On smoothers for multigrid of the second kind

We study smoothers for the multigrid method of the second kind arising from Fredholm integral equations. Our model problems use nonlocal governing operators that enforce local boundary conditions. For discretization, we utilize the Nyström method with the trapezoidal rule. We find the eigenvalues of matrices associated to periodic, antiperiodic, and Dirichlet problems in terms of the nonlocality parameter and mesh size. Knowing explicitly the spectrum of the matrices enables us to analyze the behavior of smoothers. Although spectral analyses exist for finding effective smoothers for 1D elliptic model problems, to the best of our knowledge, a guiding spectral analysis is not available for smoothers of a multigrid of the second kind. We fill this gap in the literature. The Picard iteration has been the default smoother for a multigrid of the second kind. Jacobi‐like methods have not been considered as viable options. We propose two strategies. The first one focuses on the most oscillatory mode and aims to damp it effectively. For this choice, we show that weighted‐Jacobi relaxation is equivalent to the Picard iteration. The second strategy focuses on the set of oscillatory modes and aims to damp them as quickly as possible, simultaneously. Although the Picard iteration is an effective smoother for model nonlocal problems under consideration, we show that it is possible to find better than ones using the second strategy. We also shed some light on internal mechanism of the Picard iteration and provide an example where the Picard iteration cannot be used as a smoother.     

Tracing pore-space heterogeneities in X-type zeolites by diffusion studies

Pore-space homogeneity of zeolite NaX was probed by pulsed field gradient (PFG) NMR diffusion studies with n-butane as a guest molecule. At a loading of 0.75 molecules per supercage, a wide spectrum of diffusivities was observed. Guest molecules in the (well-shaped) zeolite crystallites were thus found to experience pore spaces of quite different properties. After loading enhancement to 3 molecules per supercage, however, molecular propagation ideally followed the laws of normal diffusion in homogeneous media. At sufficiently high guest concentrations, sample heterogeneity was thus found to be of no perceptible influence on the guest mobilities anymore.     

Equiatomic binary phases of Copper-Rare Earth Elements. An overview of monocuprides from first-principles calculations

Equiatomic binary phases of copper with rare earth (RE) elements exhibit either primitive cubic ( ) or orthorhombic (Pnma) structures and in some cases both. By using density functional theory (DFT), we calculated the enthalpies of formation along the series of RE elements combined equimolarly with copper. For RE from Sc to Lu, the calculated enthalpies of formation fall in the range −49.8 kJ/mol for LuCu to −9.1 kJ/mol for the least thermodynamically stable CeCu. Except NdCu, all the other cubic or orthorhombic compounds exhibit lattice stability. Either forms of NdCu indicated lattice instability. Along the Sc-group, the hypothetical primitive cubic and orthorhombic forms of LuCu are found thermodynamically and mechanically stable. The overall trend of the formation enthalpies as a function of the Meyer Periodic Number is consistent with the energy trend of the 4 f-orbital filling as moving from Sc to Lu monocuprides. In addition, the calculated Gibbs free energies indicate that the thermodynamic stability is largely due to the entropic contributions. All standard DFT calculations were also repeated with DFT+U to better describe the correlation between the 5d–4f and 3d shells of RECu compounds. It has been found that DFT+U slightly affects the enthalpies of formation of RECu binaries. Moreover, DFT+U shifts up the f-band energies of RECu with light RE elements (such as La, Ce and Pr) and in contrast lowers them in the case of RECu with heavy RE elements from Nd to Lu.