763 research outputs found
Finite element differential forms on cubical meshes
We develop a family of finite element spaces of differential forms defined on
cubical meshes in any number of dimensions. The family contains elements of all
polynomial degrees and all form degrees. In two dimensions, these include the
serendipity finite elements and the rectangular BDM elements. In three
dimensions they include a recent generalization of the serendipity spaces, and
new H(curl) and H(div) finite element spaces. Spaces in the family can be
combined to give finite element subcomplexes of the de Rham complex which
satisfy the basic hypotheses of the finite element exterior calculus, and hence
can be used for stable discretization of a variety of problems. The
construction and properties of the spaces are established in a uniform manner
using finite element exterior calculus.Comment: v2: as accepted by Mathematics of Computation after minor revisions;
v3: this version corresponds to the final version for Math. Comp., after
copyediting and galley proof
Finite element differential forms on curvilinear cubic meshes and their approximation properties
We study the approximation properties of a wide class of finite element
differential forms on curvilinear cubic meshes in n dimensions. Specifically,
we consider meshes in which each element is the image of a cubical reference
element under a diffeomorphism, and finite element spaces in which the shape
functions and degrees of freedom are obtained from the reference element by
pullback of differential forms. In the case where the diffeomorphisms from the
reference element are all affine, i.e., mesh consists of parallelotopes, it is
standard that the rate of convergence in L2 exceeds by one the degree of the
largest full polynomial space contained in the reference space of shape
functions. When the diffeomorphism is multilinear, the rate of convergence for
the same space of reference shape function may degrade severely, the more so
when the form degree is larger. The main result of the paper gives a sufficient
condition on the reference shape functions to obtain a given rate of
convergence.Comment: 17 pages, 1 figure; v2: changes in response to referee reports; v3:
minor additional changes, this version accepted for Numerische Mathematik;
v3: very minor updates, this version corresponds to the final published
versio
Spaces of finite element differential forms
We discuss the construction of finite element spaces of differential forms
which satisfy the crucial assumptions of the finite element exterior calculus,
namely that they can be assembled into subcomplexes of the de Rham complex
which admit commuting projections. We present two families of spaces in the
case of simplicial meshes, and two other families in the case of cubical
meshes. We make use of the exterior calculus and the Koszul complex to define
and understand the spaces. These tools allow us to treat a wide variety of
situations, which are often treated separately, in a unified fashion.Comment: To appear in: Analysis and Numerics of Partial Differential
Equations, U. Gianazza, F. Brezzi, P. Colli Franzone, and G. Gilardi, eds.,
Springer 2013. v2: a few minor typos corrected. v3: a few more typo
correction
A simplicial gauge theory
We provide an action for gauge theories discretized on simplicial meshes,
inspired by finite element methods. The action is discretely gauge invariant
and we give a proof of consistency. A discrete Noether's theorem that can be
applied to our setting, is also proved.Comment: 24 pages. v2: New version includes a longer introduction and a
discrete Noether's theorem. v3: Section 4 on Noether's theorem has been
expanded with Proposition 8, section 2 has been expanded with a paragraph on
standard LGT. v4: Thorough revision with new introduction and more background
materia
Trimmed Serendipity Finite Element Differential Forms
We introduce the family of trimmed serendipity finite element differential
form spaces, defined on cubical meshes in any number of dimensions, for any
polynomial degree, and for any form order. The relation between the trimmed
serendipity family and the (non-trimmed) serendipity family developed by Arnold
and Awanou [Math. Comp. 83(288) 2014] is analogous to the relation between the
trimmed and (non-trimmed) polynomial finite element differential form families
on simplicial meshes from finite element exterior calculus. We provide degrees
of freedom in the general setting and prove that they are unisolvent for the
trimmed serendipity spaces. The sequence of trimmed serendipity spaces with a
fixed polynomial order r provides an explicit example of a system described by
Christiansen and Gillette [ESAIM:M2AN 50(3) 2016], namely, a minimal compatible
finite element system on squares or cubes containing order r-1 polynomial
differential forms.Comment: Improved results, detailed comparison to prior and contemporary work,
and further explanation of computational benefits have been added since the
original version. This version has been accepted for publication in
Mathematics of Computatio
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
Nonstandard finite element de Rham complexes on cubical meshes
We propose two general operations on finite element differential complexes on
cubical meshes that can be used to construct and analyze sequences of
"nonstandard" convergent methods. The first operation, called DoF-transfer,
moves edge degrees of freedom to vertices in a way that reduces global degrees
of freedom while increasing continuity order at vertices. The second operation,
called serendipity, eliminates interior bubble functions and degrees of freedom
locally on each element without affecting edge degrees of freedom. These
operations can be used independently or in tandem to create nonstandard
complexes that incorporate Hermite, Adini and "trimmed-Adini" elements. We show
that the resulting elements provide convergent, non-conforming methods for
problems requiring stronger regularity and satisfy a discrete Korn inequality.
We discuss potential benefits of applying these elements to Stokes, biharmonic
and elasticity problems.Comment: 31 page
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