Local modifications of a computational domain are often performed in order to simplify the meshing process and to reduce computational costs and memory requirements. However, removing geometrical features of a domain often introduces a non-negligible error in the solution of a differential problem in which it is defined. In this work, we extend the results from Buffa et al. (Math Models Methods Appl Sci. 2022; 32(02):359-402.) by studying the case of domains containing an arbitrary number of distinct Neumann features, and by performing an analysis on Poisson's, linear elasticity, and Stokes' equations. We introduce a simple, computationally cheap, reliable, and efficient a posteriori estimator of the geometrical defeaturing error. Moreover, we also introduce a geometric refinement strategy that accounts for the defeaturing error: starting from a fully defeatured geometry, the algorithm determines at each iteration step which features need to be added to the geometrical model to reduce the defeaturing error. These important features are then added to the (partially) defeatured geometrical model at the next iteration, until the solution attains a prescribed accuracy. A wide range of two- and three-dimensional numerical experiments are finally reported to illustrate this work.