Dokument

Summary:
This thesis is concerned with the construction and application of a new class of functions called quarklets. With the intention of constructing an adaptive hp-method based on wavelets, they do arise out of the latter through an enrichment with polynomials. The starting point for the construction is the real axis. There, we derive frames for the Sobolev space H^s(R). Through a boundary adaptation, tensorization and the application of a scale-dependent extension operator we are even able to construct quarklet frames on very general domains in multiple spatial dimensions. With these frames at hand we can discretize linear elliptic operator equations in a stable way. The discrete system can be handled with an adaptive numerical scheme. For this purpose it is necessary to show the compressibility of the stiffness matrix. We do this for the prototypical example of the Poisson equation independently of the dimension and in this way we are able to prove the optimality of the adaptive scheme. By the latter we mean that the approximation rate of the best n-term quarklet approximation is realized by the scheme. Finally, we carry out some numerical experiments in one and two spatial dimensions, where the theoretical findings are validated in practice and moreover, the value of the quarklets in the numerical scheme becomes visible by analysing the quarklet coefficients of the approximate solutions.

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