Computable a-posteriori error estimates for finite element solutions are derived in an asymptotic form for h 0 where h measures the size of the elements. The approach has similarity to the residual method but differs from it in the use of norms of negative Sobolev spaces corresponding to the given bilinear (energy) form. For clarity the presentation is restricted to one-dimensional model problems. More specifically, the source, eigenvalue, and parabolic problems are considered involving a linear, self-adjoint operator of the second order. Generalizations to more general one-dimensional problems are straightforward, and the results also extend to higher space dimensions; but this involves some additional considerations. The estimates can be used for a practical a-posteriori assessment of the accuracy of a computed finite element solution, and they provide a basis for the design of adaptive finite element solvers.

A‐posteriori error estimates for the finite element method

http://www.icranet.org/misc/Scientific_Report_2009/Reports/08Xue_b.pdf

Electron-positron pairs in physics and astrophysics: from heavy nuclei to black holes

From the interaction of physics and astrophysics we are witnessing in these years a splendid synthesis of theoretical, experimental and observational results originating from three fundametal physical processes. They were originally proposed by Dirac, by Breit and Wheeler and by Sauter, Heisenberg, Euler and Schwinger. The vacuum polarization process in strong electromagnetic field, pioneered by Sauter, Heisenberg, Euler and Schwinger, introduced the concept of critical electric field. It has been searched without success for more than forty years by heavy-ion collisions in many of the leading particle accelerators worldwide. The novel situation today is that these same processes can be studied on a much more grandiose scale during the gravitational collapse leading to the formation of a black hole being observed in Gamma Ray Bursts. This report is dedicated to the scientific race in act. The theoretical and experimental work developed in Earth-based laboratories is confronted with the theoretical interpretation of space-based observations of phenomena originating on cosmological scales. What has become clear in the last ten years is that all the three above mentioned processes, duly extended in the general relativistic framework, are necessary for the understanding of the physics of the gravitational collapse to a black hole. Vice versa, the natural arena where these processes can be observed in mutual interaction and on an unprecedented scale, is indeed the realm of relativistic astrophysics.

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A nonlinear analysis, based on lubrication theory, is presented for the adjustment under surface tension of an initially uniform annular film of viscous fluid confined within a circular cylindrical pipe The film surrounds a thread of another viscous fluid Small axisymmetric interfacial disturbances of sufficiently long wavelength are found to grow, leading to the break-up of the initially continuous outer film into a number of isolated rings of fixed length on the pipe wall The implications for the rupture of fluid threads surrounded by moderately thin films in confined geometries are discussed

Nonlinear adjustment of a thin annular film of viscous fluid surrounding a thread of another within a circular cylindrical pipe

The first step in investigating the dynamics of a continuous-time system described by an ordinary differential equation is to integrate to obtain trajectories. In this paper, we attempt to elucidate the dynamics of the most commonly used family of numerical integration schemes, Runge–Kutta methods, by the application of the techniques of dynamical systems theory to the maps produced in the numerical analysis.

/pdf/the-dynamics-of-runge-kutta-methods-46up4c1ad9.pdf

The dynamics of runge–kutta methods

Modern numerical methods for ordinary differential equations

An internetworking system operating over an ATM backbone. The physical internetworking devices within the system are shared to provide the internetworking functions while servicing two or more distinct and isolated user networks. This is accomplished by logically partitioning the devices into distinct sub-elements which provide all or part of the internetworking functions. These sub-elements are uniquely allocated to independent realms which are then assigned to specific user networks.

Grant Hall

Papers

Modern numerical methods for ordinary differential equations

Method and apparatus for forwarding packets