Hostname: page-component-6898cfcb55-zkb6p Total loading time: 0 Render date: 2026-11-11T09:36:10.232Z Has data issue: false hasContentIssue false
  • English
  • Français

Model Adaptation Enriched with an Anisotropic Mesh Spacing for Nonlinear Equations: Application to Environmental and CFD Problems

Published online by Cambridge University Press:  28 May 2015

Stefano Micheletti,
Simona Perotto and
Filippo David
Stefano Micheletti*
Affiliation:
MOX-Modeling and Scientific Computing, Department of Mathematics “F. Brioschi”, Politecnico of Milano, Via Bonardi 9, I-20133 Milano, Italy
Simona Perotto*
Affiliation:
MOX-Modeling and Scientific Computing, Department of Mathematics “F. Brioschi”, Politecnico of Milano, Via Bonardi 9, I-20133 Milano, Italy
Filippo David*
Affiliation:
STMicroelectronics, Via Tolomeo 1, I-20100 Cornaredo (MI), Italy
*
Corresponding author.Email address:Stefano.micheletti@polimi.it
Corresponding author.Email address:simona.perotto@polimi.it
Corresponding author.Email address:filippo.david@st.com
Get access

Abstract

Goal of this paper is to suitably combine a model with an anisotropic mesh adaptation for the numerical simulation of nonlinear advection-diffusion-reaction systems and incompressible flows in ecological and environmental applications. Using the reduced-basis method terminology, the proposed approach leads to a noticeable computational saving of the online phase with respect to the resolution of the reference model on nonadapted grids. The search of a suitable adapted model/mesh pair is to be meant, instead, in an offline fashion.

Keywords

65N1565N5065J1535J60Model adaptationanisotropic mesh adaptationgoal-oriented analysisadvection-diffusion-reaction systemsNavier-Stokes equationsfinite elements

Information

Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 6 , Issue 3 , August 2013, pp. 447 - 478
Copyright
Copyright © Global Science Press Limited 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1] Barrett, J. W. and Elliott, C. M., Finite element approximation of the Dirichlet problem using the boundary penalty method, Numer. Math., 49 (1986), pp. 343–366.Google Scholar
[2] Becker, R. and Rannacher, R., An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer., 10 (2001), pp. 1–102.Google Scholar
[3] Braack, M. and Ern, A., A posteriori control of modeling errors and discretization errors, Multi-scale Model. Simul., 1(2) (2003), pp. 221–238.Google Scholar
[4] Braack, M. and Ern, A., Coupling multimodelling with local mesh refinement for the numerical computation of laminar flames, Combust. Theory Model., 8 (2004), pp. 771–788.CrossRefGoogle Scholar
[5] Brooks, A. N. and 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) (1982), pp. 199–259.CrossRefGoogle Scholar
[6] Bruckstein, A. M., Donoho, D. L. and Elad, M., From sparse solutions of systems of equations to sparse modeling of signal and images, SIAM Rev., 51(1) (2009), pp. 34–81.CrossRefGoogle Scholar
[7] Cardiovascular mathematics, modeling and simulation of the circulatory system. In: For-maggia, L., Quarteroni, A. and Veneziani, A., (eds.), Modeling Simulation and Applications 1, Springer, Milano, 2009.Google Scholar
[8] Chinesta, F., Ammar, A. and Cueto, E., Recent advances and new challenges in the use of the proper generalized decomposition for solving multidimensional models, Arch. Comput. Methods Eng., 17(4) (2010), pp. 327–350.Google Scholar
[9] Ciarlet, PH., The Finite Element Method for Elliptic Problems, North-Holland Publishing Company, Amsterdam, 1978.Google Scholar
[10] Clement, PH., Approximation by finite element functions using local regularization, RAIRO Anal. Numér., 2 (1975), pp. 77–84.Google Scholar
[11] David, F. and Prandi, S., Adattività di Modello in Problemi di Diffusione-Trasporto-Reazione con Applicazioni Alla Microelettronica, Master Thesis, Politecnico di Milano, Milano (2006).Google Scholar
[12] Dedè, L., Micheletti, S. and Perotto, S., Anisotropic error control for environmental applications, Appl. Numer. Math., 58(9) (2008), pp. 1320–1339.CrossRefGoogle Scholar
[13] Formaggia, L., Micheletti, S. and Perotto, S., Anisotropic mesh adaptation in computational fluid dynamics: application to the advection diffusion reaction and the Stokes problems, Appl. Numer. Math., 51 (2004), pp. 511–533.CrossRefGoogle Scholar
[14] Formaggia, L. and Perotto, S., New anisotropic a priori error estimates, Numer. Math., 89 (2001), pp. 641–667.Google Scholar
[15] Formaggia, L. and Perotto, S., Anisotropic error estimates for elliptic problems, Numer. Math., 94 (2003), pp. 67–92.CrossRefGoogle Scholar
[16] Georgoulis, E. H., Hall, E. and Houston, P., Discontinuous Galerkin methods for advection-diffusion-reaction problems on anisotropically refined meshes, SIAM J. Sci. Comput., 30(1) (2007), pp. 246–271.Google Scholar
[17] Giles, M. B. and Süli, E., Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality, Acta Numer., 11 (2002), pp. 145–236.Google Scholar
[18] Habashi, W. G., Dompierre, J., Bourgault, Y., Ait-Ali-Yahia, D., Fortin, M. and Vallet, M.-G., Anisotropic mesh adaptation: towards user-independent, mesh-independent and solver-independent CFD. I. general principles, Internat. J. Numer. Methods Fluids, 32(6) (2000), pp. 725–744.Google Scholar
[19] Lions, J. L. and Magenes, E., Non-Homogeneous Boundary Value Problem and Application, Volume I, Springer-Verlag, Berlin, 1972.Google Scholar
[20] Lotka, A. J., Contribution to the theory of periodic reactions, J. Phys. Chem., 14(3) (1910), pp. 271–274.CrossRefGoogle Scholar
[21] Maday, Y. and Rønquist, E. M., A reduced-basis element method, C. R. Acad. Sci. Paris, Ser. I, 335 (2002), pp. 195–200.Google Scholar
[22] Micheletti, S. and Perotto, S., Output functional control for nonlinear equations driven by anisotropic mesh adaption. The Navier-Stokes equations, SIAM J. Sci. Comput., 30(6) (2008), pp. 2817–2854.CrossRefGoogle Scholar
[23] Micheletti, S. and Perotto, S., The effect of anisotropic mesh adaptation on PDE-constrained optimal control problems, SIAM J. Control Optim., 49(4) (2011), pp. 1793–1828.Google Scholar
[24] Micheletti, S. and Perotto, S., Space-time adaptation for purely diffusive problems in an anisotropic framework, Int. J. Numer. Anal. Model., 7(1) (2010), pp. 125–155.Google Scholar
[25] Micheletti, S., Perotto, S. and Picasso, M., Stabilized finite elements on anisotropic meshes: a priori error estimates for the advection-diffusion and Stokes problems, SIAM J. Numer. Anal., 41 (2003), pp. 1131–1162.Google Scholar
[26] Murray, J. D., Mathematical Biology. I. An Introduction, 3rd edition, Springer Verlag, New York 2002.Google Scholar
[27] Murray, J. D., Mathematical Biology. II. Spatial Models and Biomedical Applications, 3rd edition, Springer Verlag, New York 2003.Google Scholar
[28] Oden, J. T. and Prudhomme, S., Goal-oriented error estimation and adaptivity for the finite element method, Comput. Math. Appl., 41 (2001), pp. 735–756.Google Scholar
[29] Oruganti, S., Shi, J. and Shivaji, R., Diffusive logistic equation with constant yield harvesting, I: steady states, Trans. Amer. Math. Soc., 354(9) (2002), pp. 3601–3619.Google Scholar
[30] Perotto, S., Adaptive modeling for free-surface flows, M2AN Math. Model. Numer. Anal., 40(3) (2006), pp. 469–499.CrossRefGoogle Scholar
[31] Perotto, S., Ern, A. and Veneziani, A., Hierarchical local model reduction for elliptic problems: a domain decomposition approach, accepted for publication in Multiscale Model. Simul., (2010).Google Scholar
[32] Power, P. W., Pain, C. C., Piggott, M. D., Fang, F., Gorman, G. J., Umpleby, A. P., God-Dard, A. J. H. and Navon, I. M., Adjoint a posteriori error measures for anisotropic mesh optimisation, Comput. Math. Appl., 52(8-9) (2006), pp. 1213–1242.CrossRefGoogle Scholar
[33] Shi, J. and Shivaji, R., Persistence in reaction diffusion models with weak Allee effect, J. Math. Biol., 52 (2006), pp. 807–829.Google Scholar
[34] Zienkiewicz, O. C. and Zhu, J. Z., A simple error estimator and adaptive procedure for practical engineering analysis, Internat. J. Numer. Methods Eng., 24 (1987), pp. 337–357.Google Scholar
[35] Zienkiewicz, O. C. and Zhu, J. Z., The superconvergent patch recovery and a posteriori error estimates. Part 1: the recovery technique, Internat. J. Numer. Methods Eng., 33 (1992), pp. 1331–1364.Google Scholar