3,191 research outputs found
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 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 -dimensional manifolds and investigate some analytical properties
(such as, e.g., embeddings and best -term approximation rates) of the
resulting quasi-Banach spaces. In particular, we prove that different prominent
constructions of biorthogonal wavelet systems on domains or manifolds
which admit a decomposition into smooth patches actually generate the
same Besov-type function spaces , provided that
their univariate ingredients possess a sufficiently large order of cancellation
and regularity (compared to the smoothness parameter of the space).
For this purpose, a theory of almost diagonal matrices on related sequence
spaces 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
Multilevel Preconditioning of Discontinuous-Galerkin Spectral Element Methods, Part I: Geometrically Conforming Meshes
This paper is concerned with the design, analysis and implementation of
preconditioning concepts for spectral Discontinuous Galerkin discretizations of
elliptic boundary value problems. While presently known techniques realize a
growth of the condition numbers that is logarithmic in the polynomial degrees
when all degrees are equal and quadratic otherwise, our main objective is to
realize full robustness with respect to arbitrarily large locally varying
polynomial degrees degrees, i.e., under mild grading constraints condition
numbers stay uniformly bounded with respect to the mesh size and variable
degrees. The conceptual foundation of the envisaged preconditioners is the
auxiliary space method. The main conceptual ingredients that will be shown in
this framework to yield "optimal" preconditioners in the above sense are
Legendre-Gauss-Lobatto grids in connection with certain associated anisotropic
nested dyadic grids as well as specially adapted wavelet preconditioners for
the resulting low order auxiliary problems. Moreover, the preconditioners have
a modular form that facilitates somewhat simplified partial realizations. One
of the components can, for instance, be conveniently combined with domain
decomposition, at the expense though of a logarithmic growth of condition
numbers. Our analysis is complemented by quantitative experimental studies of
the main components.Comment: 41 pages, 11 figures; Major revision: rearrangement of the contents
for better readability, part on wavelet preconditioner adde
Besov regularity for operator equations on patchwise smooth manifolds
We study regularity properties of solutions to operator equations on
patchwise smooth manifolds such as, e.g., boundaries of
polyhedral domains . Using suitable biorthogonal
wavelet bases , we introduce a new class of Besov-type spaces
of functions
. Special attention is paid on the
rate of convergence for best -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
into , ,
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 .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
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