Adaptive refinement has proved to be a useful tool for reducing the size of the linear system of equations obtained by discretizing partial differential equations. We consider techniques for the adaptive refinement of triangulations used with the finite element method with piecewise linear functions. Several such techniques that differ mainly in the method for dividing triangles and the method for indicating which triangles have the largest error have been developed. We describe four methods for dividing triangles and eight methods for indicating errors. Angle bounds for the triangle division methods are compared. All combinations of triangle divisions and error indicators are compared in a numerical experiment using a population of eight test problems with a variety of difficulties (peaks, boundary layers, singularities, etc.). The comparison is based on the L -infinity norm of the error versus the number of vertices. It is found that all of the methods produce asymptotically optimal grids and that the number of vertices needed for a given error rarely differs by more than a factor of two.

We point out that theories describing s-wave collisions of bosonic atoms confined in one-or two-dimensional geometries can be extended to much tighter confinements than previously thought. This is achieved by replacing the scattering length by an energy-dependent scattering length which was already introduced for the calculation of energy levels under 3D confinement. This replacement accurately predicts the position of confinementinduced resonances in strongly confined geometries.Many experiments investigating the properties of cold atomic gases and Bose-Einstein condensates are now performed in tightly confining traps, such as tight optical lattices, leading to systems of reduced dimensionality [1,2,3,4]. There are many uses for such confinements. In spectroscopic measurements, they eliminate unwanted Doppler and recoil effects [5,6]. They can also be used to create tunable analogs of condensed matter systems, and give the possibility to investigate remarkable many-body regimes in low dimensions such as the Tonks-Girardeau gas [7,8,9,10]. The theory of s-wave atomic collisions in strongly confined systems has been established in Refs.[7] and [11] for 2D and 1D confinement, respectively. Both predict a confinement-induced resonance of the effective 1D or 2D interaction strength. These predictions rely on a description of the atomic interaction in terms of the scattering length only. However, in 3D confined systems, it was shown that a more refined description is needed for very tight confinement [12,13]. Similarly, in 2D confined systems, numerical calculations in Ref. [14] showed that the scattering length description of Ref.[7] may be insufficient. In this paper, we present an accurate analytical description for scattering in 1D and 2D geometries based on the findings of Refs. [13,15].We consider a gas of bosonic atoms in an optical lattice and assume that there is little tunelling between the lattice cells, so that each cell is independent. The atoms in a cell are confined by a trapping potential which will be assumed harmonic (which is true near the centre of the cell). Let us consider a pair of atoms in such a cell. For a harmonic potential, the centre-of-mass motion decouples from the relative motion and the stationary Schrödinger equation for the relative motion wave function ψ( r) reads:Here, r = (x, y, z) is the relative coordinate with separation r, µ the reduced mass, U (r) the isotropic atom-atom interaction potential, V ( r) the trapping potential, and E the relative energy.For 2D confinement (tube or wave guide geometry), the atoms are strongly confined in the xy directions and (almost) free to move in the z direction, therefore we set:where ρ = ρ and ρ is the projection of r on the xy plane. For 1D confinement (pancake geometry), the atoms are strongly confined in the z direction and (almost) free to move in the xy directions:Here, ω is the trapping frequency at the centre of the cell and we define σ = /(2µω) as the typical length scale associated to the trap in the confined directions.Any scatteri...

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