Computational Methods are developed for failure problems such as shear banding. H-Adaptive procedures for the finite element solution of transient solid mechanics problems are studied, with particular emphasis on problems involving localization due to material instability. Various types of error criteria are examined and it is shown that for problems involving plastic response or localization, an error criterion based on an L2 projection of strains is the most effective for the constant strain elements considered here. Examples of one dimensional and two dimensional localization shear band formation problems are given. Massively parallel computations are performed to study the effects of imperfections on shear band morphology Finite elements, Material instability shear bands.