Abstract
In this work we derive a formulation for discontinuous diffusion tensor for the Discrete Duality Finite Volume (DDFV) framework that is exact for affine solutions. In fact, DDFV methods can naturally handle anisotropic or non-linear problems on general distorded meshes. Nonetheless, a special treatment is required when the diffusion tensor is discontinuous across an internal interfaces shared by two control volumes of the mesh. In such a case, two different gradients are considered in the two subdiamonds centered at that interface and the flux conservation is imposed through an auxiliary variable at the interface.
MSC2010: 65N08, 76S05
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Andreianov, B., Bendahmane, M., Hubert, F.: On 3D DDFV discretization of gradient and divergence operators. Part II. (2011). HAL, http://hal.archives-ouvertes.fr/hal-00567342
Andreianov, B., Bendahmane, M., Hubert, F., Krell, S.: On 3D DDFV discretization of gradient and divergence operators. Part I. (2011) HAL, http://hal.archives-ouvertes.fr/hal-00355212.
Andreianov, B., Bendahmane, M., Karlsen, K.: A gradient reconstruction formula for finite-volume schemes and discrete duality. FVCA5, Wiley, (2008).
Andreianov, B., Hubert, F., Krell, S.: Benchmark 3D: a version of the DDFV scheme with cell/vertex unknowns on general meshes, this volume (2011).
Boyer, F., Hubert, F.: Benchmark on anisotropic problems, the DDFV discrete duality finite volumes and m-DDFV schemes. In: R. Eymard, J.M. Hérard (eds.) FVCA5, Wiley, (2008).
Coudière, Y., Hubert, F.: A 3D discrete duality finite volume method for nonlinear elliptic equation (2010) HAL, URL: http://hal.archives-ouvertes.fr/hal-00456837/fr.
Coudière, Y., Hubert, F., Manzini, G.: Benchmark 3D: CeVeFE-DDFV, a discrete duality scheme with cell/vertex/face+edge unknowns. (this volume) (2011).
Coudière, Y., Pierre, C., Rousseau, O., Turpault, R.: A 2D/3D discrete duality finite volume scheme. Application to ECG simulation. International Journal on Finite Volumes (2009) 6(1).
Domelevo, K., Omnès, P.: A finite volume method for the laplace equation on almost arbitrary two-dimensional grids. M2AN, Math. Model. Numer. Anal. (2005) 39(6), 1203–1249.
Herbin, R., Hubert, F.: Benchmark on discretization schemes for anisotropic diffusion problems on general grids. FVCA5, Wiley, (2008).
Hermeline, F.: Approximation of diffusion operators with discontinuous tensor coefficients on distorted meshes. Comp. Meth. Appl. Mech. Eng. (2003) 192(16), 1939–1959.
Hermeline, F.: A finite volume method for approximating 3D diffusion operators on general meshes. Journal of computational Physics (2009) 228(16), 5763–5786.
Pierre, C.: Modélisation et simulation de l’activité électrique du coeur dans le thorax, analyse numérique et méthodes de volumes finis. Ph.D. thesis, Université de Nantes (2005).
Acknowledgements
The work of the first two authors was supported by Groupement de Recherche MOMAS. The work of the third author was partially supported by the Italian MIUR through the program PRIN2008.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Coudière, Y., Hubert, F., Manzini, G. (2011). A CeVeFE DDFV scheme for discontinuous anisotropic permeability tensors. In: Fořt, J., Fürst, J., Halama, J., Herbin, R., Hubert, F. (eds) Finite Volumes for Complex Applications VI Problems & Perspectives. Springer Proceedings in Mathematics, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20671-9_30
Download citation
DOI: https://doi.org/10.1007/978-3-642-20671-9_30
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-20670-2
Online ISBN: 978-3-642-20671-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)