This thesis investigates the high-order hierarchical finite element method, also known as the finite element p-version, as a computationally-efficient technique for generating numerical solutions to the cardiac monodomain equation. We first present it as a uniform-order method, and through an a priori error bound we explain why the associated cardiac cell model must be thought of as a PDE and approximated to high-order in order to obtain the accuracy that the p-version is capable of. We perform simulations demonstrating that the achieved error agrees very well with the a priori error bound. Further, in terms of solution accuracy for time taken to solve the linear system that arises in the finite element discretisation, it is more efficient that the state-of-the-art piecewise linear finite element method. We show that piecewise linear FEM actually introduces quite significant amounts of error into the numerical approximations, particularly in the direction perpendicular to the cardiac fibres with physiological conductivity values, and that without resorting to extremely fine meshes with elements considerably smaller than 70 micrometres, we can not use it to obtain high-accuracy solutions. In contrast, the p-version can produce extremely high accuracy solutions on meshes with elements around 300 micrometres in diameter with these conductivities. Noting that most of the numerical error is due to under-resolving the wave-front in the transmembrane potential, we also construct an adaptive high-order scheme which controls the error locally in each element by adjusting the finite element polynomial basis degree using an analytically-derived a posteriori error estimation procedure. This naturally tracks the location of the wave-front, concentrating computational effort where it is needed most and increasing computational efficiency. The scheme can be controlled by a user-defined error tolerance parameter, which sets the target error within each element as a proportion of the local magnitude of the solution as measured in the H^1 norm. This numerical scheme is tested on a variety of problems in one, two and three dimensions, and is shown to provide excellent error control properties and to be likely capable of boosting efficiency in cardiac simulation by an order of magnitude. The thesis amounts to a proof-of-concept of the increased efficiency in solving the linear system using adaptive high-order finite elements when performing single-thread cardiac simulation, and indicates that the performance of the method should be investigated in parallel, where it can also be expected to provide considerable improvement. In general, the selection of a suitable preconditioner is key to ensuring efficiency; we make use of a variety of different possibilities, including one which can be expected to scale very well in parallel, meaning that this is an excellent candidate method for increasing the efficiency of cardiac simulation using high-performance computing facilities.

<p>Quantum dynamics is the study of time-dependent phenomena in fundamental processes of atomic and molecular systems. This thesis focuses on systems where nature reveals its quantum aspect; e.g. in vibrational resonance structures, in wave packet revivals and in matter wave interferometry. Grid based numerical methods for solving the time-dependent Schrödinger equation are implemented for simulating time resolved molecular vibrations and to compute photo-electron spectra, without the necessity of diagonalizing a large matrix to find eigenvalues and eigenvectors.</p><p>Pump-probe femtosecond laser spectroscopy on the sodium potassium molecule, showing a vibrational period of 450 fs, is theoretically simulated. We find agreement with experiment by inclusion of the finite length laser pulse and finite temperature effects.</p><p>Complicated resonance structures observed experimentally in photo-electron spectra of hydrogen- and deuterium chloride is analyzed by a numerical computation of the spectra. The dramatic difference in the two spectra arises from non-adiabatic interactions, i.e. the interplay between nuclear and electron dynamics. We suggest new potential curves for the 3²Σ⁺ and 4²Σ⁺ states in HCI⁺.</p><p>It is possible to guide slow atoms along magnetic potentials like light is guided in optical fibers. Quantum mechanics dictates that matter can show wave properties. A proposal for a multi mode matter wave interferometer on an atom chip is studied by solving the time-dependent Schrödinger equation in two dimensions. The results verifies a possible route for an experimental realization.</p><p>An improved representation for wave functions using a discrete set of coherent states is presented. We develop a practical method for computing the expansion coefficients in this non-orthogonal set. It is built on the concept of frames, and introduces an iterative method for computing a representation of the identity operator. The phase-space localization property of the coherent states gives adaptability and better sampling efficiency.</p>

Quantum dynamics is the study of time-dependent phenomena in fundamental processes of atomic and molecular systems. This thesis focuses on systems where nature reveals its quantum aspect; e.g. in vibrational resonance structures, in wave packet revivals and in matter wave interferometry. Grid based numerical methods for solving the time-dependent Schrödinger equation are implemented for simulating time resolved molecular vibrations and to compute photo-electron spectra, without the necessity of diagonalizing a large matrix to find eigenvalues and eigenvectors. Pump-probe femtosecond laser spectroscopy on the sodium potassium molecule, showing a vibrational period of 450 fs, is theoretically simulated. We find agreement with experiment by inclusion of the finite length laser pulse and finite temperature effects. Complicated resonance structures observed experimentally in photo-electron spectra of hydrogen- and deuterium chloride is analyzed by a numerical computation of the spectra. The dramatic difference in the two spectra arises from non-adiabatic interactions, i.e. the interplay between nuclear and electron dynamics. We suggest new potential curves for the 32Σ+ and 42Σ+ states in HCI+. It is possible to guide slow atoms along magnetic potentials like light is guided in optical fibers. Quantum mechanics dictates that matter can show wave properties. A proposal for a multi mode matter wave interferometer on an atom chip is studied by solving the time-dependent Schrödinger equation in two dimensions. The results verifies a possible route for an experimental realization. An improved representation for wave functions using a discrete set of coherent states is presented. We develop a practical method for computing the expansion coefficients in this non-orthogonal set. It is built on the concept of frames, and introduces an iterative method for computing a representation of the identity operator. The phase-space localization property of the coherent states gives adaptability and better sampling efficiency.