We discuss the performance of the Finite Element Ocean Model (FESOM) on locally eddy-resolving global unstructured meshes. In particular, the utility of the mesh design approach whereby mesh horizontal resolution is varied as half the Rossby radius in most of the model domain is explored. Model simulations on such a mesh (FESOM-XR) are compared with FESOM simulations on a smaller-size mesh, where refinement depends only on the pattern of observed variability (FESOM-HR). We also compare FESOM results to a simulation of the ocean model of the Max Planck Institute for Meteorology (MPIOM) on a tripolar regular grid with refinement toward the poles, which uses a number of degrees of freedom similar to FESOM-XR. The mesh design strategy, which relies on the Rossby radius and/or the observed variability pattern, tends to coarsen the resolution in tropical and partly subtropical latitudes compared to the regular MPIOM grid. Excessive variations of mesh resolution are found to affect the performance in other nearby areas, presumably through dissipation that increases if resolution is coarsened. The largest improvement shown by FESOM-XR is a reduction of the surface temperature bias in the so-called North-West corner of the North Atlantic Ocean where horizontal resolution was increased dramatically. However, other biases in FESOM-XR remain largely unchanged compared to FESOM-HR. We conclude that resolving the Rossby radius alone (with two points per Rossby radius) is insufficient, and that careful use of a priori information on eddy dynamics is required to exploit the full potential of ocean models on unstructured meshes.