28,998 research outputs found
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
PyDEC: Software and Algorithms for Discretization of Exterior Calculus
This paper describes the algorithms, features and implementation of PyDEC, a
Python library for computations related to the discretization of exterior
calculus. PyDEC facilitates inquiry into both physical problems on manifolds as
well as purely topological problems on abstract complexes. We describe
efficient algorithms for constructing the operators and objects that arise in
discrete exterior calculus, lowest order finite element exterior calculus and
in related topological problems. Our algorithms are formulated in terms of
high-level matrix operations which extend to arbitrary dimension. As a result,
our implementations map well to the facilities of numerical libraries such as
NumPy and SciPy. The availability of such libraries makes Python suitable for
prototyping numerical methods. We demonstrate how PyDEC is used to solve
physical and topological problems through several concise examples.Comment: Revised as per referee reports. Added information on scalability,
removed redundant text, emphasized the role of matrix based algorithms,
shortened length of pape
Homalg: A meta-package for homological algebra
The central notion of this work is that of a functor between categories of
finitely presented modules over so-called computable rings, i.e. rings R where
one can algorithmically solve inhomogeneous linear equations with coefficients
in R. The paper describes a way allowing one to realize such functors, e.g.
Hom, tensor product, Ext, Tor, as a mathematical object in a computer algebra
system. Once this is achieved, one can compose and derive functors and even
iterate this process without the need of any specific knowledge of these
functors. These ideas are realized in the ring independent package homalg. It
is designed to extend any computer algebra software implementing the
arithmetics of a computable ring R, as soon as the latter contains algorithms
to solve inhomogeneous linear equations with coefficients in R. Beside
explaining how this suffices, the paper describes the nature of the extensions
provided by homalg.Comment: clarified some points, added references and more interesting example
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