The p-Poisson Equation: Regularity Analysis and Adaptive Wavelet Frame Approximation

This thesis is concerned with an important class of quasilinear elliptic equations: the p-Poisson equations -div(|\nabla u|^{p-2} \nabla u) = f in Ω, where 1 =2. Equations of this type appear, inter alia, in various problems in continuum mechanics, for instance in the mathematical modelling of non-Newtonian fluids. Furthermore, the p-Poisson equations possess a certain model character for more general quasilinear elliptic problems. The central aspect of this thesis is the regularity analysis of solutions u to the p-Poisson equation in the so-called adaptivity scale B^σ_τ(L_τ(Ω)), 1/τ = σ/d + 1/p, σ > 0, of Besov spaces. It is well-known that the smoothness parameter σ determines the approximation rate of the best n-term wavelet approximation, and hence provides information on the maximal convergence rate of certain adaptive numerical wavelet methods. To derive Besov regularity estimates for solutions to the p-Poisson equation, two approaches are pursued in this work. The first approach makes use of the fact that under appropriate conditions the solutions to the p-Poisson equation admit certain higher regularity in the interior of the domain, in the sense that they are locally Hölder continuous. In general, the Hölder semi-norms may explode as one approaches the boundary of the domain, but this singular behavior can be controlled by some power of the distance to the boundary. It turns out that the combination of global Sobolev regularity and locally weighted Hölder regularity can be used to derive Besov smoothness in the adaptivity scale for solutions to the p-Poisson equation. The results of the first approach are stated in two steps. At first, a general embedding theorem is proved, which says that the intersection of a classical Sobolev space with a Hölder space having the above mentioned properties can be embedded into certain Besov spaces in the adaptivity scale. The proof of this result is based on extension arguments in connection with the characterization of Besov spaces by wavelet expansion coefficients. Subsequently, it is verified that in many cases the solutions u to the p-Poisson equation indeed satisfy the conditions of the embedding theorem, so that its application yields the desired regularity result. As it is shown, in many cases the Besov smoothness σ of the solution is significantly higher than its Sobolev smoothness, so that the development of adaptive schemes for the p-Poisson problem is completely justified. It is worthwhile noting that this universal approach is applicable for the general class of Lipschitz domains. The aim of the second approach is to make a first step in improving some of the derived Besov regularity results for solutions on polygonal domains. To this end the regularity is examined in a neighborhood of the corners of the domain, since generally the critical singularities of the solutions occur there. As it is shown, this approach leads to regularity assertions which are – in a local sense – indeed stronger in some cases than those derived with the first approach. The proofs are based on known results on the singular expansion of the solution in a neighborhood of a conical boundary point, as well as on embeddings of the intersection of Babuska-Kondratiev spaces K^l_{p,a}(Ω) with certain Besov spaces into the adaptivity scale of Besov spaces. As it is shown, in some cases the solutions to the p-Poisson equation admit arbitrary high weighted Sobolev regularity l in a neighborhood of the corners, and hence arbitrary high Besov regularity σ. Because of this fact the borderline case of this embedding for l equal infinity is analyzed in addition. It is shown that the resulting Fréchet spaces are continuously embedded into the corresponding F-spaces. It is worth mentioning that by these embeddings – independent of the p-Poisson setting – universal functional analytical tools are provided. The second central issue of this thesis is the numerical solution of the p-Poisson equation for 1 < p < 2. In this context, the focus is put on the implementation and numerical testing of a relaxed Kačanov-type iteration scheme for the approximate solution of the p-Poisson equation with homogeneous Dirichlet boundary conditions. For the numerical solution of the occurring linear elliptic subproblems an adaptive wavelet frame method is used. The resulting algorithm is studied in a series of numerical tests. Here, it turns out that in practice the implemented algorithm shows a stable convergence behavior