Complex conjugate gradient methods

Linear systems with complex coefficients arise from various physical problems. Examples are the Helmholtz equation and Maxwell equations approximated by finite difference or finite element methods, that lead to large sparse linear systems. When the continuous problem is reduced to integral equations, after discretization, one obtains a dense linear system. The resulting matrices are generally non-Hermitian but, most of the time, symmetric and consequently the classical conjugate gradient method cannot be directly applied. Usually, these linear systems have to be solved with a large number of unknowns because, for instance, in electromagnetic scattering problems the mesh size must be related to the wave length of the incoming wave. The higher the frequency of the incoming wave, the smaller the mesh size must be. When one wants to solve 3D-problems, it is no longer practical to use direct method solvers, because of the huge memory they need. So iterative methods are attractive for this kind of problems, even though their convergence cannot be always guaranteed with theoretical results. In this paper we derive several methods from a unified framework and we numerically compare these algorithms on some test problems.