An Energy-Minimization Finite-Element Approach for the Frank-Oseen Model of Nematic Liquid Crystals: Continuum and Discrete Analysis

This paper outlines an energy-minimization finite-element approach to the computational modeling of equilibrium configurations for nematic liquid crystals under free elastic effects. The method targets minimization of the system free energy based on the Frank-Oseen free-energy model. Solutions to the intermediate discretized free elastic linearizations are shown to exist generally and are unique under certain assumptions. This requires proving continuity, coercivity, and weak coercivity for the accompanying appropriate bilinear forms within a mixed finite-element framework. Error analysis demonstrates that the method constitutes a convergent scheme. Numerical experiments are performed for problems with a range of physical parameters as well as simple and patterned boundary conditions. The resulting algorithm accurately handles heterogeneous constant coefficients and effectively resolves configurations resulting from complicated boundary conditions relevant in ongoing research.Comment: 31 pages, 3 figures, 3 table

Gamma-Ray Bursts: Temporal Scales and the Bulk Lorentz Factor

For a sample of Swift and Fermi GRBs, we show that the minimum variability timescale and the spectral lag of the prompt emission is related to the bulk Lorentz factor in a complex manner: For small $\Gamma$'s, the variability timescale exhibits a shallow (plateau) region. For large $\Gamma$'s, the variability timescale declines steeply as a function of $\Gamma$ ($\delta T\propto\Gamma^{-4.05\pm0.64}$). Evidence is also presented for an intriguing correlation between the peak times, t$_p$, of the afterglow emission and the prompt emission variability timescale.Comment: Accepted for publication in Ap

Multigrid reduction-in-time convergence for advection problems: A Fourier analysis perspective

A long-standing issue in the parallel-in-time community is the poor convergence of standard iterative parallel-in-time methods for hyperbolic partial differential equations (PDEs), and for advection-dominated PDEs more broadly. Here, a local Fourier analysis (LFA) convergence theory is derived for the two-level variant of the iterative parallel-in-time method of multigrid reduction-in-time (MGRIT). This closed-form theory allows for new insights into the poor convergence of MGRIT for advection-dominated PDEs when using the standard approach of rediscretizing the fine-grid problem on the coarse grid. Specifically, we show that this poor convergence arises, at least in part, from inadequate coarse-grid correction of certain smooth Fourier modes known as characteristic components, which was previously identified as causing poor convergence of classical spatial multigrid on steady-state advection-dominated PDEs. We apply this convergence theory to show that, for certain semi-Lagrangian discretizations of advection problems, MGRIT convergence using rediscretized coarse-grid operators cannot be robust with respect to CFL number or coarsening factor. A consequence of this analysis is that techniques developed for improving convergence in the spatial multigrid context can be re-purposed in the MGRIT context to develop more robust parallel-in-time solvers. This strategy has been used in recent work to great effect; here, we provide further theoretical evidence supporting the effectiveness of this approach

Strange form factors of the nucleon in a two-component model

The strange form factors of the nucleon are studied in a two-component model consisting of a three-quark intrinsic structure surrounded by a meson cloud. A comparison with the available experimental world data from the SAMPLE, PVA4, HAPPEX and G0 collaborations shows a good overall agreement. The strange magnetic moment is found to be positive, 0.315 nm.Comment: 11 pages, 2 tables, 5 figures, accepted for publication in J. Phys. G. Revised version, new figures, extra table, new results, updated reference

Monolithic Multigrid for Magnetohydrodynamics

The magnetohydrodynamics (MHD) equations model a wide range of plasma physics applications and are characterized by a nonlinear system of partial differential equations that strongly couples a charged fluid with the evolution of electromagnetic fields. After discretization and linearization, the resulting system of equations is generally difficult to solve due to the coupling between variables, and the heterogeneous coefficients induced by the linearization process. In this paper, we investigate multigrid preconditioners for this system based on specialized relaxation schemes that properly address the system structure and coupling. Three extensions of Vanka relaxation are proposed and applied to problems with up to 170 million degrees of freedom and fluid and magnetic Reynolds numbers up to 400 for stationary problems and up to 20,000 for time-dependent problems