Best N Term Approximation Spaces for Tensor Product Wavelet Bases

We consider best N term approximation using anisotropic tensor product wavelet bases ("sparse grids"). We introduce a tensor product structure ⊗q on certain quasi-Banach spaces. We prove that the approximation spaces Aαq(L2) and Aαq(H1) equal tensor products of Besov spaces Bαq(Lq), e.g., Aαq(L2([0,1]d)) = Bαq(Lq([0,1])) ⊗q · ⊗q Bαq · ·(Lq([0,1])). Solutions to elliptic partial differential equations on polygonal/polyhedral domains belong to these new scales of Besov space

Besov regularity for operator equations on patchwise smooth manifolds

We study regularity properties of solutions to operator equations on patchwise smooth manifolds $\partial\Omega$ such as, e.g., boundaries of polyhedral domains $\Omega \subset \mathbb{R}^3$. Using suitable biorthogonal wavelet bases $\Psi$, we introduce a new class of Besov-type spaces $B_{\Psi,q}^\alpha(L_p(\partial \Omega))$ of functions $u\colon\partial\Omega\rightarrow\mathbb{C}$. Special attention is paid on the rate of convergence for best $n$-term wavelet approximation to functions in these scales since this determines the performance of adaptive numerical schemes. We show embeddings of (weighted) Sobolev spaces on $\partial\Omega$ into $B_{\Psi,\tau}^\alpha(L_\tau(\partial \Omega))$, $1/\tau=\alpha/2 + 1/2$, which lead us to regularity assertions for the equations under consideration. Finally, we apply our results to a boundary integral equation of the second kind which arises from the double layer ansatz for Dirichlet problems for Laplace's equation in $\Omega$.Comment: 42 pages, 3 figures, updated after peer review. Preprint: Bericht Mathematik Nr. 2013-03 des Fachbereichs Mathematik und Informatik, Universit\"at Marburg. To appear in J. Found. Comput. Mat

Besov Regularity of Solutions to Navier-Stokes Equations

This thesis is concerned with the regularity of solutions to Navier-Stokes and Stokes equation on domains with point singularities, namely polyhedral domains contained in R3 and general bounded Lipschitz domains in Rd, d ≥ 3 with connected boundary. The Navier-Stokes equations provide a mathematical model of the motion of a uid. These Navier-Stokes equations form the basis for the whole world of computational uid dynamics, and therefore they are considered as maybe the most important PDEs known so far. We consider the stationary (Navier-)Stokes equations. The study the Besov regularity of the solution in the scale BsƬ (LƬ (Ω))d, 1/Ƭ = s/d + 1/2 of Besov spaces. This scale is the so-called adaptivity scale. The parameter s determines the approximation order of adaptive numerical wavelet schemes and other nonlinear approximation methods when the error is measured in the L2-norm. In contrast to this the convergence order of linear schemes is determined by the classical L2-Sobolev regularity. In many papers the Besov regularity of the solution to various operator equations/partial differential equations was investigated. The proof of Besov regularity in the adaptivity scale was in many contributions performed by combining weighted Sobolev regularity results with characterizations of Besov spaces by wavelet expansions. Choosing a suitable wavelet basis the coeffcients of the wavelet expansion of the solution can be estimated by exploiting the weighted Sobolev regularity of the solution, such that a certain Besov regularity can be established. This technique was applied for the Stokes system in all papers which are part of this thesis. For achieving Besov regularity for Navier-Stokes equation we used a fixed point argument. We formulate the Navier-Stokes equation as a fixed point equation and therefore regularity results for the corresponding Stokes equation can be transferred to the non-linear case. In the first paper "Besov regularity for the Stokes and the Navier-Stokes system in polyhedral domains" we considered the stationary Stokes- and the Navier-Stokes equations in polyhedral domains. Exploiting weighted Sobolev estimates for the solution we proved that the Besov regularity of the solutions to these equations exceed their Sobolev regularity. In the second paper "Besov Regularity for the Stationary Navier-Stokes Equation on Bounded Lipschitz Domains" we have investigated the stationary (Navier-)Stokes equations on bounded Lipschitz domain. Based on weighted Sobolev estimates again we could establish a Besov regularity result for the solution to the Stokes system. By applying Banach's fixed point theorem we transferred these results to the non-linear Navier-Stokes equation. In order to apply the fixed point theorem we had to require small data and small Reynolds number

Almost diagonal matrices and Besov-type spaces based on wavelet expansions

This paper is concerned with problems in the context of the theoretical foundation of adaptive (wavelet) algorithms for the numerical treatment of operator equations. It is well-known that the analysis of such schemes naturally leads to function spaces of Besov type. But, especially when dealing with equations on non-smooth manifolds, the definition of these spaces is not straightforward. Nevertheless, motivated by applications, recently Besov-type spaces $B^\alpha_{\Psi,q}(L_p(\Gamma))$ on certain two-dimensional, patchwise smooth surfaces were defined and employed successfully. In the present paper, we extend this definition (based on wavelet expansions) to a quite general class of $d$-dimensional manifolds and investigate some analytical properties (such as, e.g., embeddings and best $n$-term approximation rates) of the resulting quasi-Banach spaces. In particular, we prove that different prominent constructions of biorthogonal wavelet systems $\Psi$ on domains or manifolds $\Gamma$ which admit a decomposition into smooth patches actually generate the same Besov-type function spaces $B^\alpha_{\Psi,q}(L_p(\Gamma))$, provided that their univariate ingredients possess a sufficiently large order of cancellation and regularity (compared to the smoothness parameter $\alpha$ of the space). For this purpose, a theory of almost diagonal matrices on related sequence spaces $b^\alpha_{p,q}(\nabla)$ of Besov type is developed. Keywords: Besov spaces, wavelets, localization, sequence spaces, adaptive methods, non-linear approximation, manifolds, domain decomposition.Comment: 38 pages, 2 figure

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

Besov regularity of solutions to the p-Poisson equation

In this paper, we study the regularity of solutions to the $p$-Poisson equation for all $1<p<\infty$. In particular, we are interested in smoothness estimates in the adaptivity scale $B^\sigma_{\tau}(L_{\tau}(\Omega))$, $1/\tau = \sigma/d+1/p$, of Besov spaces. The regularity in this scale determines the order of approximation that can be achieved by adaptive and other nonlinear approximation methods. It turns out that, especially for solutions to $p$-Poisson equations with homogeneous Dirichlet boundary conditions on bounded polygonal domains, the Besov regularity is significantly higher than the Sobolev regularity which justifies the use of adaptive algorithms. This type of results is obtained by combining local H\"older with global Sobolev estimates. In particular, we prove that intersections of locally weighted H\"older spaces and Sobolev spaces can be continuously embedded into the specific scale of Besov spaces we are interested in. The proof of this embedding result is based on wavelet characterizations of Besov spaces.Comment: 45 pages, 2 figure

Optimal Approximation of Elliptic Problems by Linear and Nonlinear Mappings I

We study the optimal approximation of the solution of an operator equation Au=f by linear and nonlinear mappings