Computational material design is progressively gaining momentum in the engineering world. Recent breakthroughs in high performance computing and emerging multiscale algorithms have facilitated the simulation of materials at different scales of observation. In particular, the multiscale study of failure phenomena becomes crucial to assess the performance of engineering materials and structures. In this thesis, a concurrent multiscale method is proposed for the failure analysis of quasi-brittle materials. Domain decomposition techniques, such as the Finite Element Tearing and Interconnecting (FETI) method, are used to partition the structure in a number of non-overlapping domains. A different treatment, from a numerical standpoint, is given to linear elastic and non-linear domains in the sense that most of the computational effort is spent in non-linear regions. Multiscale analysis is achieved by means of an adaptive refinement at those domains that are affected by damage processes. This refinement is done in terms of material scale and finite element size. It is verified that the framework is able to correctly capture the initiation and growth of non-linearity at a reduced computational cost when compared to full scale computations. The multiscale framework is found specially attractive for the study of failure in quasi-brittle materials.