58 research outputs found
Multigrid applied to singular perturbation problems
The solution of the singular perturbation problem by a multigrid algorithm is considered. Theoretical and experimental results for a number of different discretizations are presented. The theoretical and observed rates agree with the results developed in an earlier work of Kamowitz and Parter. In addition, the rate of convergence of the algorithm when the coarse grid operator is the natural finite difference analog of the fine grid operator is presented. This is in contrast to the case in the previous work where the Galerkin choice (I sup H sub h L sub h,I sup h sub H) was used for the coarse grid operators
Endomorphisms of Banach algebras of infinitely differentiable functions on compact plane sets
In this note we study the endomorphisms of certain Banach algebras of
infinitely differentiable functions on compact plane sets, associated with
weight sequences M. These algebras were originally studied by Dales, Davie and
McClure.
In a previous paper this problem was solved in the case of the unit interval
for many weights M. Here we investigate the extent to which the methods used
previously apply to general compact plane sets, and introduce some new methods.
In particular, we obtain many results for the case of the closed unit disc.
This research was supported by EPSRC grant GR/M31132Comment: 15 pages LaTe
Some observations on boundary conditions for numerical conservation laws
Four choices of outflow boundary conditions are considered for numerical conservation laws. All four methods are stable for linear problems, for which examples are presented where either a boundary layer forms or the numerical scheme, together with the boundary condition, is unstable due to the formation of a reflected shock. A simple heuristic argument is presented for determining the suitability of the boundary condition
Approximately Vanishing of Topological Cohomology Groups
In this paper, we establish the Pexiderized stability of coboundaries and
cocycles and use them to investigate the Hyers--Ulam stability of some
functional equations. We prove that for each Banach algebra , Banach
-bimodule and positive integer if and only if the -th
cohomology group approximately vanishes.Comment: 18 pages, minor correction
The Julia sets and complex singularities in hierarchical Ising models
We study the analytical continuation in the complex plane of free energy of
the Ising model on diamond-like hierarchical lattices. It is known that the
singularities of free energy of this model lie on the Julia set of some
rational endomorphism related to the action of the Migdal-Kadanoff
renorm-group. We study the asymptotics of free energy when temperature goes
along hyperbolic geodesics to the boundary of an attractive basin of . We
prove that for almost all (with respect to the harmonic measure) geodesics the
complex critical exponent is common, and compute it
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