A new approach of dealing with mesh dependence in finite element modelling of fracture processes is introduced. In particular, in brittle fracture modelling, the stress concentration is mesh dependent as the results do not stabilise when refining the mesh. This paper presents an approach based on the explicit incorporation of mesh dependence into the computations. The dependence of the relevant stress is quantified on the finite elements at the crack tip upon the element size; when the dependence approaches a power law with the required accuracy, the mesh is called scalable. If the mesh is scalable and the exponent and pre-factor are known, then the results of the computations can be scaled to the size relevant to the scale of the physical microstructure of the material; the latter while not being modelled directly ultimately controls the fracture propagation. To illustrate this new approach, four 2D examples of a single straight crack loaded under tensile and shear tractions applied either to the external boundary or to the crack faces are considered. It is shown that combining the stresses at the crack tip computed using a set of similar meshes of different densities with the crack tip asymptotic allows accurate recovery of the stress intensity factors.