A Multilevel Jacobi–Davidson Method for Polynomial PDE Eigenvalue Problems Arising in Plasma Physics

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.