1,856 research outputs found
Adaptive Finite Element Methods for Elliptic Problems with Discontinuous Coefficients
Elliptic partial differential equations (PDEs) with discontinuous diffusion
coefficients occur in application domains such as diffusions through porous
media, electro-magnetic field propagation on heterogeneous media, and diffusion
processes on rough surfaces. The standard approach to numerically treating such
problems using finite element methods is to assume that the discontinuities lie
on the boundaries of the cells in the initial triangulation. However, this does
not match applications where discontinuities occur on curves, surfaces, or
manifolds, and could even be unknown beforehand. One of the obstacles to
treating such discontinuity problems is that the usual perturbation theory for
elliptic PDEs assumes bounds for the distortion of the coefficients in the
norm and this in turn requires that the discontinuities are matched
exactly when the coefficients are approximated. We present a new approach based
on distortion of the coefficients in an norm with which
therefore does not require the exact matching of the discontinuities. We then
use this new distortion theory to formulate new adaptive finite element methods
(AFEMs) for such discontinuity problems. We show that such AFEMs are optimal in
the sense of distortion versus number of computations, and report insightful
numerical results supporting our analysis.Comment: 24 page
Approximation by Rational Functions
Making use of the Hardy-Littlewood maximal function, we give a new proof of the following theorem of Pekarski: If f\u27 is in L log L on a finite interval, then f can be approximated in the uniform norm by rational functions of degree n to an error 0(1/n) on that interval
Power-Law Statistics Of Driven Reconnection In The Magnetically Closed Corona
Numerous observations have revealed that power-law distributions are
ubiquitous in energetic solar processes. Hard X-rays, soft X-rays, extreme
ultraviolet radiation, and radio waves all display power-law frequency
distributions. Since magnetic reconnection is the driving mechanism for many
energetic solar phenomena, it is likely that reconnection events themselves
display such power-law distributions. In this work, we perform numerical
simulations of the solar corona driven by simple convective motions at the
photospheric level. Using temperature changes, current distributions, and
Poynting fluxes as proxies for heating, we demonstrate that energetic events
occurring in our simulation display power-law frequency distributions, with
slopes in good agreement with observations. We suggest that the
braiding-associated reconnection in the corona can be understood in terms of a
self-organized criticality model driven by convective rotational motions
similar to those observed at the photosphere.Comment: Accepted by Ap
New Constraints on Quantum Gravity from X-ray and Gamma-Ray Observations
One aspect of the quantum nature of spacetime is its "foaminess" at very
small scales. Many models for spacetime foam are defined by the accumulation
power , which parameterizes the rate at which Planck-scale spatial
uncertainties (and thephase shifts they produce) may accumulate over large
path-lengths. Here is defined by theexpression for the path-length
fluctuations, , of a source at distance , wherein , with being the Planck
length. We reassess previous proposals to use astronomical observations
ofdistant quasars and AGN to test models of spacetime foam. We show explicitly
how wavefront distortions on small scales cause the image intensity to decay to
the point where distant objects become undetectable when the path-length
fluctuations become comparable to the wavelength of the radiation. We use X-ray
observations from {\em Chandra} to set the constraint ,
which rules out the random walk model (with ). Much firmer
constraints canbe set utilizing detections of quasars at GeV energies with {\em
Fermi}, and at TeV energies with ground-based Cherenkovtelescopes: and , respectively. These limits on
seem to rule out , the model of some physical interest.Comment: 11 pages, 9 figures, ApJ, in pres
The Averaging Lemma
Averaging lemmas deduce smoothess of velocity averages, such as f(x) := Z f(x; v) dv; IR d ; from properties of f . A canonical example is that f is in the Sobolev space W 1=2 (L 2 (IR d )) whenever f and g(x; v) := v r x f(x; v) are in L 2 (IR d 4 The present paper shows how techniques from Harmonic Analysis such as maximal functions wavelet decompositions and interpolation can be used to prove L p versions of the averaging lemma. For example, it is shown that f; g 2 L p (IR d implies that f is in the Besov space B s p (L p (IR d )), s := min(1=p; 1=p 0 ). Examples are constructed using wavelet decompositions to show that these averaging lemmas are sharp. A deeper analysis of the averaging lemma is made near the endpoint p = 1. AMS subject classication: 35L60, 35L65, 35B65, 46B70, 46B45, 42B25. Key Words: averaging lemma, regularity, transport equations, Besov spaces 1 Introduction Averaging lemmas arise in the study of regularity of solut..
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