In this paper we prove uniform a priori estimates for transmission problems
with constant coefficients on two subdomains, with a special emphasis for the
case when the ratio between these coefficients is large. In the most part of
the work, the interface between the two subdomains is supposed to be Lipschitz.
We first study a scalar transmission problem which is handled through a
converging asymptotic series. Then we derive uniform a priori estimates for
Maxwell transmission problem set on a domain made up of a dielectric and a
highly conducting material. The technique is based on an appropriate
decomposition of the electric field, whose gradient part is estimated thanks to
the first part. As an application, we develop an argument for the convergence
of an asymptotic expansion as the conductivity tends to infinity.
martes, 6 de octubre de 2009
Uniform estimates for transmission problems with high contrast in heat conduction and electromagnetism. (arXiv:0910.1018v1 [math.NA])
Semi-Lagrangian schemes for linear and fully non-linear diffusion equations. (arXiv:0910.1046v1 [math.NA])
For linear and fully non-linear diffusion equations of
Bellman-Isaacs type, we introduce a class of monotone approximation schemes
relying on monotone interpolation. As opposed to classical numerical methods,
these schemes converge for degenerate diffusion equations having general
non-diagonal dominant coefficient matrices. Such schemes have to have a wide
stencil in general. Besides providing a unifying framework for several known
first order accurate schemes, our class of schemes also includes more efficient
versions, and a new second order scheme that converges only for essentially
monotone solutions. The methods are easy to implement and analyze, and they are
more efficient than some other known schemes. We prove stability and
convergence of the schemes in the general case, and provide error estimates in
the convex case which are robust in the sense that they apply to degenerate
equations and non-smooth solutions. The methods are extensively tested.
Runge-Kutta methods for third order weak approximation of SDEs with multidimensional additive noise. (arXiv:0910.1078v2 [math.NA])
A new class of third order Runge-Kutta methods for stochastic differential
equations with additive noise is introduced. In contrast to Platen's method,
which to the knowledge of the author has been up to now the only known third
order Runge-Kutta scheme for weak approximation, the new class of methods
affords less random variable evaluations and is also applicable to SDEs with
multidimensional noise. Order conditions up to order three are calculated and
coefficients of a four stage third order method are given. This method has
deterministic order four and minimized error constants, and needs in addition
less function evaluations than the method of Platen. Applied to a simple
example, the new method is compared numerically with Platen's method and some
well known second order methods and yields very promising results.
Robust multigrid preconditioners for the high-contrast biharmonic plate equation. (arXiv:0910.0487v1 [math.NA])
We study the high-contrast biharmonic plate equation with HCT and Morley
discretizations. We construct a preconditioner that is robust with respect to
contrast size and mesh size simultaneously based on the preconditioner proposed
by Aksoylu et al. (2008, Comput. Vis. Sci. 11, pp. 319--331). By extending the
devised singular perturbation analysis from linear finite element
discretization to the above discretizations, we prove and numerically
demonstrate the robustness of the preconditioner. Therefore, we accomplish a
desirable preconditioning design goal by using the same family of
preconditioners to solve elliptic family of PDEs with varying discretizations.
We also present a strategy on how to generalize the proposed preconditioner to
cover high-contrast elliptic PDEs of order $2k, k>2$. Moreover, we prove a
fundamental qualitative property of solution of the high-contrast biharmonic
plate equation. Namely, the solution over the highly-bending island becomes a
linear polynomial asymptotically. The effectiveness of our preconditioner is
largely due to the integration of this qualitative understanding of the
underlying PDE into its construction.
Determining nodes for semilinear parabolic equations. (arXiv:0910.0595v1 [math.AP])
We discuss the uniqueness of the equilibria of time-global solutions of
general semilinear parabolic equations by a finite set of values of these
solutions. More precisely, if the asymptotic behaviour of a time-global
solution is known on an appropriate finite set, then the asymptotic behaviour
of a time-global solution itself is entirely determined in a domain.
Numerical Implementation of Streaming Down the Gradient: Application to Fluid Modeling of Cosmic Rays. (arXiv:0909.5426v1 [astro-ph.HE] CROSS LISTED)
The equation governing the streaming of a quantity down its gradient
superficially looks similar to the simple constant velocity advection equation.
In fact, it is the same as an advection equation if there are no local extrema
in the computational domain or at the boundary. However, in general when there
are local extrema in the computational domain it is a non-trivial nonlinear
equation. The standard upwind time evolution with a CFL-limited time step
results in spurious oscillations at the grid scale. These oscillations, which
originate at the extrema, propagate throughout the computational domain and are
undamped even at late times. These oscillations arise because of unphysically
large fluxes leaving (entering) the maxima (minima) with the standard
CFL-limited explicit methods. Regularization of the equation shows that it is
diffusive at the extrema; because of this, an explicit method for the
regularized equation with $Delta t propto Delta x^2$ behaves fine. We show
that the implicit methods show stable and converging results with $Delta t
propto Delta x$; however, surprisingly, even implicit methods are not stable
with large enough timesteps. In addition to these subtleties in the numerical
implementation, the solutions to the streaming equation are quite novel:
non-differentiable solutions emerge from initially smooth profiles; the
solutions show transport over large length scales, e.g., in form of tails. The
fluid model for cosmic rays interacting with a thermal plasma (valid at space
scales much larger than the cosmic ray Larmor radius) is similar to the
equation for streaming of a quantity down its gradient, so our method will find
applications in fluid modeling of cosmic rays.
Numerical approach to Schrodinger equation in momentum space. (arXiv:0910.0574v1 [physics.comp-ph])
Time-independent Schrodinger equation in real-space is an indispensable
element of introductory quantum mechanics. In contrast, Schrodinger equation in
momentum space, being an integral equation not readily amenable to an
analytical solution, is a rarity in the quantum mechanics pedagogy. In this
tutorial, we present a numerical approach to the Schrodinger equation in
momentum space. After a suitable discretization process, we obtain the
Hamiltonian matrix that is numerically diagonalized. By considering a few
examples, we show that this method is ideal for exploring bound-states in a
localized potential in one or higher dimensions, and it complements the
traditional (analytical or numerical) treatment of Schrodinger equation in
real-space.
A Note on the Finite Variance of the Averaging Function for Polynomial System Solving
with random initial input (in a sense to be described below) was proven to be finite, and even polynomial in the size of the
input. In this paper, we prove that some other higher moments are also finite. In particular, we show that the variance is
polynomial in the size of the input.
- Content Type Journal Article
- DOI 10.1007/s10208-009-9054-4
- Authors
- Carlos Beltrán, Universidad de Cantabria Departamento de Matemáticas, Estadística y Computación Santander Spain
- Michael Shub, University of Toronto Department of Mathematics Toronto Ontario Canada M5S 2E4
- Journal Foundations of Computational Mathematics
- Online ISSN 1615-3383
- Print ISSN 1615-3375