Directional Anisotropic Multiscale Systems on Bounded Domains

Driven by an overwhelming amount of applications numerical approximation of partial differential equations was established as one of the core areas in applied mathematics. During the last decades a trend for the solution of PDEs emerged, that focuses on employing systems from applied harmonic analysis for the adaptive solution of these equations. Most notably wavelet systems have been used and lead for instance to provably optimal solvers for elliptic PDEs, [1]. Inspired by this success story also other systems with various advantages in different directions should be employed in various discretization problems. For instance, ridgelets where recently successfully used in the discretization of linear transport equations, [3]. Another famous system is that of shearlets, [5], which admits optimal representations of functions that have singularities along smooth curves. The main bottleneck in developing shearlet, or ridgelet-based PDE solvers is the fact that originally these systems are constructed as representation systems, or frames, for functions defined on Rd, while most PDEs are defined on a finite domain Ω ⊂ Rd which implies that the development of effective PDE solvers crucially depends on the construction of anisotropic representation systems on finite domains, satisfying various boundary conditions. Hence it is necessary to have a system on a bounded domain Ω, which