Wear on total knee replacements is an important criterion for their performance characteristics. Numerical simulations of such wear have seen increasing attention over the last years. They have the potential to be much faster and less expensive than the in vitro tests in use today. While it is unlikely that in silico tests will replace actual physical tests in the foreseeable future, a judicious combination of both approaches can help making both implant design and pre-clinical testing quicker and more cost-effective. The challenge today for the design of simulation methods is to obtain results that convey quantitative information and to do so quickly and reliably. This involves the choice of mathematical models as well as the numerical tools used to solve them. The correctness of the choice can only be validated by comparing with experimental results. In this article, we present finite element simulations of the wear in total knee replacements during the gait cycle standardized in the ISO 14243-1 document, used for compliance testing in several countries. As the ISO 14243-1 standard is precisely defined and publicly available, it can serve as an excellent benchmark for comparison of wear simulation methods. We use comparatively simple wear and material models, but we solve them using a new wear algorithm that combines extrapolation of the geometry changes with a contact algorithm based on nonsmooth multigrid ideas. The contact algorithm works without Lagrange multipliers and penalty parameters, achieving unparalleled stability and efficiency. We compare our simulation results with the experimental data from physical tests using two different actual total knee replacements. Even though the model is simple, we can predict the total mass loss due to wear after 5-million gait cycles, and we observe a good match between the wear patterns seen in experiments and our simulation results. When compared with a state-of-the-art penalty-based solver for the same model, we measure a roughly fivefold increase of execution speed.