60,986 research outputs found
Finite Size Scaling in 2d Causal Set Quantum Gravity
We study the -dependent behaviour of causal set quantum
gravity. This theory is known to exhibit a phase transition as the analytic
continuation parameter , akin to an inverse temperature, is varied.
Using a scaling analysis we find that the asymptotic regime is reached at
relatively small values of . Focussing on the causal set
action , we find that scales like where
the scaling exponent takes different values on either side of the phase
transition. For we find that which is consistent with
our analytic predictions for a non-continuum phase in the large regime.
For we find that , consistent with a continuum phase of
constant negative curvature thus suggesting a dynamically generated
cosmological constant. Moreover, we find strong evidence that the phase
transition is first order. Our results strongly suggest that the asymptotic
regime is reached in causal set quantum gravity for .Comment: 32 pages, 27 figures (v2 typos and missing reference fixed
Resolving phase transitions with Discontinuous Galerkin methods
We demonstrate the applicability and advantages of Discontinuous Galerkin
(DG) schemes in the context of the Functional Renormalization Group (FRG). We
investigate the -model in the large limit. It is shown that the flow
equation for the effective potential can be cast into a conservative form. We
discuss results for the Riemann problem, as well as initial conditions leading
to a first and second order phase transition. In particular, we unravel the
mechanism underlying first order phase transitions, based on the formation of a
shock in the derivative of the effective potential.Comment: 19 pages, 9 figures, corrected typos, updated references, extended
explanation
Preconditioning of weighted H(div)-norm and applications to numerical simulation of highly heterogeneous media
In this paper we propose and analyze a preconditioner for a system arising
from a finite element approximation of second order elliptic problems
describing processes in highly het- erogeneous media. Our approach uses the
technique of multilevel methods and the recently proposed preconditioner based
on additive Schur complement approximation by J. Kraus (see [8]). The main
results are the design and a theoretical and numerical justification of an
iterative method for such problems that is robust with respect to the contrast
of the media, defined as the ratio between the maximum and minimum values of
the coefficient (related to the permeability/conductivity).Comment: 28 page
A Two-Level Method for Mimetic Finite Difference Discretizations of Elliptic Problems
We propose and analyze a two-level method for mimetic finite difference
approximations of second order elliptic boundary value problems. We prove that
the two-level algorithm is uniformly convergent, i.e., the number of iterations
needed to achieve convergence is uniformly bounded independently of the
characteristic size of the underling partition. We also show that the resulting
scheme provides a uniform preconditioner with respect to the number of degrees
of freedom. Numerical results that validate the theory are also presented
- …