# Stabilization methods in relaxed micromagnetism

Stefan A. Funken; Andreas Prohl

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 39, Issue: 5, page 995-1017
- ISSN: 0764-583X

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topFunken, Stefan A., and Prohl, Andreas. "Stabilization methods in relaxed micromagnetism." ESAIM: Mathematical Modelling and Numerical Analysis 39.5 (2010): 995-1017. <http://eudml.org/doc/194296>.

@article{Funken2010,

abstract = {
The magnetization of a ferromagnetic sample solves a
non-convex variational problem, where its relaxation by convexifying
the energy density resolves relevant
macroscopic information.
The numerical analysis of the relaxed model
has to deal with a constrained convex
but degenerated, nonlocal energy functional in mixed formulation for
magnetic potential u and magnetization m.
In [C. Carstensen and A. Prohl, Numer. Math.90
(2001) 65–99], the conforming P1 - (P0)d-element in d=2,3 spatial
dimensions is shown to lead to
an ill-posed discrete problem in relaxed micromagnetism, and suboptimal
convergence.
This observation motivated a
non-conforming finite element method which leads to
a well-posed discrete problem, with solutions converging at
optimal rate.
In this work, we provide both an a priori and a posteriori error analysis for two
stabilized conforming methods which account for inter-element jumps of the
piecewise constant magnetization.
Both methods converge at optimal rate;
the new approach is applied to a macroscopic nonstationary
ferromagnetic model [M. Kružík and A. Prohl, Adv. Math. Sci. Appl. 14 (2004) 665–681 – M. Kružík and T. Roubíček, Z. Angew. Math. Phys. 55 (2004) 159–182 ].
},

author = {Funken, Stefan A., Prohl, Andreas},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Micromagnetics; stationary; nonstationary; microstructure; relaxation;
nonconvex minimization; degenerate convexity; finite elements methods;
stabilization; penalization; a priori error estimates; a posteriori
error estimates.; micromagnetics; nonconvex minimization; stabilization; a priori error; estimates; a posteriori error estimates},

language = {eng},

month = {3},

number = {5},

pages = {995-1017},

publisher = {EDP Sciences},

title = {Stabilization methods in relaxed micromagnetism},

url = {http://eudml.org/doc/194296},

volume = {39},

year = {2010},

}

TY - JOUR

AU - Funken, Stefan A.

AU - Prohl, Andreas

TI - Stabilization methods in relaxed micromagnetism

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 39

IS - 5

SP - 995

EP - 1017

AB -
The magnetization of a ferromagnetic sample solves a
non-convex variational problem, where its relaxation by convexifying
the energy density resolves relevant
macroscopic information.
The numerical analysis of the relaxed model
has to deal with a constrained convex
but degenerated, nonlocal energy functional in mixed formulation for
magnetic potential u and magnetization m.
In [C. Carstensen and A. Prohl, Numer. Math.90
(2001) 65–99], the conforming P1 - (P0)d-element in d=2,3 spatial
dimensions is shown to lead to
an ill-posed discrete problem in relaxed micromagnetism, and suboptimal
convergence.
This observation motivated a
non-conforming finite element method which leads to
a well-posed discrete problem, with solutions converging at
optimal rate.
In this work, we provide both an a priori and a posteriori error analysis for two
stabilized conforming methods which account for inter-element jumps of the
piecewise constant magnetization.
Both methods converge at optimal rate;
the new approach is applied to a macroscopic nonstationary
ferromagnetic model [M. Kružík and A. Prohl, Adv. Math. Sci. Appl. 14 (2004) 665–681 – M. Kružík and T. Roubíček, Z. Angew. Math. Phys. 55 (2004) 159–182 ].

LA - eng

KW - Micromagnetics; stationary; nonstationary; microstructure; relaxation;
nonconvex minimization; degenerate convexity; finite elements methods;
stabilization; penalization; a priori error estimates; a posteriori
error estimates.; micromagnetics; nonconvex minimization; stabilization; a priori error; estimates; a posteriori error estimates

UR - http://eudml.org/doc/194296

ER -

## References

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- W.F. Brown, Micromagnetics. Interscience, New York (1963).
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- C. Carstensen and D. Praetorius, Numerical analysis for a macroscopic model in micromagnetics. SIAM J. Numer. Anal.42 (2005) 2633–2651, electronic. Zbl1088.78009
- C. Carstensen and A. Prohl, Numerical analysis of relaxed micromagnetics by penalized finite elements. Numer. Math.90 (2001) 65–99. Zbl1004.78006
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- M. Kružík and A. Prohl, Young measure approximation in micromagnetics. Numer. Math.90 (2001) 291–307. Zbl0994.65078
- M. Kružík and A. Prohl, Macroscopic modeling of magnetic hysteresis. Adv. Math. Sci. Appl.14 (2004) 665–681. Zbl1105.74034
- M. Kružík and A. Prohl, Recent developments in modeling, analysis and numerics of ferromagnetism. SIAM Rev. (accepted, 2005). Zbl1126.49040
- M. Kružík and T. Roubíček, Microstructure evolution model in micromagnetics. Z. Angew. Math. Phys.55 (2004) 159–182. Zbl1059.82047
- M. Kružík and T. Roubíček, Interactions between demagnetizing field and minor-loop development in bulk ferromagnets. J. Magn. Magn. Mater.277 (2004) 192–200.
- P. Pedregal, Parametrized Measures and Variational Principles. Birkhäuser (1997).
- A. Prohl, Computational micromagnetism. Teubner (2001).
- R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley-Teubner (1996). Zbl0853.65108

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