The Penalized Lebesgue Constant for Surface Spline Interpolation. (arXiv:0911.1815v1 [math.CA])

Problems involving approximation from scattered data where data is arranged
quasi-uniformly have been treated by RBF methods for decades. Treating data
with spatially varying density has not been investigated with the same
intensity, and is far less well understood. In this article we consider the
stability of surface spline interpolation (a popular type of RBF interpolation)
for data with nonuniform arrangements. Using techniques similar to those
recently employed by Hangelbroek, Narcowich and Ward to demonstrate the
stability of interpolation from quasi-uniform data on manifolds, we show that
surface spline interpolation on R^d is stable, but in a stronger, local sense.
We also obtain pointwise estimates showing that the Lagrange function decays
very rapidly, and at a rate determined by the local spacing of datasites. These
results, in conjunction with a Lebesgue lemma, show that surface spline
interpolation enjoys the same rates of convergence as those of the local
approximation schemes recently developed by DeVore and Ron.

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Original source : http://arxiv.org/abs/0911.1815...

A convergent mixed method for the Stokes approximation of viscous compressible flow. (arXiv:0911.1870v1 [math.NA])

We propose a mixed finite element method for the motion of a strongly
viscous, ideal, and isentropic gas. At the boundary we impose a Navier-slip
condition such that the velocity equation can be posed in mixed form with the
vorticity as an auxiliary variable. In this formulation we design a finite
element method, where the velocity and vorticity is approximated with the div-
and curl- conforming Nedelec elements, respectively, of the first order and
first kind. The mixed scheme is coupled to a standard piecewise constant upwind
discontinuous Galerkin discretization of the continuity equation. For the time
discretization, implicit Euler time stepping is used. Our main result is that
the numerical solution converges to a weak solution as the discretization
parameters go to zero. The convergence analysis is inspired by the continuous
analysis of Feireisl and Lions for the compressible Navier-Stokes equations.
Tools used in the analysis include an equation for the effective viscous flux
and various renormalizations of the density scheme.

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Original source : http://arxiv.org/abs/0911.1870...