FV Upwind Stabilization of FE Discretizations for Advection–Diffusion Problems

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 77)

Abstract

We apply a novel upwind stabilization of a mixed hybrid finite element method of lowest order to advection–diffusion problems with dominant advection and compare it with a finite element scheme stabilized by finite volume upwinding. Both schemes are locally mass conservative and employ an upwind-weighting formula in the discretization of the advective term. Numerical experiments indicate that the upwind-mixed method is competitive with the finite volume method. It prevents the appearance of spurious oscillations and produces nonnegative solutions for strongly advection-dominated problems, while the amount of artificial diffusion is lower than that of the finite volume method. This makes the method attractive for applications in which too much numerical diffusion is critical and may lead to false predictions; e.g., if highly nonlinear reactive processes take place only in thin interaction regions.

References

  1. 1.
    Bause, M., Knabner, P.: Numerical simulation of contaminant biodegradation by higher order methods and adaptive time stepping. Comput. Vis. Sci. 7, 61–78 (2004)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer, Berlin (2013)CrossRefMATHGoogle Scholar
  3. 3.
    Brunner, F., Radu, F.A., Bause, M., Knabner, P.: Optimal order convergence of a modified BDM\(_1\) mixed finite element scheme for reactive transport in porous media. Adv. Water Resour. 35, 163–171 (2012)CrossRefGoogle Scholar
  4. 4.
    Brunner, F., Radu, F.A., Knabner, P.: Analysis of an upwind-mixed hybrid finite element method for transport problems. SIAM J. Num. Anal. 52(1), 83–102 (2014)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Hoffmann, J.: Reactive transport and mineral dissolution/precipitation in porous media: efficient solution algorithms, benchmark computations and existence of global solutions. Ph.D. thesis, Friedrich–Alexander University of Erlangen–Nuremberg (2010)Google Scholar
  6. 6.
    Knabner, P., Angermann, L.: Numerical Methods for Elliptic and Parabolic Partial Differential Equations. Springer, New York (2003)MATHGoogle Scholar
  7. 7.
    Ohlberger, M., Rohde, C.: Adaptive finite volume approximations of weakly coupled convection dominated parabolic systems. IMA J. Numer. Anal. 22, 253–280 (2002)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Radu, F.A., Suciu, N., Hoffmann, J., Vogel, A., Kolditz, O., Park, C.H., Attinger, S.: Accuracy of numerical simulations of contaminant transport in heterogeneous aquifers: a comparative study. Adv. Water Resour. 34(1), 47–61 (2011)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Erlangen–NurembergErlangenGermany

Personalised recommendations