AbstractCommon envelope evolution (CEE) occurs in some binary systems involving asymptotic giant branch (AGB) or red giant branch (RGB) stars, and understanding this process is crucial for understanding the origins of various transient phenomena. CEE has been shown to be highly asymmetrical and global 3D simulations are needed to help understand the dynamics. We perform and analyze hydrodynamic CEE simulations with the adaptive mesh refinement (AMR) code AstroBEAR, and focus on the role of accretion onto the companion star. We bracket the range of accretion rates by comparing a model that removes mass and pressure using a subgrid accretion prescription with one that does not. Provided a pressure-release valve, such as a bipolar jet, is available, super-Eddington accretion could be common. Finally, we summarize new results pertaining to the energy budget, and discuss the overall implications relating to the feasibility of unbinding the envelope in CEE simulations.