Uniformly convergent finite element and finite difference methods for singularly perturbed ordinary differential equations

This thesis is concerned with uniformly convergent finite element and finite difference methods for numerically solving singularly perturbed two-point boundary value problems. We examine the following four problems: (i) high order problem of reaction-diffusion type; (ii) high order problem of convection-diffusion type; (iii) second order interior turning point problem; (iv) semilinear reaction-diffusion problem. Firstly, we consider high order problems of reaction-diffusion type and convection-diffusion type. Under suitable hypotheses, the coercivity of the associated bilinear forms is proved and representation results for the solutions of such problems are given. It is shown that, on an equidistant mesh, polynomial schemes cannot achieve a high order of convergence which is uniform in the perturbation parameter. Piecewise polynomial Galerkin finite element methods are then constructed on a Shishkin mesh. High order convergence results, which are uniform in the perturbation parameter, are obtained in various norms. Secondly, we investigate linear second order problems with interior turning points. Piecewise linear Galerkin finite element methods are generated on various piecewise equidistant meshes designed for such problems. These methods are shown to be convergent, uniformly in the singular perturbation parameter, in a weighted energy norm and the usual L2 norm. Finally, we deal with a semilinear reaction-diffusion problem. Asymptotic properties of solutions to this problem are discussed and analysed. Two simple finite difference schemes on Shishkin meshes are applied to the problem. They are proved to be uniformly convergent of second order and fourth order respectively. Existence and uniqueness of a solution to both schemes are investigated. Numerical results for the above methods are presented.

Kinetics and coverage dependent reaction mechanisms of the copper atomic layer deposition from copper dimethylamino-2-propoxide and diethylzinc

Atomic layer deposition (ALD) has been recognized as a promising method to deposit conformal and uniform thin film of copper for future electronic devices. However, many aspects of the reaction mechanism and the surface chemistry of copper ALD remain unclear. In this paper, we employ plane wave density functional theory (DFT) to study the transmetalation ALD reaction of copper dimethylamino-2-propoxide [Cu(dmap)2] and diethylzinc [Et2Zn] that was realized experimentally by Lee et al. [ Angew. Chem., Int. Ed. 2009, 48, 4536−4539]. We find that the Cu(dmap)2 molecule adsorbs and dissociates through the scission of one or two Cu–O bonds into surface-bound dmap and Cu(dmap) fragments during the copper pulse. As Et2Zn adsorbs on the surface covered with Cu(dmap) and dmap fragments, butane formation and desorption was found to be facilitated by the surrounding ligands, which leads to one reaction mechanism, while the migration of ethyl groups to the surface leads to another reaction mechanism. During both reaction mechanisms, ligand diffusion and reordering are generally endothermic processes, which may result in residual ligands blocking the surface sites at the end of the Et2Zn pulse, and in residual Zn being reduced and incorporated as an impurity. We also find that the nearby ligands play a cooperative role in lowering the activation energy for formation and desorption of byproducts, which explains the advantage of using organometallic precursors and reducing agents in Cu ALD. The ALD growth rate estimated for the mechanism is consistent with the experimental value of 0.2 Å/cycle. The proposed reaction mechanisms provide insight into ALD processes for copper and other transition metals.