  We present an elegant method of determining the eigensolutions of the
induction and the dynamo equation in a fluid embedded in a vacuum. The magnetic
field is expanded in a complete set of functions. The new method is based on
the biorthogonality of the adjoint electric current and the vector potential
with an inner product defined by a volume integral over the fluid domain. The
advantage of this method is that the velocity and the dynamo coefficients of
the induction and the dynamo equation do not have to be differentiated and thus
even numerically determined tabulated values of the coefficients produce
reasonable results. We provide test calculations and compare with published
results obtained by the classical treatment based on the biorthogonality of the
magnetic field and its adjoint. We especially consider dynamos with mean-field
coefficients determined from direct numerical simulations of the geodynamo and
compare with initial value calculations and the full MHD simulations.
