Convergence of preconditioned conjugate gradient method applied to driven microwave problems

Driven microwave problems can be solved with the finite-element method formulated in terms of the electric field, as well as the vector and scalar potentials. It is known that the latter gives faster convergence of the preconditioned conjugate gradient method than the former. This can be understood from the following facts: namely, the preconditioned finite-element matrix of the former method can contain small negative eigenvalues which make the matrix condition worse. On the other hand, in the latter, such eigenvalues are shown to be composed of zeros and normalized ones.