominated transport problems. Mecánica Computacional 29, 2011–2025 (2010)
Brooks, A., Hughes, T.: Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 32, 199–259 (1982)
Cawood, M., Ervin, V., Layton, W., Maubach, J.: Adaptive defect correction methods for convection dominated, convection diffusion problems. J. Comput. Appl. Math. 116(1), 1–21 (2000)
Galeão, A., do Carmo, E.D.: A consistent approximate upwind Petrov-Galerkin method for convection-dominated. Comput. Methods Appl. Mech. Eng. 10, 83–95 (1988)
Geuzaine, C., Remacle, J.F.: GMSH: a 3-D finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Meth. Eng. 79(11), 1309–1331 (2009)
Guermond, J.L.: Stabilization of Galerkin approximations of transport equation by subgrid modeling. Math. Model. Num. Anal. 33, 1293–1316 (1999)
Guermond, J.L.: Subgrid stabilization of Galerkin approximations of linear monotone operators. IMA J. Numer. Anal. 21, 165–197 (2001)
Hughes, T.J.R., Scovazzi, G., Franca, L.P.: Multiscale and Stabilized Methods, pp. 1–64. American Cancer Society (2017). https://doi.org/10.1002/9781119176817.ecm2051
Hughes, T., Feijoo, G., Luca, M., Jean-Baptiste, Q.: The variational multiscale method - a paradigm for computational mechanics. Comput. Methods Appl. Mech. Eng. 166, 3–24 (1998)
Iliescu, T.: Genuinely nonlinear models for convection-dominated problems. Comput. Math. Appl. 48(10–11), 1677–1692 (2004)
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. Academy of Sciences of the Czech Republic (2006)
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(17–20), 2197–2215 (2007)
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)
Johnson, C., Navert, U., Pitkäranta, J.: Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Eng. 45, 285–312 (1984)
Knopp, T., Lube, G., Rapin, G.: Stabilized finite element methods with shock capturing for advection-diffusion problems. Comput. Methods Appl. Mech. Eng. 191(27), 2997–3013 (2002). https://doi.org/10.1016/S0045-7825(02)00222-0
Mallet, M.: A finite element method for computational fluid dynamics. Ph.D. thesis, Department of Civil Engineering, Stanford University (1985)
Santos, I.P., Almeida, R.C.: A nonlinear subgrid method for advection-diffusion problems. Comput. Methods Appl. Mech. Eng. 196, 4771–4778 (2007)
Santos, I.P., Malta, S.M., Valli, A.M., Catabriga, L., Almeida, R.C.: Convergence analysis of a new dynamic diffusion method. Comput. Math. Appl. 98, 1–9 (2021). https://doi.org/10.1016/j.camwa.2021.06.012
Valli, A., Catabriga, L., Santos, I., Coutinho, A., Almeida, R.: Multiscale dynamic diffusion method to solve advection-diffusion problems. In: XXXVI Ibero-Latin American Congress on Computational Methods in Engineering, Rio de Janeiro, RJ (2015)
Valli, A.M., Almeida, R.C., Santos, I.P., Catabriga, L., Malta, S.M., Coutinho, A.L.: A parameter-free dynamic diffusion method for advection-diffusion-reaction problems. Comput. Math. Appl. 75(1), 307–321 (2018)
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This work was supported by the Foundation for Research Support of Espírito Santo (FAPES) under Grant 181/2017.
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Valli, A.M.P., Santos, I.P., Malta, S.M.C., Catabriga, L., Almeida, R.C. (2021). A Variant of the Nonlinear Multiscale 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_4
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