Electro Discharge Machining (EDM) is a critical process for manufacturing high-precision components, particularly in aerospace, biomedical, and die/mold industries. Accurate numerical simulations of EDM finishing operations are essential for optimizing performance and ensuring surface quality. This study presents an adaptive numerical simulation framework utilizing an embedded approach for modeling EDM finishing processes. The proposed method integrates heat transfer and material removal to predict the thermal and material response during machining. A key innovation lies in the usage of an embedded mesh technique coupled with dynamic adaptive mesh refinement that allows to adjust computational resolution based on a temperature field based error estimator. This allows to significantly reduce computational cost. Validation is performed using experimental data from finishing operations on steel workpieces generated through additive manufacturing.