20th GAMM-Seminar Leipzig on
Numerical Methods for Non-Local Operators

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  20th GAMM-Seminar
January, 22th-24th, 2004
  Winterschool on hierarchical matrices  
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  Abstract Dirk Praetorius, Fri, 11.00-11.30 Previous Contents Next  
  Effective Simulation of a Macroscopic Model in Micromagnetics
Dirk Praetorius (University Wien)

The large body limit in the Landau-Lifshitz equations of micromagnetics [1] yields a macroscopic model without exchange energy and convexified side conditions for the macroscopic magnetisation vectors. Its Euler Lagrange equations (P) read: Given a magnetic body tex2html_wrap_inline76, d=2,3, an exterior field tex2html_wrap_inline80 and the convexified anisotropy density tex2html_wrap_inline82, find a magnetization tex2html_wrap_inline84 and a Lagrange multiplier tex2html_wrap_inline86 such that a.e. in tex2html_wrap_inline88
The potential tex2html_wrap_inline90 solves the (Maxwell) equations
in the entire space. It therefore appears natural to recast the associated far field energy into an integral operator tex2html_wrap_inline92 which maps tex2html_wrap_inline94 to the corresponding stray field tex2html_wrap_inline96.

The proposed numerical scheme involves the operator tex2html_wrap_inline92 and replaces pointwise side-condition tex2html_wrap_inline100 by a penalization strategy. Given a triangulation tex2html_wrap_inline102, the induced space of piecewise constant functions tex2html_wrap_inline104 on tex2html_wrap_inline88, and a penalization parameter tex2html_wrap_inline108, the discrete penalized problem tex2html_wrap_inline110 reads: Find tex2html_wrap_inline112 such that for all tex2html_wrap_inline114
with tex2html_wrap_inline116.

Numerical aspects addressed in the presentation include the integration of the matrices with quadrature rules and hierarchical matrices as well as a  priori and a posteriori error control and adaptive mesh-design.

[1] A. DESIMONE: Energy Minimizers for Large Ferromagnetic Bodies, Arch. Rational Mech. Anal. 125 (1993), 99-143.

[2] D. PRAETORIUS: Analysis, Numerik und Simulation eines relaxierten Modellproblems zum Mikromagnetismus, Doctorial thesis, Vienna University of Technology 2003.

[3] C. CARSTENSEN, D. PRAETORIUS: Numerical Analysis for a Macroscopic Model in Micromagnetics, submitted 2003.

This is a joint work with C. Carstensen (Vienna University of Technology).

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Concept, Design and Realisation
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