155,472 research outputs found
Interpolation of nonlinear maps
Let and be complex Banach couples and assume that
with norms satisfying for
some . For any , denote by
and the complex interpolation spaces and by
, the open ball of radius in
, centered at zero. Then for any analytic map such that and
are continuous and bounded by constants and , respectively, the
restriction of to , is
shown to be a map with values in which is analytic and bounded by
Rate of convergence for a Galerkin scheme approximating a two-scale reaction-diffusion system with nonlinear transmission condition
We study a two-scale reaction-diffusion system with nonlinear reaction terms
and a nonlinear transmission condition (remotely ressembling Henry's law) posed
at air-liquid interfaces. We prove the rate of convergence of the two-scale
Galerkin method proposed in Muntean & Neuss-Radu (2009) for approximating this
system in the case when both the microstructure and macroscopic domain are
two-dimensional. The main difficulty is created by the presence of a boundary
nonlinear term entering the transmission condition. Besides using the
particular two-scale structure of the system, the ingredients of the proof
include two-scale interpolation-error estimates, an interpolation-trace
inequality, and improved regularity estimates.Comment: 14 pages, table of content
Fast and accurate clothoid fitting
An effective solution to the problem of Hermite interpolation with a
clothoid curve is provided. At the beginning the problem is naturally
formulated as a system of nonlinear equations with multiple solutions that is
generally difficult to solve numerically. All the solutions of this nonlinear
system are reduced to the computation of the zeros of a single nonlinear
equation. A simple strategy, together with the use of a good and simple guess
function, permits to solve the single nonlinear equation with a few iterations
of the Newton--Raphson method.
The computation of the clothoid curve requires the computation of Fresnel and
Fresnel related integrals. Such integrals need asymptotic expansions near
critical values to avoid loss of precision. This is necessary when, for
example, the solution of interpolation problem is close to a straight line or
an arc of circle. Moreover, some special recurrences are deduced for the
efficient computation of asymptotic expansion.
The reduction of the problem to a single nonlinear function in one variable
and the use of asymptotic expansions make the solution algorithm fast and
robust.Comment: 14 pages, 3 figures, 9 Algorithm Table
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