7,032 research outputs found
Discretization of Linear Problems in Banach Spaces: Residual Minimization, Nonlinear Petrov-Galerkin, and Monotone Mixed Methods
This work presents a comprehensive discretization theory for abstract linear
operator equations in Banach spaces. The fundamental starting point of the
theory is the idea of residual minimization in dual norms, and its inexact
version using discrete dual norms. It is shown that this development, in the
case of strictly-convex reflexive Banach spaces with strictly-convex dual,
gives rise to a class of nonlinear Petrov-Galerkin methods and, equivalently,
abstract mixed methods with monotone nonlinearity. Crucial in the formulation
of these methods is the (nonlinear) bijective duality map.
Under the Fortin condition, we prove discrete stability of the abstract
inexact method, and subsequently carry out a complete error analysis. As part
of our analysis, we prove new bounds for best-approximation projectors, which
involve constants depending on the geometry of the underlying Banach space. The
theory generalizes and extends the classical Petrov-Galerkin method as well as
existing residual-minimization approaches, such as the discontinuous
Petrov-Galerkin method.Comment: 43 pages, 2 figure
Whipping Instabilities in Electrified Liquid Jets
A liquid jet may develop different types of instabilities, like the so-called
Rayleigh-Plateau instability, which breaks the jet into droplets. However,
another type of instabilities may appear when we electrify a liquid jet and
induce some charge at his surface. Among them, the most common is the so-called
Whipping Instability, which is characterized by violent and fast lashes of the
jet. In the submitted fluid dynamic video(see
http://hdl.handle.net/1813/11422), we will show an unstable charged glycerine
jet in a dielectric liquid bath, which permits an enhanced visualization of the
instability. For this reason, it is probably the first time that these
phenomena are visualized with enough clarity to analyze features as the effect
of the feeding liquid flow rate through the jet or as the surprising
spontaneous stabilization at some critical distance to the ground electrode.Comment: 3 pages, no figures, links to videos, Submission to the 26th Gallery
of Fluid Motion (2009
Extended patchy ecosystems may increase their total biomass through self-replication
Patches of vegetation consist of dense clusters of shrubs, grass, or trees,
often found to be circular characteristic size, defined by the properties of
the vegetation and terrain. Therefore, vegetation patches can be interpreted as
localized structures. Previous findings have shown that such localized
structures can self-replicate in a binary fashion, where a single vegetation
patch elongates and divides into two new patches. Here, we extend these
previous results by considering the more general case, where the plants
interact non-locally, this extension adds an extra level of complexity and
shrinks the gap between the model and real ecosystems, where it is known that
the plant-to-plant competition through roots and above-ground facilitating
interactions have non-local effects, i.e. they extend further away than the
nearest neighbor distance. Through numerical simulations, we show that for a
moderate level of aridity, a transition from a single patch to periodic pattern
occurs. Moreover, for large values of the hydric stress, we predict an opposing
route to the formation of periodic patterns, where a homogeneous cover of
vegetation may decay to spot-like patterns. The evolution of the biomass of
vegetation patches can be used as an indicator of the state of an ecosystem,
this allows to distinguish if a system is in a self-replicating or decaying
dynamics. In an attempt to relate the theoretical predictions to real
ecosystems, we analyze landscapes in Zambia and Mozambique, where vegetation
forms patches of tens of meters in diameter. We show that the properties of the
patches together with their spatial distributions are consistent with the
self-organization hypothesis. We argue that the characteristics of the observed
landscapes may be a consequence of patch self-replication, however, detailed
field and temporal data is fundamental to assess the real state of the
ecosystems.Comment: 38 pages, 12 figures, 1 tabl
Critical Behavior in the Gravitational Collapse of a Scalar Field with Angular Momentum in Spherical Symmetry
We study the critical collapse of a massless scalar field with angular
momentum in spherical symmetry. In order to mimic the effects of angular
momentum we perform a sum of the stress-energy tensors for all the scalar
fields with the same eigenvalue, l, of the angular momentum operator and
calculate the equations of motion for the radial part of these scalar fields.
We have found that the critical solutions for different values of l are
discretely self-similar (as in the original l=0 case). The value of the
discrete, self-similar period, Delta_l, decreases as l increases in such a way
that the critical solution appears to become periodic in the limit. The mass
scaling exponent, gamma_l, also decreases with l.Comment: 10 pages, 8 figure
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