Optimal adaptive finite element and wavelet methods for p-Poisson equations

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This project is concerned with the design of adaptive strategies for certain classes of quasilinear problems, in particular p-Poisson equations, their convergence analysis, and the proof of optimality in terms of the number of degrees of freedom and the algebraic complexity, respectively. Our approach is based on an adaptive regularization of so-called Kacanov iterations, whose regularization parameter is tuned according to a posteriori error estimators which have also the function of guiding adaptive discretizations. We shall focus on both, adaptive finite element and wavelet methods.The motivation is twofold: on the one hand, appropriate reliable error estimators for finite element discretizations have already been defined and studied for the p-Poisson equation, and we expect that we will be able to „port“ this knowledge to wavelet methods for which, in this particular problem, reliable error estimators are not yet available. On the other hand, the strong analytical properties of wavelets can usually be exploited to derive more simply and sometimes more rigorously a convergence and optimality analysis for adaptive wavelet schemes compared to finite element approaches; moreover, the understanding of Besov regularity of solutions of any type of known elliptic equations so far considered has been based on the use of wavelets. Let us stress the fact that Besov regularity of solutions is a fundamental issue when it comes to address the rate of convergence or the complexity of both adaptive finite element and wavelet methods. In addition to the analysis of the adaptive methods for p-Poisson equations, we also plan to perform extensive numerical simulations in order to demonstrate the validity of the theoretical results.