When solving localized structural failure problems, the standard displacement-based finite element formulation from solid mechanics has demonstrated to generate spuriously mesh dependent localization paths. Fracture results obtained with this method present a pathological bias governed by the orientation of the FE mesh adopted in the calculation. This critical issue has been addressed in the past by the authors through the adoption of the mixed strain/displacement finite element formulation to calculate the solid mechanics problem. This approach has shown to guarantee the local convergence of the computations in terms of strains and stresses. This constitutes a fundamental property when aiming at achieving mesh bias objective results in localized structural failure analyses. This work presents the Adaptive Formulation Refinement (AFR) method to address the problem in a more computationally efficient manner. Its basis is performing the numerical analyses starting from the standard displacement-based FE approach, and adaptively activating the mixed formulation exclusively in the areas of the domain where damage appears while the standard FE method is maintained elsewhere. This allows for very substantial reductions in computational cost, while the mesh-objectivity of the results achieved by the mixed formulation is preserved. The AFR methodology is adopted together with an octree-based Adaptive Mesh Refinement (AMR) approach in order to further increase the cost-effectiveness of the computations. The proposed approach based on the combination of the AFR and AMR strategies permits the cost-efficient mesh-objective analysis of fracture propagation while preserving the required mesh resolution and quality of the results. This is demonstrated via a comprehensive set of numerical simulations of benchmark problems and laboratory experiments reported in the literature in 2D and 3D. Analyses demonstrate the aptness of the strategy to replicate experimental results in terms of force-displacement curves, fracture paths and collapse mechanisms with accuracy.