3,322 research outputs found
The gradient discretisation method for linear advection problems
We adapt the Gradient Discretisation Method (GDM), originally designed for
elliptic and parabolic partial differential equations, to the case of a linear
scalar hyperbolic equations. This enables the simultaneous design and
convergence analysis of various numerical schemes, corresponding to the methods
known to be GDMs, such as finite elements (conforming or non-conforming,
standard or mass-lumped), finite volumes on rectangular or simplicial grids,
and other recent methods developed for general polytopal meshes. The scheme is
of centred type, with added linear or non-linear numerical diffusion. We
complement the convergence analysis with numerical tests based on the
mass-lumped P1 conforming and non conforming finite element and on the hybrid
finite volume method
A unified analysis of elliptic problems with various boundary conditions and their approximation
We design an abstract setting for the approximation in Banach spaces of
operators acting in duality. A typical example are the gradient and divergence
operators in Lebesgue--Sobolev spaces on a bounded domain. We apply this
abstract setting to the numerical approximation of Leray-Lions type problems,
which include in particular linear diffusion. The main interest of the abstract
setting is to provide a unified convergence analysis that simultaneously covers
(i) all usual boundary conditions, (ii) several approximation methods. The
considered approximations can be conforming, or not (that is, the approximation
functions can belong to the energy space of the problem, or not), and include
classical as well as recent numerical schemes. Convergence results and error
estimates are given. We finally briefly show how the abstract setting can also
be applied to other models, including flows in fractured medium, elasticity
equations and diffusion equations on manifolds. A by-product of the analysis is
an apparently novel result on the equivalence between general Poincar{\'e}
inequalities and the surjectivity of the divergence operator in appropriate
spaces
A unified approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume methods
We investigate the connections between several recent methods for the
discretization of anisotropic heterogeneous diffusion operators on general
grids. We prove that the Mimetic Finite Difference scheme, the Hybrid Finite
Volume scheme and the Mixed Finite Volume scheme are in fact identical up to
some slight generalizations. As a consequence, some of the mathematical results
obtained for each of the method (such as convergence properties or error
estimates) may be extended to the unified common framework. We then focus on
the relationships between this unified method and nonconforming Finite Element
schemes or Mixed Finite Element schemes, obtaining as a by-product an explicit
lifting operator close to the ones used in some theoretical studies of the
Mimetic Finite Difference scheme. We also show that for isotropic operators, on
particular meshes such as triangular meshes with acute angles, the unified
method boils down to the well-known efficient two-point flux Finite Volume
scheme
Non-adiabatic oscillations of fast-rotating stars: the example of Rasalhague
Early-type stars generally tend to be fast rotators. In these stars, mode
identification is very challenging as the effects of rotation are not well
known. We consider here the example of Ophiuchi, for which dozens of
oscillation frequencies have been measured. We model the star using the
two-dimensional structure code ESTER, and we compute both adiabatic and
non-adiabatic oscillations using the TOP code. Both calculations yield very
complex spectra, and we used various diagnostic tools to try and identify the
observed pulsations. While we have not reached a satisfactory mode-to-mode
identification, this paper presents promising early results.Comment: 4 pages, 3 figures. SF2A 2017 proceeding
Sequential resonant tunneling in quantum cascade lasers
A model of sequential resonant tunneling transport between two-dimensional
subbands that takes into account explicitly elastic scattering is investigated.
It is compared to transport measurements performed on quantum cascade lasers
where resonant tunneling processes are known to be dominating. Excellent
agreement is found between experiment and theory over a large range of current,
temperature and device structures
Gravitational duality near de Sitter space
Gravitational instantons ''Lambda-instantons'' are defined here for any given
value Lambda of the cosmological constant. A multiple of the Euler
characteristic appears as an upper bound for the de Sitter action and as a
lower bound for a family of quadratic actions. The de Sitter action itself is
found to be equivalent to a simple and natural quadratic action. In this paper
we also describe explicitly the reparameterization and duality invariances of
gravity (in 4 dimensions) linearized about de Sitter space. A noncovariant
doubling of the fields using the Hamiltonian formalism leads to first order
time evolution with manifest duality symmetry. As a special case we recover the
linear flat space result of Henneaux and Teitelboim by a smooth limiting
process.Comment: 13 pages, no figure - v2 contains only small redactional changes (one
reference added) and is essentially the published versio
On the "Causality Argument" in Bouncing Cosmologies
We exhibit a situation in which cosmological perturbations of astrophysical
relevance propagating through a bounce are affected in a scale-dependent way.
Involving only the evolution of a scalar field in a closed universe described
by general relativity, the model is consistent with causality. Such a specific
counter-example leads to the conclusion that imposing causality is not
sufficient to determine the spectrum of perturbations after a bounce provided
it is known before. We discuss consequences of this result for string motivated
scenarios.Comment: 4 pages, 1 figure, ReVTeX, to appear in Phys. Rev. Let
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