High-voltage insulators are stressed by the applied electrical field as well as by other enviromental factors. As a result of this stress, the surface of the insulating material gets aged and the dielectric material looses its hydrophobic and insulating characteristics. The contamination of the object with water droplets accelerates the aging process. Micro-surface discharges between droplets on a wet surface are an important source of aging. In order to improve the understanding of ageing phenomena it is advisable to simulate single droplets on an insulating surface. First steps are the calculations of the electric field strength and the force density surrounding the droplets. For the discretization we use the Finite Integration Theory (FIT) which was especially developed to solve Maxwell's equations (see [1]). We solve our insulator problem as an electro-quasistatic problem (see [2]). FIT leads to a complex symmetric system of linear equations, the complex Poisson's equation, with a sparse matrix with band width seven. Large differences in the order of magnitude  of the relative permittivity and of the conductivity lead to a large condition number of the linear system. We are going to investigate  several iterative methods to solve the discretized complex symmetric potential problem. The Krylov-subspace methods are an important class of iterative methods for the solution of the insulator problem. Numerical studies showed that it is recommendable to use the Jacobi-preconditioned Krylov-subspace methods (see [3]).  An iterative method by A. Bunse-Gerstner and R. Stoever  (see [4]) which exploits the complex symmetric structure of the system-matrix is also investigated. Another method by Axelsson solves the problem as a real one (see [5]). We compare these algorithms and calculate the electric field strength and the force density for an insulator example.

*research supported by Deutsche Forschungsgemeinschaft (DFG)

[1] T. Weiland, "A discretization method for the solution of Maxwell's equation for six-component fields", Electron. Commun. AEÜ, 31(3), pp. 116 - 120, 1977.
[2] U. v. Rienen and M. Clemens and T. Weiland, "Computation of Low-frequency Electromagnetic Fields", ZAMM, Applied Mathematics and Mechanics, 76, pp. 567 - 568, 1996. 
[3] M. Clemens and R. Schumann and U. v. Rienen and T. Weiland, "Modern Krylov Subspace Methods in Electromagnetic Field Computation Using the Finite Integration Theory", Applied Computational Electromagnetics Society Journal, Vol.11, No.1, pp. 70 - 84, 1996. 
[4] A. Bunse-Gerstner and Ronald Stoever, "On a Conjugate Gradient-Type Method for Solving Complex Symmetric Linear Systems", Linear Algebra and its Applications, 1992. 
[5] O. Axelsson, "Real Valued Iterative Methods for Solving Complex Linear Systems", University Nijmegen, January 1999.