[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

A Posteriori Analysis of a Positive Streamwise Invariant Discretization of a Convection-Diffusion Equation

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

Abstract

We consider the finite element discretization of a convection-diffusion equation, where the convection term is handled via a fluctuation splitting algorithm. We prove a posteriori error estimates which allow us to perform mesh adaptivity in order to optimize the discretization of these equations. Numerical results confirm the interest of such an approach.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abgrall, R.: Residual distribution schemes: current status and future trends. Comput. Fluids 35, 641–669 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abgrall, R., Mezine, M.: Construction of second-order accurate monotone and stable residual distribution schemes for steady problems. J. Comput. Phys. 195, 474–507 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abgrall, R., Roe, P.L.: High order fluctuation schemes on triangular meshes. J. Sci. Comput. 19, 3–36 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bernardi, C., Hecht, F., Verfürth, R.: A posteriori error analysis of the method of characteristics. Math. Models Methods Appl. Sci. 21, 1355–1376 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bernardi, C., Maday, Y., Rapetti, F.: Discrétisations variationnelles de problèmes aux limites elliptiques. In: Collection Mathématiques et Applications, vol. 45. Springer, Berlin (2004)

    Google Scholar 

  6. Burman, E.: A unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty. SIAM J. Numer. Anal. 43, 2012–2033 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chacón Rebollo, T., Gómez Mármol, M., Narbona Reina, G.: Numerical analysis of the PSI solution of advection–diffusion problems through a Petrov–Galerkin formulation. Math. Models Methods Appl. Sci. 17, 1905–1936 (2004)

    Article  Google Scholar 

  8. Clément, P.: Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9(R2), 77–84 (1975)

    Google Scholar 

  9. Cockburn, B.: An introduction to the discontinuous Galerkin method for advection-dominated problems. In: Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. Lecture Notes in Math., vol. 1697, pp. 151–268. Springer, Berlin (1998)

    Chapter  Google Scholar 

  10. Deconinck, H., Struijs, R., Bourgeois, G., Roe, P.L.: Compact advection schemes on unstructured meshes. In: Comput. Fluid Dynamics. VKI Lecture Series, vol. 1993-04 (1993)

    Google Scholar 

  11. Deconinck, H., Struijs, R., Roe, P.L.: Compact advection schemes on unstructured grids. In: Comput. Fluid Dynamics. VKI Lecture Series, vol. 1993-04 (1993)

    Google Scholar 

  12. Després, B.: Lax theorem and finite volume schemes. Math. Comput. 73, 1203–1234 (2004)

    MATH  Google Scholar 

  13. Douglas, J. Jr., Russell, T.F.: Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SIAM J. Numer. Anal. 19, 871–885 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  14. Eriksson, K., Larson, M.G., Målqvist, A.: A posteriori error analysis of the boundary penalty method. Chalmers Finite Element Center Preprints 09 (2004)

  15. Ern, A., Stephansen, A.F.: A posteriori energy-norm error estimates for advection-diffusion equations approximated by weighted interior penalty methods. J. Comput. Math. 26, 488–510 (2008)

    MathSciNet  MATH  Google Scholar 

  16. Ern, A., Stephansen, A.F., Vohralík, M.: Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusion-reaction problems. J. Comput. Appl. Math. 234, 114–130 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Frey, P.J., George, P.-L.: Maillages, applications aux éléments finis. Hermès, Paris (1999)

    Google Scholar 

  18. Hecht, F., Pironneau, O.: FreeFem++ (2011). www.freefem.org

  19. Lesaint, P., Raviart, P.-A.: On a finite element method for solving the neutron transport equation. In: Mathematical Aspects of Finite Elements in Partial Differential Equations. Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison (1974), pp. 89–123

    Google Scholar 

  20. Nitsche, J.: Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Semin. Univ. Hamb. 36, 9–15 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  21. Perthame, B.: Convergence of N-schemes for linear advection equations. Trends in applications of mathematics to mechanics. Pitman Monogr. Surv. Pure Appl. Math. 77, 323–333 (1985)

    MathSciNet  Google Scholar 

  22. Pironneau, O.: On the transport-diffusion algorithm and its applications to the Navier–Stokes equations. Numer. Math. 38, 309–332 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  23. Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer, Berlin (1994)

    MATH  Google Scholar 

  24. Raviart, P.-A., Thomas, J.-M.: A mixed finite element method for second order elliptic problems. In: Mathematical Aspects of Finite Element Methods. Lecture Notes in Mathematics, vol. 606, pp. 292–315. Springer, Berlin (1977)

    Chapter  Google Scholar 

  25. Reed, W., Hill, T.: Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos National Laboratory, Los Alamos, NM (1973)

  26. Roe, P.L.: Fluctuations and signals—a framework for numerical evolution problems. In: Numerical Methods for Fluid Dynamics. Proc. Conf., Reading (1982), pp. 219–257

    Google Scholar 

  27. Smith, R.M., Hutton, A.G.: The numerical treatment of advection: a performance comparison of current methods. Numer. Heat Transf., Part A, Appl. 5, 439–461 (1982)

    Google Scholar 

  28. Struijs, R.: A multidimensional upwind discretization method for the Euler equations on unstructured grids. Ph.D. Thesis, Technische Universiteit Delft (1994)

  29. Struijs, R., Deconinck, H., Roe, P.L.: Fluctuation splitting schemes for the 2d Euler equations. In: Comput. Fluid Dynamics. VKI Lecture Series, vol. 1991-01 (1991)

    Google Scholar 

  30. Verfürth, R.: A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley & Teubner, New York (1996)

    MATH  Google Scholar 

  31. Vohralík, M.: A posteriori error estimates for lowest-order mixed finite element discretizations of convection-diffusion-reaction equations. SIAM J. Numer. Anal. 45, 1570–1599 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. Vohralík, M.: Residual flux-based a posteriori error estimates for finite volume and related locally conservative methods. Numer. Math. 111, 121–158 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  33. Vohralík, M.: Guaranteed and fully robust a posteriori error estimates for conforming discretizations of diffusion problems with discontinuous coefficients. J. Sci. Comput. 46, 397–438 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions, C.N.R.S. & Université Pierre et Marie Curie—Paris 6, boîte 187, 4 place Jussieu, 75252, Paris Cedex 05, France

    Christine Bernardi

  2. Departamento de Ecuaciones Diferenciales y Análisis Numerico, Universidad de Sevilla, Tarfia s/n, 41012, Sevilla, Spain

    Tomás Chacón Rebollo & Marco Restelli

Authors
  1. Christine Bernardi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tomás Chacón Rebollo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marco Restelli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christine Bernardi.

Additional information

Research partially supported by Junta de Andalucía Excellence grant P07-FQM-02538.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bernardi, C., Chacón Rebollo, T. & Restelli, M. A Posteriori Analysis of a Positive Streamwise Invariant Discretization of a Convection-Diffusion Equation. J Sci Comput 51, 349–374 (2012). https://doi.org/10.1007/s10915-011-9514-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-011-9514-2

Keywords

Search

Navigation