SIAM J. SCI. COMPUT. c  2010 Society for Industrial and Applied Mathematics Vol. 32, No. 6, pp. 3151–3169 A MULTILEVEL JACOBI–DAVIDSON METHOD FOR POLYNOMIAL PDE EIGENVALUE PROBLEMS ARISING IN PLASMA PHYSICS ∗ MARLIS HOCHBRUCK † AND DOMINIK L ¨ OCHEL † Abstract. The simulation of drift instabilities in the plasma edge leads to cubic polynomial PDE eigenvalue problems with parameter dependent coefficients. The aim is to determine the wave number which leads to the maximum growth rate of the amplitude of the wave. This requires the solution of a large number of PDE eigenvalue problems. Since we are only interested in a smooth eigenfunction corresponding to the eigenvalue with largest imaginary part, the Jacobi–Davidson method can be applied. Unfortunately, a naive implementation of this method is much too expensive for the large number of problems that have to be solved. In this paper we will present a multilevel approach for the construction of an appropriate initial search space. We will also discuss the efficient solution of the correction equation, and we will show how optimal scaling helps to accelerate the convergence. Key words. polynomial eigenvalue problem, PDE eigenvalue problem, Jacobi–Davidson method, multilevel method, parameter dependent problem, optimal scaling, drift instabilities AMS subject classification. 65L15 DOI. 10.1137/090774604 1. Introduction. In this paper we consider polynomial eigenvalue problems of the form (1.1) p 0 (ω,θ)φ(θ)+ p 1 (ω,θ) ∂ ∂θ φ(θ)+ p 2 (ω,θ) ∂ 2 ∂θ 2 φ(θ)=0, where (1.2) p 0 (ω,θ)= d 0  j=0 a j (θ)ω j , p 1 (ω,θ)= d 1  j=0 b j (θ)ω j , p 2 (ω,θ)= d 2  j=0 c j (θ)ω j . Here, the coefficients a j ,b j ,c j are smooth, complex valued functions of θ, which might depend on certain other parameters, and d j denotes the degree of the polynomials p j . The application we are interested in is the study of instabilities in the plasma edge of a tokamak, which is a magnetic fusion device. One of the problems to make mag- netic fusion efficient is to reduce energy losses resulting from the unavoidable transport of plasma towards the wall. These losses arise due to microinstabilities in the plasma edge; see [15, 18, 19, 20, 36] for details. In [15, 33] a model for the simulation of the so-called anomalous transport is derived, which—after several simplifications—leads to a cubic eigenvalue problem of the form (1.1) with d 0 = 3 and d 1 = d 2 = 2. The eigenvalue of interest is the one with maximum imaginary part, since it contributes most to the losses. All functions a j ,b j ,c j are 2π-periodic in θ. The 2π-periodic eigen- function φ corresponds to the envelope of the electric potential perturbation for a certain wave number K , which is one of the parameters entering the coefficient func- tions a j ,b j , and c j . To be more precise, the physicists aim at finding the wave number ∗ Received by the editors October 22, 2009; accepted for publication (in revised form) June 21, 2010; published electronically October 21, 2010. This work has been supported by the Deutsche Forschungsgemeinschaft through the research training group GRK 1203. http://www.siam.org/journals/sisc/32-6/77460.html † Fakult¨ at f¨ ur Mathematik, Karlsruher Institut f¨ ur Technologie (KIT), Kaiserstraße 93, D–76133 Karlsruhe, Germany (marlis.hochbruck@kit.edu, dominik.loechel@kit.edu). 3151