Our main goal is the robust numerical simulation of brittle fractures, implementing a provably convergent and geometrically unbiased adaptive frame scheme based on shearlet discretizations.

# C03

Shearlet approximation of brittle fracture evolutions

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A brittle material, subjected to a force, first deforms itself elastically, then it breaks without any intermediate phase. A model of brittle fractures was proposed by Francfort and Marigo, where the displacement is typically a smooth function except on a relatively smooth jump set determining the fracture. This approach has the advantage not to require a pre-defined crack path, but has the drawback that any mesh discretization is a geometrical bias. Despite encouraging numerical results, FEM are expected to retain a certain geometrical bias and so far a proof of convergence of the proposed algorithm remains out of reach. Differently from finite elements, shearlets are frames with rotation invariance and optimal nonlinear approximation properties for the class of functions, which are smooth except on smooth lower dimensional sets. In this project we intend to compare anisotropic and adaptive mesh refinements with adaptive frame methods based on shearlets. In particular, by taking advantage of the shearlet property of optimally approximating piecewise smooth functions, we aim at reaching not only a proof of the convergence of the frame adaptive algorithms but also their optimal complexity.
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**Group:**C. Computation-
**Principal Investigators:**Prof. Dr. Massimo Fornasier, Prof. Dr. Gitta Kutyniok -
**Investigators:**Dr. Markus Hansen, Dr. Philipp Petersen **Universities:**TU Berlin, TU München**Term:**since 2012