138 research outputs found
Analytic regularity for a singularly perturbed system of reaction-diffusion equations with multiple scales: proofs
We consider a coupled system of two singularly perturbed reaction-diffusion
equations, with two small parameters , each
multiplying the highest derivative in the equations. The presence of these
parameters causes the solution(s) to have \emph{boundary layers} which overlap
and interact, based on the relative size of and . We
construct full asymptotic expansions together with error bounds that cover the
complete range . For the present case of analytic
input data, we derive derivative growth estimates for the terms of the
asymptotic expansion that are explicit in the perturbation parameters and the
expansion order
On the suboptimality of the p-version interior penalty discontinuous Galerkin method
We address the question of the rates of convergence of the p-version interior penalty discontinuous Galerkin method (p-IPDG) for second order elliptic problems with non-homogeneous Dirichlet boundary conditions. It is known that the p-IPDG method admits slightly suboptimal a-priori bounds with respect to the polynomial degree (in the Hilbertian Sobolev space setting). An example for which the
suboptimal rate of convergence with respect to the polynomial degree is both proven theoretically and
validated in practice through numerical experiments is presented. Moreover, the performance of p-
IPDG on the related problem of p-approximation of corner singularities is assessed both theoretically and numerically, witnessing an almost doubling of the convergence rate of the p-IPDG method
Multiscale Partition of Unity
We introduce a new Partition of Unity Method for the numerical homogenization
of elliptic partial differential equations with arbitrarily rough coefficients.
We do not restrict to a particular ansatz space or the existence of a finite
element mesh. The method modifies a given partition of unity such that optimal
convergence is achieved independent of oscillation or discontinuities of the
diffusion coefficient. The modification is based on an orthogonal decomposition
of the solution space while preserving the partition of unity property. This
precomputation involves the solution of independent problems on local
subdomains of selectable size. We deduce quantitative error estimates for the
method that account for the chosen amount of localization. Numerical
experiments illustrate the high approximation properties even for 'cheap'
parameter choices.Comment: Proceedings for Seventh International Workshop on Meshfree Methods
for Partial Differential Equations, 18 pages, 3 figure
The upper triangular solutions to the three-state constant quantum Yang-Baxter equation
In this article we present all nonsingular upper triangular solutions to the
constant quantum Yang-Baxter equation
in the three state
case, i.e. all indices ranging from 1 to 3. The upper triangular ansatz implies
729 equations for 45 variables. Fortunately many of the equations turned out to
be simple allowing us to start breaking the problem into smaller ones. In the
end we had a total of 552 solutions, but many of them were either inherited
from two-state solutions or subcases of others. The final list contains 35
nontrivial solutions, most of them new.Comment: 24 Pages in LaTe
Variational Multiscale Stabilization and the Exponential Decay of Fine-scale Correctors
This paper addresses the variational multiscale stabilization of standard
finite element methods for linear partial differential equations that exhibit
multiscale features. The stabilization is of Petrov-Galerkin type with a
standard finite element trial space and a problem-dependent test space based on
pre-computed fine-scale correctors. The exponential decay of these correctors
and their localisation to local cell problems is rigorously justified. The
stabilization eliminates scale-dependent pre-asymptotic effects as they appear
for standard finite element discretizations of highly oscillatory problems,
e.g., the poor approximation in homogenization problems or the pollution
effect in high-frequency acoustic scattering
On thin plate spline interpolation
We present a simple, PDE-based proof of the result [M. Johnson, 2001] that
the error estimates of [J. Duchon, 1978] for thin plate spline interpolation
can be improved by . We illustrate that -matrix
techniques can successfully be employed to solve very large thin plate spline
interpolation problem
Permutation-type solutions to the Yang-Baxter and other n-simplex equations
We study permutation type solutions to n-simplex equations, that is,
solutions whose R matrix can be written as a product of delta- functions
depending linearly on the indices. With this ansatz the D^{n(n+1)} equations of
the n-simplex equation reduce to an [n(n+1)/2+1]x[n(n+1)/2+1] matrix equation
over Z_D. We have completely analyzed the 2-, 3- and 4-simplex equations in the
generic D case. The solutions show interesting patterns that seem to continue
to still higher simplex equations.Comment: 20 pages, LaTeX2e. to appear in J. Phys. A: Math. Gen. (1997
A Family of Nonlinear Fourth Order Equations of Gradient Flow Type
Global existence and long-time behavior of solutions to a family of nonlinear
fourth order evolution equations on are studied. These equations
constitute gradient flows for the perturbed information functionals with
respect to the -Wasserstein metric. The value of ranges from
, corresponding to a simplified quantum drift diffusion model, to
, corresponding to a thin film type equation.Comment: 33 pages, no figure
A three-scale domain decomposition method for the 3D analysis of debonding in laminates
The prediction of the quasi-static response of industrial laminate structures
requires to use fine descriptions of the material, especially when debonding is
involved. Even when modeled at the mesoscale, the computation of these
structures results in very large numerical problems. In this paper, the exact
mesoscale solution is sought using parallel iterative solvers. The LaTIn-based
mixed domain decomposition method makes it very easy to handle the complex
description of the structure; moreover the provided multiscale features enable
us to deal with numerical difficulties at their natural scale; we present the
various enhancements we developed to ensure the scalability of the method. An
extension of the method designed to handle instabilities is also presented
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