  The optimal dividend problem by De Finetti (1957) has been recently
generalized to the spectrally negative L\'evy model where the implementation of
optimal strategies draws upon the computation of scale functions and their
derivatives. This paper proposes a phase-type fitting approximation of the
optimal strategy. We consider spectrally negative L\'evy processes with
phase-type jumps as well as meromorphic L\'evy processes (Kuznetsov et al.,
2010a), and use their scale functions to approximate the scale function for a
general spectrally negative L\'evy process. We obtain analytically the
convergence results and illustrate numerically the effectiveness of the
approximation methods using examples with the spectrally negative L\'evy
process with i.i.d. Weibull-distributed jumps, the \beta-family and CGMY
process.
