Inner products in covolume and mimetic methods

Kathryn A. Trapp

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

  • Volume: 42, Issue: 6, page 941-959
  • ISSN: 0764-583X

Abstract

top
A class of compatible spatial discretizations for solving partial differential equations is presented. A discrete exact sequence framework is developed to classify these methods which include the mimetic and the covolume methods as well as certain low-order finite element methods. This construction ensures discrete analogs of the differential operators that satisfy the identities and theorems of vector calculus, in particular a Helmholtz decomposition theorem for the discrete function spaces. This paper demonstrates that these methods differ only in their choice of discrete inner product. Finally, certain uniqueness results for the covolume inner product are shown.

How to cite

top

Trapp, Kathryn A.. "Inner products in covolume and mimetic methods." ESAIM: Mathematical Modelling and Numerical Analysis 42.6 (2008): 941-959. <http://eudml.org/doc/250375>.

@article{Trapp2008,
abstract = { A class of compatible spatial discretizations for solving partial differential equations is presented. A discrete exact sequence framework is developed to classify these methods which include the mimetic and the covolume methods as well as certain low-order finite element methods. This construction ensures discrete analogs of the differential operators that satisfy the identities and theorems of vector calculus, in particular a Helmholtz decomposition theorem for the discrete function spaces. This paper demonstrates that these methods differ only in their choice of discrete inner product. Finally, certain uniqueness results for the covolume inner product are shown. },
author = {Trapp, Kathryn A.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Compatible discretization; discrete Helmholtz orthogonality; discrete exact sequence; mimetic method; covolume method.; compatible discretization; covolume method; finite element methods},
language = {eng},
month = {7},
number = {6},
pages = {941-959},
publisher = {EDP Sciences},
title = {Inner products in covolume and mimetic methods},
url = {http://eudml.org/doc/250375},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Trapp, Kathryn A.
TI - Inner products in covolume and mimetic methods
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2008/7//
PB - EDP Sciences
VL - 42
IS - 6
SP - 941
EP - 959
AB - A class of compatible spatial discretizations for solving partial differential equations is presented. A discrete exact sequence framework is developed to classify these methods which include the mimetic and the covolume methods as well as certain low-order finite element methods. This construction ensures discrete analogs of the differential operators that satisfy the identities and theorems of vector calculus, in particular a Helmholtz decomposition theorem for the discrete function spaces. This paper demonstrates that these methods differ only in their choice of discrete inner product. Finally, certain uniqueness results for the covolume inner product are shown.
LA - eng
KW - Compatible discretization; discrete Helmholtz orthogonality; discrete exact sequence; mimetic method; covolume method.; compatible discretization; covolume method; finite element methods
UR - http://eudml.org/doc/250375
ER -

References

top
  1. D.N. Arnold, Differential complexes and numerical stability, in Proceedings of the International Congress of Mathematicians, Vol. I, Higher Ed. Press, Beijing (2002) 137–157.  
  2. M. Berndt, K. Lipnikov, D. Moulton and M. Shashkov, Convergence of mimetic finite difference discretizations of the diffusion equation. East-West J. Numer. Math9 (2001) 253–316.  Zbl1014.65114
  3. P. Bochev and J.M. Hyman, Principles of mimetic discretizations of differential operators, in Compatible Spatial Discretizations, D. Arnold, P. Bochev, R. Lehoucq, R. Nicolaides and M. Shashkov Eds., IMA Volumes in Mathematics and its Applications142, Springer, New York (2006).  Zbl1110.65103
  4. A. Bossavit, Generating whitney forms of polynomial degree one and higher. IEEE Trans. Magn.38 (2002) 341–344.  
  5. R. Hiptmair, Canonical construction of finite elements. Math. Comp.68 (1999) 1325–1346.  Zbl0938.65132
  6. A. Hirani, Discrete Exterior Calculus. Ph.D. thesis, California Institute of Technology, USA (2003).  
  7. J.M. Hyman and M. Shashkov, The adjoint operators for the natural discretizations for the divergence, gradient, and curl on logically rectangular grids. IMACS J. Appl. Num. Math.25 (1997) 1–30.  Zbl1005.65024
  8. J.M. Hyman and M. Shashkov, Natural discretizations for the divergence, gradient, and curl on logically rectangular grids. Comput. Math. Appl.33 (1997) 81–104.  Zbl0868.65006
  9. J.M. Hyman and M. Shashkov, Mimetic discretizations for Maxwell's equations. J. Comp. Phys.151 (1999) 881–909.  Zbl0956.78015
  10. J.M. Hyman and M. Shashkov, The orthogonal decomposition theorems for mimetic finite difference methods. SIAM J. Numer. Anal.36 (1999) 788–818.  Zbl0972.65077
  11. J.C. Nedelec, Mixed finite elements in 3 . Numer. Math.35 (1980) 315–341.  Zbl0419.65069
  12. J.C. Nedelec, A new family of mixed finite elements in 3 . Numer. Math.50 (1986) 57–81.  Zbl0625.65107
  13. R.A. Nicolaides, Direct discretization of planar div-curl problems. SIAM J. Numer. Anal.29 (1992) 32–56.  Zbl0745.65063
  14. R. Nicolaides and K. Trapp, Covolume discretizations of differential forms, in Compatible Spatial Discretizations, D. Arnold, P. Bochev, R. Lehoucq, R. Nicolaides and M. Shashkov Eds., IMA Volumes in Mathematics and its Applications142, Springer, New York (2006).  Zbl1110.65024
  15. R.A. Nicolaides and D.Q. Wang, Convergence analysis of a covolume scheme for Maxwell's equations in three dimensions. Math. Comp.67 (1998) 947–963.  Zbl0907.65116
  16. R.A. Nicolaides and X. Wu, Covolume solutions of three-dimensional div-curl equations. SIAM J. Numer. Anal.34 (1997) 2195–2203.  Zbl0889.35006
  17. P.A. Raviart and J.M. Thomas, A mixed finite elemnt method for second order elliptic problems, in Springer Lecture Notes in Mathematics606, Springer-Verlag (1977) 292–315.  
  18. K. Trapp, A Class of Compatible Discretizations with Applications to Div-Curl Systems. Ph.D. thesis, Carnegie Mellon University, USA (2004).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.