Abstract
The relation between physics, its description in terms of partial differential equations and geometry is explored in this paper. Geometry determines the correct weak formulation in finite element methods and also dictates which basis functions should be employed to obtain discrete well-posedness.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
D. Arnold, R. Falk & R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numerica,15, pp. 1–155, 2006.
D. Arnold, R. Falk & R. Winther, Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Amer. Math. Soc., 47, pp. 281–354, 2010.
D. Arnold & G. Awanou, Finite element differential forms on cubical meshes, Arxiv preprint math/1204.2595, to appear in: Mathematics of Computation, 2013. http://arxiv.org/abs/1204.2595
P. Bochev & J. Hyman, Principles of mimetic discretizations of differential operators, IMA Volumes In Mathematics and its Applications, Springer, 142, pp. 89–114, 2006.
P. Bochev, A discourse on variational and geometric aspects of stability of discretizations, in VKI Lecture Series: 33rd computational fluid dynamics course - novel methods for solving convection dominated systems March 24–28, 2003.
A.Bossavit, Discretization of electromagnetic problems in Handbook of Numerical Analysis, Vol. 13, Elsevier, pp. 105–197, 2005.
A. Bossavit & F. Rapetti, Whitney elements, from manifolds to fields, proceedings ICOSAHOM 2012, 2013.
F. Brezzi & A. Buffa, Innovative mimetic discretizations for electromagnetic problems, Journal of Computational and Applied Mathematics, 234, pp. 1980–1987, 2010.
F. Brezzi, A. Buffa & K. Lipnikov, Mimetic finite differences for elliptic problems, Mathematical Modelling and Numerical Analysis, 43, pp. 277–296, 2009.
F. Brezzi & M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Verlag, 1991.
A. Buffa, G. Sangalli & R. Vázquez, Isogeometric analysis in electromagnetics: B-splines approximation, Computer Methods in Applied Mechanics and Engineering 199 (17–20), pp. 1143–1152, 2010.
M. Desbrun, A. Hirani, M. Leok, J. Marsden, Discrete exterior calculus, Arxiv preprint math/0508341, 2005.
H. Flanders, Differential forms with applications to the physical sciences, Dover Publications, New York, 1989.
Th. Frankel, The Geometry of Physics. An Introduction. 2nd edition, Cambridge University Press, 2011.
J.A. Evans & T.J.R. Hughes, Isogeometric divergence-conforming B-splines for the unsteady Navier-Stokes equations, Journal of Computational Physics 241, pp. 141–167, 2013.
J.A. Evans & T.J.R. Hughes, Isogeometric divergence-conforming B-splines for the Darcy-Stokes-Brinkman equations, Mathematical Models and Methods in Applied Sciences 23 (4), pp. 671–741, 2013.
E.S. Gawlik, P. Mullen, D. Pavlov, J.E. Marsden & M. Desbrun, Geometric, variational discretization of continuum theories, Physica D: Nonlinear Phenomena 240 (21), pp. 1724–1760, 2011.
M.I. Gerritsma, Edge functions for spectral element methods, Spectral and High Order Methods for Partial differential equations, Eds J.S. Hesthaven & E.M. Rønquist, Lecture Notes in Computational Science and Engineering, 76, pp. 199–207, 2011.
M.I. Gerritsma, An Introduction to a Compatible Spectral Discretization Method, Mechanics of Advanced Materials and Structures, 19 pp. 48–67, 2012.
J. Harrison, Geometric Hodge star operator with applications to the theorems of Gauss and Green, Math. Proc. Camb Phil. Soc., 140, pp. 135–155, 2006.
R.R. Hiemstra, R.H.M. Huijsmans & M.I. Gerritsma, High order gradient, curl and divergence conforming spaces, with an application to compatible IsoGeometric Analysis, submitted to JCP, 2012.
R.R. Hiemstra, & M.I. Gerritsma, High order methods with exact conservation properties, proceedings ICOSAHOM 2012
R. Hiptmair, Discrete hodge operators, Numerische Mathematik, 90, pp. 265–289, 2001.
R. Hiptmair, Higher order Whitney forms, Geometric Methods for Computational Electromagnetics, 42, pp. 271–299, 2001.
A.N. Hirani, Discrete Exterior Calculus, PhD thesis, California Institute of Technology, 2003. http://thesis.library.caltech.edu/1885/3/thesis_hirani.pdf
A.N. Hirani, K. Kalyanaraman & E.B.van der Zee, Delaunay Hodge star, CAD Computer Aided Design 45 (2), pp. 540–544, 2013.
J.M. Hyman & J.C. Scovel, Deriving mimetic difference approximations to differential operators using algebraic topology, Math. Comp. 52, No.186, pp. 471–494, 1989.
J.M. Hyman, M. Shashkov & S. Steinberg, The numerical solution of diffusion problems in strongly heterogeous non-isotropic materials, Journal of Computational Physics, 132, 1, pp. 30–148, 1997.
J.J. Kreeft & M.I. Gerritsma, Mixed Mimetic Spectral Element Method for Stokes Flow: A Pointwise Divergence-Free Solution, Journal of Computational Physics, 240, pp. 284–309, 2013..
J.J. Kreeft & M.I. Gerritsma, A priori error estimates for compatible spectral discretization of the Stokes problem for all admissible boundary conditions, arXiv preprint arXiv:1206.2812, 2012. http://arxiv.org/abs/1206.2812
J.J. Kreeft & M.I. Gerritsma, Higher-order compatible discretization on hexahedrals, proceedings ICOSAHOM 2012. http://arxiv.org/abs/1304.7018
J.J. Kreeft, A. Palha & M.I. Gerritsma, Mimetic framework on curvilinear quadrilaterals of arbitrary order, Arxiv preprint arXiv:1111.4304, pp. 1–69, 2011. http://arxiv.org/abs/1111.4304
C. Mattiussi, The finite volume, finite element, and finite difference methods as numerical methods for physical field problems, Advances in Imaging and electron physics, 133, pp. 1–147, 2000.
A. Palha, P. Rebelo, R.R. Hiemstra, J.J. Kreeft & M.I. Gerritsma, Physics-compatible discretization techniques on single and dual grids, with application to the Poisson equation of volume forms, submitted to JCP, http://arxiv.org/abs/1304.6908, 2012.
A. Palha, P. Rebelo & M.I. Gerritsma, Mimetic Spectral Element advection, proceedings ICOSAHOM 2012. http://arxiv.org/abs/1304.6926
D. Pavlov, P. Mullen, Y. Tong, E. Kanso, J.E. Marsden & M. Desbrun, Structure-preserving discretization of incompressible fluids, Physica D: Nonlinear Phenomena 240 (6), pp. 443–458, 2011.
J. Blair Perot, Conservation properties of unstructured staggered mesh schemes, Journal of Computational Physics, 159, pp. 58–89, 2000.
J. Blair Perot, Discrete Conservation Properties of Unstructured Mesh Schemes, Annual Review of Fluid Mechanics, Annual Reviews, 43, pp. 299–318, 2011.
F. Rapetti & A. Bossavit, Whitney forms of higher degree, SIAM J. Numer. Anal., 47, pp. 2369–2386, 2009.
F. Rapetti & A. Bossavit, Geometrical localisation of the degrees of freedom for Whitney elements of higher order, IET Science, Measurement and Technology 1 (1), pp. 63–66, 2007.
P. Rebelo, A. Palha & M.I. Gerritsma, Mixed Mimetic Spectral Element method applied to Darcy’s problem, proceedings ICOSAHOM 2012 http://arxiv.org/abs/1304.7147
N. Robidoux & S. Steinberg, A discrete vector calculus in tensor grids, Computational Methods in Applied Mathematics 11 (1), pp. 23–66, 2011.
Tonti, E, On the formal structure of physical theories, preprint of the Italian National Research Council, 1975. http://www.dic.univ.trieste.it/perspage/tonti/DEPOSITO/CNR.pdf
D. Toshniwal, R.H.M. Huijsmans & M.I. Gerritsma, A Geometric approach towards momentum conservation, proceedings ICOSAHOM 2012. http://arxiv.org/abs/1304.6991
Acknowledgements
The authors want to thank the reviewers for their critical remarks and useful suggestions.
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
Gerritsma, M., Hiemstra, R., Kreeft, J., Palha, A., Rebelo, P., Toshniwal, D. (2014). The Geometric Basis of Numerical Methods. In: Azaïez, M., El Fekih, H., Hesthaven, J. (eds) Spectral and High Order Methods for Partial Differential Equations - ICOSAHOM 2012. Lecture Notes in Computational Science and Engineering, vol 95. Springer, Cham. https://doi.org/10.1007/978-3-319-01601-6_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-01601-6_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-01600-9
Online ISBN: 978-3-319-01601-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)