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.
Mots clés : waves, Maxwell Klein Gordon, non-linear constraints, finite elements, convergence analysis
@article{M2AN_2011__45_4_739_0,
author = {Christiansen, Snorre H. and Scheid, Claire},
title = {Convergence of a constrained finite element discretization of the {Maxwell} {Klein} {Gordon} equation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {739--760},
publisher = {EDP-Sciences},
volume = {45},
number = {4},
year = {2011},
doi = {10.1051/m2an/2010100},
mrnumber = {2804657},
zbl = {1282.78036},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2010100/}
}
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 PY - 2011 SP - 739 EP - 760 VL - 45 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2010100/ DO - 10.1051/m2an/2010100 LA - en ID - M2AN_2011__45_4_739_0 ER -
%0 Journal Article %A Christiansen, Snorre H. %A Scheid, Claire %T Convergence of a constrained finite element discretization of the Maxwell Klein Gordon equation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2011 %P 739-760 %V 45 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2010100/ %R 10.1051/m2an/2010100 %G en %F M2AN_2011__45_4_739_0
Christiansen, Snorre H.; Scheid, Claire. Convergence of a constrained finite element discretization of the Maxwell Klein Gordon equation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 4, pp. 739-760. doi : 10.1051/m2an/2010100. http://www.numdam.org/articles/10.1051/m2an/2010100/
[1] and , Sobolev Spaces - Pure and Applied Mathematics Series. Second edition, Elsevier (2003). | MR | Zbl
[2] , and , Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15 (2006) 1-155. | MR | Zbl
[3] , and , Finite element approximations of wave maps into spheres. SIAM J. Numer. Anal. 46 (2007) 61-87. | MR | Zbl
[4] , 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. | MR | Zbl
[5] , and , On the stability of the L2 projection in H1(Ω). Math. Comput. 71 (2001) 147-156. | MR | Zbl
[6] and , The Mathematical Theory of Finite Element Methods. Second edition, Springer (2002). | MR | Zbl
[7] , Résolution des équations intégrales pour la diffraction d'ondes accoustiques et électromagnétiques. Ph.D. thesis, École polytechnique, France (2002).
[8] , Discrete Fredholm properties and convergence estimates for the Electric Field Integral Equation. Math. Comput. 73 (2004) 143-167. | MR | Zbl
[9] , Constraint preserving schemes for gauge invariant wave equations. SIAM J. Sci. Comput. 31 (2009) 1448-1469. | MR | Zbl
[10] and , On constraint preservation in numerical simulations of Yang-Mills equations. SIAM J. Sci. Comput. 28 (2006) 75-101. | MR | Zbl
[11] and , Smoothed projections in finite element exterior calculus. Math. Comput. 77 (2007) 813-829. | MR | Zbl
[12] , 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. | MR | Zbl
[13] and , The stability in Lp and W1p of the L2-projection onto finite element function spaces. Math. Comput. 48 (1987) 521-532. | MR | Zbl
[14] , and , The stability in Lq of the L2-projection into finite element function spaces. Numer. Math. 23 (1975) 193-197. | MR | Zbl
[15] , 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. | MR | Zbl
[16] and , The Cauchy problem for coupled Yang-Mills and scalar fields in the temporal gauge. Commun. Math. Phys. 82 (1981) 1-28. | MR | Zbl
[17] and , Finite Element approximation of the Navier-Stokes equations. Springer-Verlag, Berlin (1986). | MR | Zbl
[18] , On a discrete compactness property for the Nédélec finite elements. J. Fac. Sci. Univ. Tokyo, Sect. 1A Math. 36 (1989) 479-490. | MR | Zbl
[19] , Mathematical challenges of general relativity. Rend. Mat. Appl. 27 (2007) 105-122. | MR | Zbl
[20] and , On the Maxwell-Klein-Gordon equation with finite energy. Duke Math. J. 74 (1994) 19-44. | MR | Zbl
[21] and , Finite energy solutions of the Yang-Mills equations in R3+1. Ann. Math. 142 (1995) 39-119. | MR | Zbl
[22] and , Analysis Graduate Studies in Mathematics 14. Second edition, AMS (2001). | MR | Zbl
[23] and , Problèmes aux limites non homogènes et applications 1. Dunod, Paris (1968). | MR | Zbl
[24] and , Uniqueness of Finite Energy solutions for Maxwell-Dirac and Maxwell-Klein-Gordon equations. Commun. Math. Phys. 243 (2003) 123-136. | MR | Zbl
[25] , Finite Element Methods for Maxwell's Equations. Oxford Science Publication (2003). | Zbl
[26] , A posteriori error estimates for Maxwell equations. Math. Comput. 77 (2008) 633-649. | Zbl
[27] and , Finite-energy global well-posedness of the Maxwell-Klein-Gordon system in Lorenz gauge. Commun. Partial Differ. Equ. 35 (2010) 1029-1057. | MR | Zbl
[28] and , Geometric wave equations, Courant Lecture Notes in Mathematics 2. New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence (1998). | MR | Zbl
[29] , On Dirichlet Boundary Value Problem. Springer-Verlag (1972). | Zbl
[30] , Compact sets in the space Lp(0,T;B). Ann. Mat. Pura. Appl. 146 (1987) 65-96. | MR | Zbl
[31] , Local well-posedness of the Yang-Mills equation in the temporal gauge below the energy norm. J. Differ. Equ. 189 (2003) 366-382. | MR | Zbl
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