A mathematically challenging model of dynamic wetting as a process of interface formation has been, for the first time, fully incorporated into a numerical code based on the finite element method and applied, as a test case, to the problem of capillary rise. The motivation for this work comes from the fact that, as discovered experimentally more than a decade ago, the key variable in dynamic wetting flows - the dynamic contact angle - depends not just on the velocity of the three-phase contact line but on the entire flow field/geometry. Hence, to describe this effect, it becomes necessary to use the mathematical model that has this dependence as its integral part. A new physical effect, termed the 'hydrodynamic resist to dynamic wetting', is discovered where the influence of the capillary's radius on the dynamic contact angle, and hence on the global flow, is computed. The capabilities of the numerical framework are then demonstrated by comparing the results to experiments on the unsteady capillary rise, where excellent agreement is obtained. Practical recommendations on the spatial resolution required by the numerical scheme for a given set of non-dimensional similarity parameters are provided, and a comparison to asymptotic results available in limiting cases confirms that the code is converging to the correct solution. The appendix gives a user-friendly step-by-step guide specifying the entire implementation and allowing the reader to easily reproduce all presented results, including the benchmark calculations. © 2012 Elsevier Inc.