# A Multiharmonic Solver for Nonlinear Parabolic Problems

## Dipl.-Ing. Michael Kolmbauer

**Nov. 3, 2009, 2:30 p.m. P 215**

This diploma thesis is concerned with the numerical solution of nonlinear, scalar potential parabolic partial differential equations in one dimension. Due to a time harmonic excitation, the resulting periodicity of the solution can be capitalised to switch from the time domain to the frequency domain by approximating the solution by a truncated fourier series and solving the resulting system of partial differential equations in the fourier coefficients. Solving this time independent system by finite element method yields a large-scale coupled system of nonlinear equations. This nonlinearity is dealt with by a linearisation in terms of Newton iteration. Hence the construction of an appropriate preconditioner for the arising Jacobi system is inevitable for the efficiency of the entire solver. In order to tackle this challenge a Cai-Xu preconditioner based on GMRES method is used.

In addition we present a spectral analysis for the Cai-Xu preconditioner applied to the linear problem in order to study the associated convergence rates for two special cases.

We perform numerical tests and report the results, both for the linear and nonlinear case.