Advances in Numerical Mathematics

Prohl, Andreas

In this work, we study numerical issues related to a common mathematical model which describes ferromagnetic materials, both in a stationary and non­ stationary context. Electromagnetic effects are accounted for in an extended model to study nonstationary magneto-electronics. The last part deals with the numerical analysis of the commonly used Ericksen-Leslie model to study the fluid flow of nematic liquid crystals which find applications in display technologies, for example. All these mathematical models to describe different microstructural phe­ nomena share common features like (i) strong nonlinearities, and (ii) non­ convex side constraints (i.e., I m I = 1, almost everywhere in w C JRd, for the order parameter m : w -+ JRd). One key issue in numerical modeling of such problems is to make sure that the non-convex constraint is fulfilled for computed solutions. We present and analyze different solution strategies to deal with the variational problem of stationary micromagnetism, which builds part I of the book: direct minimization, convexification, and relaxation using Young measure-valued solutions. In particular, we address the following points: • Direct minimization: A spatial triangulation 'generates an artificial exchange energy contribution' in the discretized minimizing problem which may pollute physically relevant exchange energy contributions; its minimizers exhibit multiple scales (with branching structures near the boundary of the ferromagnet) and are difficult to be computed efficiently. We exploit this observation to construct an adaptive scheme which better resolves multiple scale structures (cubic ferromagnets).

Details

Published by: Vieweg+Teubner Verlag

Publication Date: 2001-12-11

Format: Paperback

ISBN-13: 9783519003588

DOI: 10.1007/978-3-663-09498-2

Dimensions: 244cm x170cm

Pages: 304