4. Conclusions
We have presented a discrete calculus of differential forms and applied it to several partial differential equations of current interest. It is of interest that our techniques apply on smooth manifolds in any finite number of dimensions. Interesting possibilities remain for future work, including applications to manifolds with indefinite inner products — related to time discretization — and deriving new error estimates in the differential forms setting.
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
Preview
Unable to display preview. Download preview PDF.
References
P. Bochev. A discourse on variational and geometric aspects of stability of discretizations. In: 33rd Computational Fluid Dynamics Lecture Series, VKI LS 2003-05, edited by H. Deconinck, ISSN0377-8312. Von Karman Institute for Fluid Dynamics, Chaussee de Waterloo, 72, B-1640 Rhode Saint Genese, Belgium. 90 pages.
W. Boothby. An Introduction to Differentiable Manifolds and Riemannian Geometry, Second Edition. Volume 120 in Pure and Applied Mathematics. Editors: S. Eilenberg and H. Bass. Academic Press, Inc., New York, 1986.
Q. Du, V. Faber, and M. Gunzburger. Centroidal Voronoi Tessellations: Applications and Algorithms. SIAM Review 41(4):637–676, 1999.
H. Flanders. Differential Forms with Applications to the Physical Sciences. Academic Press, Inc., New York, 1963.
T. Frankel. The Geometry of Physics: An Introduction, Second Edition. Cambridge University Press, Cambridge, UK, 1997.
R.A. Nicolaides. Direct discretization of planar div-curl problems. SIAM J. Numer. Anal. 29(1):32–56, 1992.
R.A. Nicolaides and D.Q. Wang. Convergence analysis of a covolume scheme for Maxwell’s equations in three dimensions. Mathematics of Computation 67(223):947–963, 1998.
R.A. Nicolaides and X. Wu. Covolume Solutions of three-dimensional div-curl equations. SIAM J. Numer. Anal. 34(6): 2195–2203, 1997.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer Science+Business Media, LLC
About this paper
Cite this paper
Nicolaides, R.A., Trapp, K.A. (2006). Covolume Discretization of Differential Forms. In: Arnold, D.N., Bochev, P.B., Lehoucq, R.B., Nicolaides, R.A., Shashkov, M. (eds) Compatible Spatial Discretizations. The IMA Volumes in Mathematics and its Applications, vol 142. Springer, New York, NY. https://doi.org/10.1007/0-387-38034-5_8
Download citation
DOI: https://doi.org/10.1007/0-387-38034-5_8
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-30916-3
Online ISBN: 978-0-387-38034-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)