Quantifying the fractional change in a predicted quantity of interest with successive mesh refinement is an attractive and widely used but limited approach to assessing numerical error and uncertainty in physics-based computational modeling. Herein, we introduce the concept of a scalar multiplier to clarify the connection between fractional change and a more rigorous and accepted estimate of numerical uncertainty, the grid convergence index (GCI). Specifically, we generate lookup tables for as a function of observed order of accuracy and mesh refinement factor. We then illustrate the limitations of relying on fractional change alone as an acceptance criterion for mesh refinement using a case study involving the radial compression of a Nitinol stent. Results illustrate that numerical uncertainty is often many times larger than the observed fractional change in a mesh pair, especially in the presence of small mesh refinement factors or low orders of accuracy. We strongly caution against relying on fractional change alone as an acceptance criterion for mesh refinement studies, particularly in any high-risk applications requiring absolute prediction of quantities of interest. When computational resources make the systematic refinement required for calculating GCI impractical, submodeling approaches as demonstrated herein can be used to rigorously quantify discretization error at a comparatively minimal computational cost. To facilitate future quantitative mesh refinement studies, lookup tables herein provide a useful tool for guiding the selection of mesh refinement factor and element order.