Convergence of a constrained finite element discretization of the Maxwell Klein Gordon equation*

Snorre H. Christiansen; Claire Scheid

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

  • Volume: 45, Issue: 4, page 739-760
  • ISSN: 0764-583X

Abstract

top
As an example of a simple constrained geometric non-linear wave equation, we study a numerical approximation of the Maxwell Klein Gordon equation. We consider an existing constraint preserving semi-discrete scheme based on finite elements and prove its convergence in space dimension 2 for initial data of finite energy.

How to cite

top

Christiansen, Snorre H., and Scheid, Claire. "Convergence of a constrained finite element discretization of the Maxwell Klein Gordon equation*." ESAIM: Mathematical Modelling and Numerical Analysis 45.4 (2011): 739-760. <http://eudml.org/doc/197412>.

@article{Christiansen2011,
abstract = { As an example of a simple constrained geometric non-linear wave equation, we study a numerical approximation of the Maxwell Klein Gordon equation. We consider an existing constraint preserving semi-discrete scheme based on finite elements and prove its convergence in space dimension 2 for initial data of finite energy. },
author = {Christiansen, Snorre H., Scheid, Claire},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Waves; Maxwell Klein Gordon; non-linear constraints; finite elements; convergence analysis; geometric wave equation, Maxwell-Klein-Gordon equation},
language = {eng},
month = {2},
number = {4},
pages = {739-760},
publisher = {EDP Sciences},
title = {Convergence of a constrained finite element discretization of the Maxwell Klein Gordon equation*},
url = {http://eudml.org/doc/197412},
volume = {45},
year = {2011},
}

TY - JOUR
AU - Christiansen, Snorre H.
AU - Scheid, Claire
TI - Convergence of a constrained finite element discretization of the Maxwell Klein Gordon equation*
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2011/2//
PB - EDP Sciences
VL - 45
IS - 4
SP - 739
EP - 760
AB - As an example of a simple constrained geometric non-linear wave equation, we study a numerical approximation of the Maxwell Klein Gordon equation. We consider an existing constraint preserving semi-discrete scheme based on finite elements and prove its convergence in space dimension 2 for initial data of finite energy.
LA - eng
KW - Waves; Maxwell Klein Gordon; non-linear constraints; finite elements; convergence analysis; geometric wave equation, Maxwell-Klein-Gordon equation
UR - http://eudml.org/doc/197412
ER -

References

top
  1. R.A Adams and J.J.F. Fournier, Sobolev Spaces – Pure and Applied Mathematics Series. Second edition, Elsevier (2003).  
  2. D.N. Arnold, R.S. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications. Acta Numer.15 (2006) 1–155.  
  3. S. Bartels, X. Fenga and A. Prohl, Finite element approximations of wave maps into spheres. SIAM J. Numer. Anal.46 (2007) 61–87.  
  4. A. Bossavit, Mixed finite elements and the complex of Whitney forms, in The mathematics of finite elements and applicationsVI, J. Whiteman Ed., Academic Press, London (1988) 137–144.  
  5. J.H. Bramble, J.E. Pasciak and O. Steinbach, On the stability of the L2 projection in H1(Ω). Math. Comput.71 (2001) 147–156.  
  6. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Second edition, Springer (2002).  
  7. S.H. Christiansen, Résolution des équations intégrales pour la diffraction d'ondes accoustiques et électromagnétiques. Ph.D. thesis, École polytechnique, France (2002).  
  8. S.H. Christiansen, Discrete Fredholm properties and convergence estimates for the Electric Field Integral Equation. Math. Comput.73 (2004) 143–167.  
  9. S.H. Christiansen, Constraint preserving schemes for gauge invariant wave equations. SIAM J. Sci. Comput.31 (2009) 1448–1469.  
  10. S.H. Christiansen and R. Winther, On constraint preservation in numerical simulations of Yang-Mills equations. SIAM J. Sci. Comput.28 (2006) 75–101.  
  11. S.H. Christiansen and R. Winther, Smoothed projections in finite element exterior calculus. Math. Comput.77 (2007) 813–829.  
  12. P.G. Ciarlet, Basic error estimates for elliptic problems, in Handbook of numerical analysisII, P.G. Ciarlet and J.-L. Lions Eds., North Holland (1991) 17–351.  
  13. M. Crouzeix and V. Thomée, The stability in Lp and W1p of the L2-projection onto finite element function spaces. Math. Comput.48 (1987) 521–532.  
  14. J. Douglas Jr., T. Dupont and L. Wahlbin, The stability in Lq of the L2-projection into finite element function spaces. Numer. Math.23 (1975) 193–197.  
  15. F. Dubois, Discrete vector potential representation of a divergence free vector field in three-dimensional domains: Numerical analysis of a model problem. SIAM J. Numer. Anal.27 (1990) 1103–1141.  
  16. J. Ginibre and G. Velo, The Cauchy problem for coupled Yang-Mills and scalar fields in the temporal gauge. Commun. Math. Phys.82 (1981) 1–28.  
  17. V. Girault and P.-A. Raviart, Finite Element approximation of the Navier-Stokes equations. Springer-Verlag, Berlin (1986).  
  18. F. Kikuchi, On a discrete compactness property for the Nédélec finite elements. J. Fac. Sci. Univ. Tokyo, Sect. 1A Math.36 (1989) 479–490.  
  19. S. Klainerman, Mathematical challenges of general relativity. Rend. Mat. Appl.27 (2007) 105–122.  
  20. S. Klainerman and M. Machedon, On the Maxwell-Klein-Gordon equation with finite energy. Duke Math. J.74 (1994) 19–44.  
  21. S. Klainerman and M. Machedon, Finite energy solutions of the Yang-Mills equations in R3+1. Ann. Math.142 (1995) 39–119.  
  22. E.H. Lieb and M. Loss, Analysis Graduate Studies in Mathematics14. Second edition, AMS (2001).  
  23. J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications1. Dunod, Paris (1968).  
  24. N. Masmoudi and K. Nakanishi, Uniqueness of Finite Energy solutions for Maxwell-Dirac and Maxwell-Klein-Gordon equations. Commun. Math. Phys.243 (2003) 123–136.  
  25. P. Monk, Finite Element Methods for Maxwell's Equations. Oxford Science Publication (2003).  
  26. J. Schöberl, A posteriori error estimates for Maxwell equations. Math. Comput.77 (2008) 633–649.  
  27. S. Selberg and A. Tesfahun, Finite-energy global well-posedness of the Maxwell-Klein-Gordon system in Lorenz gauge. Commun. Partial Differ. Equ.35 (2010) 1029–1057.  
  28. J. Shatah and M. Struwe, Geometric wave equations, Courant Lecture Notes in Mathematics2. New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence (1998).  
  29. C.G. Simader, On Dirichlet Boundary Value Problem. Springer-Verlag (1972).  
  30. J. Simon, Compact sets in the space Lp(0,T;B). Ann. Mat. Pura. Appl.146 (1987) 65–96.  
  31. T. Tao, Local well-posedness of the Yang-Mills equation in the temporal gauge below the energy norm. J. Differ. Equ.189 (2003) 366–382.  

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.