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

Published online by Cambridge University Press:  21 February 2011

Snorre H. Christiansen  and
Claire Scheid
Snorre H. Christiansen
Affiliation:
CMA c/o Dept. Math, University of Oslo, P.O. Box 1053 Blindern, 0316 Oslo, Norway. snorrec@math.uio.no
Claire Scheid
Affiliation:
CMA c/o Dept. Math, University of Oslo, P.O. Box 1053 Blindern, 0316 Oslo, Norway. snorrec@math.uio.no Laboratoire Jean Alexandre Dieudonné, Université de Nice Sophia Antipolis, 06108 Nice Cedex 02, France. Claire.Scheid@unice.fr
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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.

Keywords

WavesMaxwell Klein Gordonnon-linear constraintsfinite elementsconvergence analysis
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 4 , July 2011 , pp. 739 - 760
Copyright
© EDP Sciences, SMAI, 2011

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References

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