11,466 research outputs found
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
Complexes of Discrete Distributional Differential Forms and their Homology Theory
Complexes of discrete distributional differential forms are introduced into
finite element exterior calculus. Thus we generalize a notion of Braess and
Sch\"oberl, originally studied for a posteriori error estimation. We construct
isomorphisms between the simplicial homology groups of the triangulation, the
discrete harmonic forms of the finite element complex, and the harmonic forms
of the distributional finite element complexes. As an application, we prove
that the complexes of finite element exterior calculus have cohomology groups
isomorphic to the de Rham cohomology, including the case of partial boundary
conditions. Poincar\'e-Friedrichs-type inequalities will be studied in a
subsequent contribution.Comment: revised preprint, 26 page
Convergence and Optimality of Adaptive Mixed Methods on Surfaces
In a 1988 article, Dziuk introduced a nodal finite element method for the
Laplace-Beltrami equation on 2-surfaces approximated by a piecewise-linear
triangulation, initiating a line of research into surface finite element
methods (SFEM). Demlow and Dziuk built on the original results, introducing an
adaptive method for problems on 2-surfaces, and Demlow later extended the a
priori theory to 3-surfaces and higher order elements. In a separate line of
research, the Finite Element Exterior Calculus (FEEC) framework has been
developed over the last decade by Arnold, Falk and Winther and others as a way
to exploit the observation that mixed variational problems can be posed on a
Hilbert complex, and Galerkin-type mixed methods can be obtained by solving
finite dimensional subproblems. In 2011, Holst and Stern merged these two lines
of research by developing a framework for variational crimes in abstract
Hilbert complexes, allowing for application of the FEEC framework to problems
that violate the subcomplex assumption of Arnold, Falk and Winther. When
applied to Euclidean hypersurfaces, this new framework recovers the original a
priori results and extends the theory to problems posed on surfaces of
arbitrary dimensions. In yet another seemingly distinct line of research,
Holst, Mihalik and Szypowski developed a convergence theory for a specific
class of adaptive problems in the FEEC framework. Here, we bring these ideas
together, showing convergence and optimality of an adaptive finite element
method for the mixed formulation of the Hodge Laplacian on hypersurfaces.Comment: 22 pages, no figures. arXiv admin note: substantial text overlap with
arXiv:1306.188
Discrete exterior calculus (DEC) for the surface Navier-Stokes equation
We consider a numerical approach for the incompressible surface Navier-Stokes
equation. The approach is based on the covariant form and uses discrete
exterior calculus (DEC) in space and a semi-implicit discretization in time.
The discretization is described in detail and related to finite difference
schemes on staggered grids in flat space for which we demonstrate second order
convergence. We compare computational results with a vorticity-stream function
approach for surfaces with genus 0 and demonstrate the interplay between
topology, geometry and flow properties. Our discretization also allows to
handle harmonic vector fields, which we demonstrate on a torus.Comment: 21 pages, 9 figure
Differential Calculi on Commutative Algebras
A differential calculus on an associative algebra A is an algebraic analogue
of the calculus of differential forms on a smooth manifold. It supplies A with
a structure on which dynamics and field theory can be formulated to some extent
in very much the same way we are used to from the geometrical arena underlying
classical physical theories and models. In previous work, certain differential
calculi on a commutative algebra exhibited relations with lattice structures,
stochastics, and parametrized quantum theories. This motivated the present
systematic investigation of differential calculi on commutative and associative
algebras. Various results about their structure are obtained. In particular, it
is shown that there is a correspondence between first order differential
calculi on such an algebra and commutative and associative products in the
space of 1-forms. An example of such a product is provided by the Ito calculus
of stochastic differentials.
For the case where the algebra A is freely generated by `coordinates' x^i,
i=1,...,n, we study calculi for which the differentials dx^i constitute a basis
of the space of 1-forms (as a left A-module). These may be regarded as
`deformations' of the ordinary differential calculus on R^n. For n < 4 a
classification of all (orbits under the general linear group of) such calculi
with `constant structure functions' is presented. We analyse whether these
calculi are reducible (i.e., a skew tensor product of lower-dimensional
calculi) or whether they are the extension (as defined in this article) of a
one dimension lower calculus. Furthermore, generalizations to arbitrary n are
obtained for all these calculi.Comment: 33 pages, LaTeX. Revision: A remark about a quasilattice and Penrose
tiling was incorrect in the first version of the paper (p. 14
Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds
We develop exterior calculus approaches for partial differential equations on
radial manifolds. We introduce numerical methods that approximate with spectral
accuracy the exterior derivative , Hodge star , and their
compositions. To achieve discretizations with high precision and symmetry, we
develop hyperinterpolation methods based on spherical harmonics and Lebedev
quadrature. We perform convergence studies of our numerical exterior derivative
operator and Hodge star operator
showing each converge spectrally to and . We show how the
numerical operators can be naturally composed to formulate general numerical
approximations for solving differential equations on manifolds. We present
results for the Laplace-Beltrami equations demonstrating our approach.Comment: 22 pages, 13 figure
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