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.

Manipulation of magnetic anisotropy in nanostructures

Of late, the magnetic properties of micro/nano-structures have attracted intense research interest both fundamentally and technologically particularly to address the question that how the manipulation in the different layers of nanostructures, geometry of a patterned structure can affect the overall magnetic properties, while generating novel applications such as in magnetic sensors, storage devices, integrated inductive components and spintronic devices. Depending on the applications, materials with high, medium or low magnetic anisotropy and their possible manipulation are required. The most dramatic manifestation in this respect is the chance to manipulate the magnetic anisotropy over the intrinsic preferential direction of the magnetization, which can open up more functionality particularly for device applications. Types of magnetic anisotropies of different nanostructured materials and their manipulation techniques are investigated in this work. Detail experimental methods for the quantitative determination of magnetic anisotropy in nanomodulated Ni45Fe55 thin film are studied. Magnetic field induced in-plane rotations within the nanomodulated Ni45Fe55 continuous films revealed various rotational symmetries of magnetic anisotropy due to dipolar interactions showing a crossover from lower to higher fold of symmetry as a function of modulation geometry. In a second approach, the control of exchange anisotropy at ferromagnetic (FM) – aniferomagnetic (AFM) interface in multifferoic nanocomposite materials, where two different phase/types of materials were simultaneously synthesized, was investigated. The third part of this work was to study the electroplated thin films of metal alloy nanocomposite for enhanced exchange anisotropy. In this work a unique observation of an anti-clock wise as well as a clock wise hysteresis loop formation in the Ni,Fe solid solution with very low coercivity and large positive exchange anisotropy/exchange bias have been investigated. Hence, controllable positive and negative exchange anisotropy has been observed for the first time which has high potential applications such as in MRAM devices.