A new shock-fitting technique for unstructured two- and three-dimensional meshes has been recently proposed and developed by the authors. In the present paper, both global and local a posteriori grid-convergence analysis is used to quantitatively measure the discretization error and order of convergence of the numerical solutions obtained using this new unstructured shock-fitting technique. Specifically, the analysis has considered the numerical solutions of two different flows characterized by the presence of strong shocks: a transonic source flow and an hypersonic flow past a circular cylinder. It is shown that the shock-fitting technique allows to compute numerical solutions that converge, both pointwise and in a global sense, with an observed order of accuracy that is very close to the design order of the spatial discretization scheme and with very small discretization errors