Reliability of efficient approximation schemes for material discontinuities described by functions ofbounded variation

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Spaces of functions of bounded variations provide an attractive framework to describe material discontinuities such as damage and fracture. Recently, suitable notions of solution and general existence theories for corresponding evolutionary model problems have been established and numerical methods for discretizing and iteratively solving variational problems involving the total variation norm have been developed and analyzed.In the first funding period of the project, abstract existence results for coupled rate-dependent/rate-independent systems, delamination processes in visco-elastodynamics, and phasefield descriptions of damage evolutions have been investigated analytically. Algorithmic contributions have been made to the iterative solution of model problems on functions of bounded variation, the adaptive approximation of discontinuous functions based on fully computable a~posteriori error estimates, and the convergent finite element simulation of a BV-regularized damage model. Within a second funding period these results will be combined, refined, and extended to complex models describing damage and fracture evolution. The envisaged models capture material discontinuities in BV either directly or as a scaling limit. The planned research includes the derivation of a priori and a posteriori error estimates, the construction of adaptive approximation schemes intertwined with results based on existence theory and Gamma-convergence, and their implementation and application to specific benchmark problems in mechanics. Particular attention will be paid to the reliability of efficient methods, e.g., convergence of suitable time-stepping and adaptive approximation schemes based on variational techniques.