Int. J. Numer. Anal. Mod., 8 (2011), pp. 641-666. An extended finite element method for 2D edge elements F. Lefevre 1, S. Lohrengel 1, S. Nicaise 21 Laboratoire de Mathematiques, Universite de Reims Champagne-Ardenne, Moulin de la Housse - B.P. 1039, 51687 Reims Cedex 2, France 2 LAMAV, Universite de Valenciennes et du Hainaut Cambresis, Le Mont Houy, 59313 Valenciennes Cedex 9, France Received by the editors January 4, 2011 and, in revised form, May 31, 2011. Abstract A new eXtended Finite Element Method based on two-dimensional edge elements is presented and applied to solve the time-harmonic Maxwell equations in domains with cracks. Error analysis is performed and shows the method to be convergent with an order of at least $O(h^{1/2-\eta})$. The implementation of the method is discussed and numerical tests illustrate its performance. AMS subject classifications: 65N30, 65N15, 78M10 Key words: Maxwell's equations, domains with cracks, XFEM, singularities of solutions.