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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bause, M., Knabner, P.: Numerical simulation of contaminant biodegradation by higher order methods and adaptive time stepping. Comput. Vis. Sci. 7, 61–78 (2004)
Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer, Berlin (2013)
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)
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)
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)
Knabner, P., Angermann, L.: Numerical Methods for Elliptic and Parabolic Partial Differential Equations. Springer, New York (2003)
Ohlberger, M., Rohde, C.: Adaptive finite volume approximations of weakly coupled convection dominated parabolic systems. IMA J. Numer. Anal. 22, 253–280 (2002)
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)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Brunner, F., Frank, F., Knabner, P. (2014). FV Upwind Stabilization of FE Discretizations for Advection–Diffusion Problems. In: Fuhrmann, J., Ohlberger, M., Rohde, C. (eds) Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects. Springer Proceedings in Mathematics & Statistics, vol 77. Springer, Cham. https://doi.org/10.1007/978-3-319-05684-5_16
Download citation
DOI: https://doi.org/10.1007/978-3-319-05684-5_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-05683-8
Online ISBN: 978-3-319-05684-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)