Flexible boundary-condition methods couple an isolated defect to bulk through the bulk lattice Green's function. Direct computation of the lattice Green's function requires projecting out the singular subspace of uniform displacements and forces for the infinite lattice. We calculate the convergence rates for elastically isotropic and anisotropic cases for three different techniques: relative d...

It is well known now that kernel density estimators are not consistent when estimating a density near the finite end points of the support of the density to be estimated. This is due to boundary effects that occur in nonparametric curve estimation problems. A number of proposals have been made in the kernel density estimation context with some success. As of yet there appears to be no single do...

The original version of this article unfortunately contained a formatting mistake in tables and figures placement.

We consider the hydrodynamic limit of a stationary Boltzmann equation in a unit plate with in-flow boundary. The classical theory claims that the solution can be approximated by the sum of interior solution which satisfies steady incompressible Navier-Stokes-Fourier system, and boundary layer derived from Milne problem. In this paper, we construct counterexamples to disprove such formulation in...

We study the modeling and simulation of steady-state measurements of light scattered by a turbid medium taken at the boundary. In particular, we implement the recently introduced corrected diffusion approximation in two spatial dimensions to model these boundary measurements. This implementation uses expansions in plane wave solutions to compute boundary conditions and the additive boundary lay...

In a recent paper we have introduced a postprocessing procedure for the Galerkin method for dissipative evolution partial differential equations with periodic boundary conditions. The postprocessing technique uses approximate inertial manifolds to approximate the high modes (the small scale components) in the exact solutions in terms of the Galerkin approximations, which in this case play the r...

The aim of this paper is to review and classify the different methods that have been developed to enable stable long time simulations of the VlasovMaxwell equations and the Maxwell equations with sources. These methods can be classified in two types: field correction methods and sources correction methods. The field correction methods introduce new unknowns in the equations, for which additiona...

In this article, we derive approximations and effective boundary laws for solutions uε of the Poisson equation on a domain Ωε ⊂ Rn whose boundary differs from the smooth boundary of a domain Ω ⊂ R n by rapid oscillations of size ε. First, we construct a boundary layer correction which yields an O(ε3/2) approximation in the energy norm, and an O(ε2) approximation in the L2-norm if Ω is bounded. ...