"New Smoothness Spaces on Domains and Their Discrete Characterization"

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Since the discovery of wavelets, the past decades have seen an explosion concerning the design of novel representation systems for functions or distributions. The main intent of that work has been to find representation systems which are optimal for the sparse approximation of various signal classes with the main applications lying in the area of signal processing. As an example we mention the efficient detection of directional information that can be performed by shearlet or curvelet or ridgelet systems which possess vastly superior approximation properties as compared to standard discretization methods such as finite elements or wavelets. Having the various spectacular results concerning the approximation properties of these new representation systems in mind, a natural next step would be to employ them also for the numerical treatment of operator equations.However, the development of numerical methods based on these new representation system is currently facing the bottleneck that to date no useful constructions of these representation systems on bounded domains exist. The aim of this project is to remove this bottleneck by constructing and analyzing new discrete representation systems on finite domains which on the one hand enjoy the same optimal approximation properties as, for example, shearlets while still forming a stable discretization of the energy space for a large class of PDEs (for example Sobolev spaces). Our focus will be on the development of a comprehensive theory for the adaption of function spaces to finite domains. We will be especially interested in the discrete characterization of such spaces, the regularity theory of various PDEs on such spaces and the compressibility properties of Galerkin matrices of various PDEs with respect to these discretizations. These studies are expected to lay the ground work for the subsequent development and implementation of a large class of novel discretization methods for operator equations which outperform current wavelet- or finite-element-based methods for a large class of important problems such as reaction-diffusion equations, linear transport equations or elliptic PDEs with discontinuous diffusion coefficients.