Andreas Prohl Libros

This work examines numerical challenges related to a mathematical model for ferromagnetic materials in both stationary and non-stationary contexts, incorporating electromagnetic effects for nonstationary magneto-electronics. The latter sections focus on the numerical analysis of the Ericksen-Leslie model, which is essential for studying the fluid flow of nematic liquid crystals used in display technologies. These models share key features, including strong nonlinearities and non-convex side constraints, such as the requirement that the order parameter m satisfies |m| = 1 almost everywhere. A critical aspect of numerical modeling is ensuring that computed solutions adhere to these non-convex constraints. Various solution strategies for the variational problem of stationary micromagnetism are presented, including direct minimization, convexification, and relaxation through Young measure-valued solutions. Notably, direct minimization faces challenges due to spatial triangulation introducing artificial exchange energy contributions that can obscure relevant physical interactions. The minimizers often display complex multiple scales, particularly near the boundaries of ferromagnets, complicating efficient computation. To address this, we propose an adaptive scheme designed to better capture these intricate multiple scale structures in cubic ferromagnets.

Projection methods had been introduced in the late sixties by A. Chorin and R. Teman to decouple the computation of velocity and pressure within the time-stepping for solving the nonstationary Navier-Stokes equations. Despite the good performance of projection methods in practical computations, their success remained somewhat mysterious as the operator splitting implicitly introduces a nonphysical boundary condition for the pressure. The objectives of this monograph are twofold. First, a rigorous error analysis is presented for existing projection methods by means of relating them to so-called quasi-compressibility methods (e. g. penalty method, pressure stabilzation method, etc.). This approach highlights the intrinsic error mechanisms of these schemes and explains the reasons for their limitations. Then, in the second part, more sophisticated new schemes are constructed and analyzed which are exempted from most of the deficiencies of the classical projection and quasi-compressibility methods. '... this book should be mandatory reading for applied mathematicians specializing in computational fluid dynamics.' J.-L. Guermond. Mathematical Reviews, Ann Arbor