European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS 2004 P. Neittaanm¨ aki, T. Rossi, S. Korotov, E. O˜ nate, J. P´ eriaux, and D. Kn¨ orzer (eds.) Jyv¨ askyl¨ a, 24–28 July 2004 INSTABILITY OF INTERACTING FAULTS: NONLINEAR EIGENVALUE ANALYSIS Ioan R. Ionescu ? , and Sylvie Wolf † ? Laboratoire de Math´ ematiques, Universit´ e de Savoie, Campus Scientifique, 73376 Le Bourget-du-Lac Cedex, France e-mail: ionescu@univ-savoie.fr , web page: http://www.lama.univ-savoie.fr/ IONESCU † Laboratoire de G´ eophysique Interne, Universit´ e Joseph Fourier, BP 53X, 38041 Grenoble Cedex, France e-mail : Sylvie.Wolf@obs.ujf-grenoble.fr , web page: http://www-lgit.obs.ujf-grenoble.fr/users/swolf/swolf.html Key words: domains with cracks, slip-dependent friction, wave equation, earthquake initiation, nonlinear eigenvalue problem, Rayleigh quotient, unilateral conditions, mixed finite element method Abstract. We analyze the evolution of a system of finite faults by considering the nonlin- ear eigenvalue problems associated to static and dynamic solutions on unbounded domains. We restrict our investigation to the first eigenvalue (Rayleigh quotient). We point out its physical significance through a stability analysis and we give an efficient numerical algo- rithm able to compute it together with the corresponding eigenfunction. We consider the anti-plane shearing on a system of finite faults under a slip-dependent friction in a linear elastic domain, not necessarily bounded. For the dynamic problem, we discuss the existence of solutions with an exponential growth, to deduce a (dynamic) nonlinear eigenvalue problem. We prove the existence of a first dynamic eigenvalue and we analyze its behavior with respect to the friction parameter. We deduce a mixed finite element discretization of the nonlinear spectral problem and we give a numerical algorithm to approach the first eigenvalue/eigenfunction. Finally we give some numerical results which include convergence tests, on a two-faults system, and a comparison between the nonlinear spectral results and the time evolution results. 1