Gradient Schemes for Linear and Non-linear Elasticity Equations

The Gradient Scheme framework provides a unified analysis setting for many different families of numerical methods for diffusion equations. We show in this paper that the Gradient Scheme framework can be adapted to elasticity equations, and provides error estimates for linear elasticity and convergence results for non-linear elasticity. We also establish that several classical and modern numerical methods for elasticity are embedded in the Gradient Scheme framework, which allows us to obtain convergence results for these methods in cases where the solution does not satisfy the full $H^2$-regularity or for non-linear models

Application of projection algorithms to differential equations: boundary value problems

The Douglas-Rachford method has been employed successfully to solve many kinds of non-convex feasibility problems. In particular, recent research has shown surprising stability for the method when it is applied to finding the intersections of hypersurfaces. Motivated by these discoveries, we reformulate a second order boundary valued problem (BVP) as a feasibility problem where the sets are hypersurfaces. We show that such a problem may always be reformulated as a feasibility problem on no more than three sets and is well-suited to parallelization. We explore the stability of the method by applying it to several examples of BVPs, including cases where the traditional Newton's method fails

Calculated Vibrational Properties of Ubisemiquinones

Density functional theory has been used to calculate harmonic normal mode vibrational frequencies for unlabeled and isotopelabeled ubisemiquinones in both the gas phase and in several solvents. It is shown that four methoxy group conformations are likely to be present in solution at room temperature. Boltzmann weighted infrared and Raman spectra for the four conformers were calculated, and composite spectra that are the sum of the Boltzmann weighted spectra were produced. These composite spectra were compared to experimental FTIR and resonance Raman spectra, and it is shown that the calculated band frequencies, relative band intensities, and 13C and 18O isotope-induced band shifts are in excellent agreement with experiment. The calculations show that the C=O and C=C modes of ubisemiquinone strongly mix with methoxy methyl CH bending vibrations, and that the degree of mixing is altered upon isotope labeling, resulting in complicated changes in mode frequencies, intensities, and composition upon isotope labeling. Upon consideration of the calculated potential energy distributions of the normal modes of ubisemiquinone, and how they change upon isotope labeling, an explanation of some puzzling features in previously published Raman spectra is provided

Calculated Vibrational Properties of Ubisemiquinones

Density functional theory has been used to calculate harmonic normal mode vibrational frequencies for unlabeled and isotopelabeled ubisemiquinones in both the gas phase and in several solvents. It is shown that four methoxy group conformations are likely to be present in solution at room temperature. Boltzmann weighted infrared and Raman spectra for the four conformers were calculated, and composite spectra that are the sum of the Boltzmann weighted spectra were produced. These composite spectra were compared to experimental FTIR and resonance Raman spectra, and it is shown that the calculated band frequencies, relative band intensities, and 13C and 18O isotope-induced band shifts are in excellent agreement with experiment. The calculations show that the C=O and C=C modes of ubisemiquinone strongly mix with methoxy methyl CH bending vibrations, and that the degree of mixing is altered upon isotope labeling, resulting in complicated changes in mode frequencies, intensities, and composition upon isotope labeling. Upon consideration of the calculated potential energy distributions of the normal modes of ubisemiquinone, and how they change upon isotope labeling, an explanation of some puzzling features in previously published Raman spectra is provided