27,457 research outputs found
Finite element methods for surface PDEs
In this article we consider finite element methods for approximating the solution of partial differential equations on surfaces. We focus on surface finite elements on triangulated surfaces, implicit surface methods using level set descriptions of the surface, unfitted finite element methods and diffuse interface methods. In order to formulate the methods we present the necessary geometric analysis and, in the context of evolving surfaces, the necessary transport formulae. A wide variety of equations and applications are covered. Some ideas of the numerical analysis are presented along with illustrative numerical examples
Finite element error analysis of wave equations with dynamic boundary conditions: estimates
norm error estimates of semi- and full discretisations, using
bulk--surface finite elements and Runge--Kutta methods, of wave equations with
dynamic boundary conditions are studied. The analysis resides on an abstract
formulation and error estimates, via energy techniques, within this abstract
setting. Four prototypical linear wave equations with dynamic boundary
conditions are analysed which fit into the abstract framework. For problems
with velocity terms, or with acoustic boundary conditions we prove surprising
results: for such problems the spatial convergence order is shown to be less
than two. These can also be observed in the presented numerical experiments
Cosmological parameters from large scale structure - geometric versus shape information
The matter power spectrum as derived from large scale structure (LSS) surveys
contains two important and distinct pieces of information: an overall smooth
shape and the imprint of baryon acoustic oscillations (BAO). We investigate the
separate impact of these two types of information on cosmological parameter
estimation, and show that for the simplest cosmological models, the broad-band
shape information currently contained in the SDSS DR7 halo power spectrum (HPS)
is by far superseded by geometric information derived from the baryonic
features. An immediate corollary is that contrary to popular beliefs, the upper
limit on the neutrino mass m_\nu presently derived from LSS combined with
cosmic microwave background (CMB) data does not in fact arise from the possible
small-scale power suppression due to neutrino free-streaming, if we limit the
model framework to minimal LambdaCDM+m_\nu. However, in more complicated
models, such as those extended with extra light degrees of freedom and a dark
energy equation of state parameter w differing from -1, shape information
becomes crucial for the resolution of parameter degeneracies. This conclusion
will remain true even when data from the Planck surveyor become available. In
the course of our analysis, we introduce a new dewiggling procedure that allows
us to extend consistently the use of the SDSS HPS to models with an arbitrary
sound horizon at decoupling. All the cases considered here are compatible with
the conservative 95%-bounds \sum m_\nu < 1.16 eV, N_eff = 4.8 \pm 2.0.Comment: 18 pages, 4 figures; v2: references added, matches published versio
Structure-preserving mesh coupling based on the Buffa-Christiansen complex
The state of the art for mesh coupling at nonconforming interfaces is
presented and reviewed. Mesh coupling is frequently applied to the modeling and
simulation of motion in electromagnetic actuators and machines. The paper
exploits Whitney elements to present the main ideas. Both interpolation- and
projection-based methods are considered. In addition to accuracy and
efficiency, we emphasize the question whether the schemes preserve the
structure of the de Rham complex, which underlies Maxwell's equations. As a new
contribution, a structure-preserving projection method is presented, in which
Lagrange multiplier spaces are chosen from the Buffa-Christiansen complex. Its
performance is compared with a straightforward interpolation based on Whitney
and de Rham maps, and with Galerkin projection.Comment: 17 pages, 7 figures. Some figures are omitted due to a restricted
copyright. Full paper to appear in Mathematics of Computatio
Numerical simulation of conservation laws with moving grid nodes: Application to tsunami wave modelling
In the present article we describe a few simple and efficient finite volume
type schemes on moving grids in one spatial dimension combined with appropriate
predictor-corrector method to achieve higher resolution. The underlying finite
volume scheme is conservative and it is accurate up to the second order in
space. The main novelty consists in the motion of the grid. This new dynamic
aspect can be used to resolve better the areas with large solution gradients or
any other special features. No interpolation procedure is employed, thus
unnecessary solution smearing is avoided, and therefore, our method enjoys
excellent conservation properties. The resulting grid is completely
redistributed according the choice of the so-called monitor function. Several
more or less universal choices of the monitor function are provided. Finally,
the performance of the proposed algorithm is illustrated on several examples
stemming from the simple linear advection to the simulation of complex shallow
water waves. The exact well-balanced property is proven. We believe that the
techniques described in our paper can be beneficially used to model tsunami
wave propagation and run-up.Comment: 46 pages, 7 figures, 7 tables, 94 references. Accepted to
Geosciences. Other author's papers can be downloaded at
http://www.denys-dutykh.com
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