The aim of this project is to establish reliable and efficient numerical methods for models of solids with spatial discontinuities caused by the evolution of dissipative processes such as plasticification, damage or fracture. In particular, the project focuses on such prototypical models that use the class of BV-functions to mathematically describe the discontinuities, that are guaranteed to converge to a solution of the infnite-dimensional model and for which iterative solution methods can be constructed. Emphasis is on unregularized numerical approaches that lead to sharp approximations of discontinuities on coarse grids and rigorous convergence proofs. The main objectives of the research project are the development, analysis, and implementation of finite element methods for model problems describing discontinuities in BV. This includes the derivation of a priori and a posteriori error estimates as well as the construction of adaptive and extended approximation methods for BV-prototype models such as the Rudin-Osher-Fatemi and the Mumford-Shah model. The techniques will be transferred to analytically justified and closely related models for the description of rate-independent inelastic processes, in particular perfect plasticity, damage and fracture. The methods and results will be applied and transferred to particular model scenarios and benchmark problems in mechanics.