Elastic guided waves are of interest for inspecting structures due to their ability to propagate over long distances. However, when the guiding structure is embedded into a solid matrix, usually considered as unbounded, waveguides are open and waves can be trapped or leaky. In the latter case, the leakage of energy into the surrounding medium yields attenuation along the axis of the waveguide, which can strongly limit the application of guided wave techniques. Analytical tools have been developed for studying open waveguides but they are limited to simple geometry (plates, cylinders). With numerical methods, one of the difficulty is that leaky modes attenuate along the axis (complex wavenumber) and exponentially grow along the transverse direction. A simple procedure used with existing codes consists in using absorbing layers of artificially growing viscoelasticity, but large layers are often required. The goal of this work is to propose a numerical approach for computing modes in open elastic waveguides combining the so-called semi-analytical finite element method and a perfectly matched layer technique. Two-dimensional problems are considered. Numerical solutions are compared to analytical results. The efficiency of both perfectly matched and absorbing layer techniques is evaluated.