1,117 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
Boundary conditions for the Einstein-Christoffel formulation of Einstein's equations
Specifying boundary conditions continues to be a challenge in numerical
relativity in order to obtain a long time convergent numerical simulation of
Einstein's equations in domains with artificial boundaries. In this paper, we
address this problem for the Einstein--Christoffel (EC) symmetric hyperbolic
formulation of Einstein's equations linearized around flat spacetime. First, we
prescribe simple boundary conditions that make the problem well posed and
preserve the constraints. Next, we indicate boundary conditions for a system
that extends the linearized EC system by including the momentum constraints and
whose solution solves Einstein's equations in a bounded domain
Mathematicians take a stand
We survey the reasons for the ongoing boycott of the publisher Elsevier. We
examine Elsevier's pricing and bundling policies, restrictions on dissemination
by authors, and lapses in ethics and peer review, and we conclude with thoughts
about the future of mathematical publishing.Comment: 5 page
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
Nonconforming tetrahedral mixed finite elements for elasticity
This paper presents a nonconforming finite element approximation of the space
of symmetric tensors with square integrable divergence, on tetrahedral meshes.
Used for stress approximation together with the full space of piecewise linear
vector fields for displacement, this gives a stable mixed finite element method
which is shown to be linearly convergent for both the stress and displacement,
and which is significantly simpler than any stable conforming mixed finite
element method. The method may be viewed as the three-dimensional analogue of a
previously developed element in two dimensions. As in that case, a variant of
the method is proposed as well, in which the displacement approximation is
reduced to piecewise rigid motions and the stress space is reduced accordingly,
but the linear convergence is retained.Comment: 13 pages, 2 figure
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