  We develop a self-consistent finite element method to study spontaneous
emission at nanoscale proximity of plasmonic waveguides. In the model, it is
assumed that only one guided mode is dominatingly excited by the quantum
emitter. With such one dominating mode assumption, the cross section of the
plasmonic waveguide can be arbitrary. We apply our numerical method to
calculate the coupling of a quantum emitter to a cylindrical nanowire and a
rectangular waveguide, and compare the cylindrical nanowire to previous work
valid in quasistatic approximation. The fraction of the energy coupled to the
plasmonic mode can be calculated exactly, which can be used to determine the
single optical plasmon generation efficiency for a quantum emitter. For a gold
nanowire we observe agreement with the quasistatic approximation for radii
below 20 nm, but for larger radii the total decay rate is up to 10 times
larger. For the rectangular waveguide we estimate an optimized value for the
spontaneous emission factor \beta of up to 80%.
