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 $L_\infty$ 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 $L_q$ norm with $q<\infty$ 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 $\alpha$, which parameterizes the rate at which Planck-scale spatial uncertainties (and thephase shifts they produce) may accumulate over large path-lengths. Here $\alpha$ is defined by theexpression for the path-length fluctuations, $\delta \ell$, of a source at distance $\ell$, wherein $\delta \ell \simeq \ell^{1 - \alpha} \ell_P^{\alpha}$, with $\ell_P$ 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 $\alpha \gtrsim 0.58$, which rules out the random walk model (with $\alpha = 1/2$). Much firmer constraints canbe set utilizing detections of quasars at GeV energies with {\em Fermi}, and at TeV energies with ground-based Cherenkovtelescopes: $\alpha \gtrsim 0.67$ and $\alpha \gtrsim 0.72$, respectively. These limits on $\alpha$ seem to rule out $\alpha = 2/3$, 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..