The discovery that ultra-intense laser pulses (I > 10{sup 18} W/cm{sup 2}) can produce short pulse, high energy proton beams has renewed interest in the fundamental mechanisms that govern particle acceleration from laser-solid interactions. Experiments have shown that protons present as hydrocarbon contaminants on laser targets can be accelerated up to energies > 50 MeV. Different theoretical models that explain the observed results have been proposed. One model describes a front-surface acceleration mechanism based on the ponderomotive potential of the laser pulse. At high intensities (I > 10{sup 18} W/cm{sup 2}), the quiver energy of an electron oscillating in the electric field of the laser pulse exceeds the electron rest mass, requiring the consideration of relativistic effects. The relativistically correct ponderomotive potential is given by U{sub p} = ([1 + I{lambda}{sup 2}/1.3 x 10{sup 18}]{sup 1/2} - 1) m{sub o}c{sup 2}, where I{lambda}{sup 2} is the irradiance in W{micro}m{sup 2}/cm{sup 2} and m{sub o}c{sup 2} is the electron rest mass.At laser irradiance of I{lambda}{sup 2} {approx} 10{sup 20} W{micro}m{sup 2}/cm{sup 2}, the ponderomotive potential can be of order several MeV. A few recent experiments--discussed in Chapter 3 of this thesis--consider this ponderomotive potential sufficiently strong to accelerate protons from the …

This dissertation documents an object oriented framework which can be used to solve any linear wave equation. The linear wave equations are expressed in the differential forms language. This differential forms expression allows a strict discrete interpretation of the system. The framework is implemented using the Galerkin Finite Element Method to define the discrete differential forms and operators. Finite element basis functions including standard scalar Nodal and vector Nedelec basis functions are used to implement the discrete differential forms resulting in a mixed finite element system. Discretizations of scalar and vector wave equations in the time and frequency domains will be demonstrated in both differential forms and vector calculi. This framework conserves energy, maintains physical continuity, is valid on unstructured grids, conditionally stable and second order accurate. Examples including linear electrodynamics, acoustics, elasticity and magnetohydrodynamics are demonstrated.