Many important problem classes are governed by anisotropic features such as singularities concentrated on lower dimensional embedded manifolds. Therefore, analyzing the intrinsic geometrical features of the underlying object is essentially in many applications.
In the first part of this talk, presented by Wang-Q Lim, we will present constructions of compactly supported directional systems. We also show that they can be effectively used to extract anisotropic structures from given data.
The second part of the talk, presented by Gerrit Welper, is concerned with preparing for the use of such $L_2$ stable shearlet-frames for solving unsymmetric partial differential equations whose solutions exhibit strong anisotropic features. Here we shall confine the discussion to the model problem of linear convection. Specifically, we will present a a well posed variational formulation for the convection problem where the solution space is $L_2$ in order to be able to make use of the shearlet frames. The key point is to identify a proper test space. On one hand this leads to finite dimensional stable discretizations. On the other hand, we shall show how to construct a frame for that test space. This will be used to formulate adaptive solution strategies similar to those developed earlier for elliptic problems.