Shearlet Galerkin for transport equations: Implementation and stability

Abstract

We provide implementation details for the framework of a solver for the advectionreaction equation using piecewise linear shearlet frames. It is hoped that the directional features of shearlets will provide a sparse representation of the solution under an adaptive setting. We investigate the stability of the resulting stiffness matrices, and find that the effective condition numbers are too high for iterative solvers, even with a modest number of degrees of freedom. This is a strong hint of instability in relevant norms.