The recently developed level-set-DEM is able to seamlessly handle arbitrarily shaped grains and their contacts through a discrete level-set representation of grains' volume and a node-based discretization of their bounding surfaces. Heretofore, the convergence properties of LS-DEM with refinement of these discretizations have not been examined. Here, we examine these properties and show that the original LS-DEM diverges upon surface discretization refinement due to its force-based discrete contact formulation. Next, we fix this issue by adopting a continuum-based contact formulation wherein the contact interactions are traction-based, and show that the adapted LS-DEM is fully discretization convergent. Lastly, we discuss the significance of convergence in capturing the physical response, as well as a few other convergence-related topics of practical importance.