By Andreas Prohl

ISBN-10: 3519003589

ISBN-13: 9783519003588

ISBN-10: 3663094987

ISBN-13: 9783663094982

Ferromagnetic fabrics are everyday as recording media.

Their magnetic styles are defined by means of the well-accepted version of Landau and Lifshitz. during the last years, assorted recommendations habe been constructed to take on the comparable non-convex minimization challenge: direct minimization, convexification, and leisure through the use of younger measures. Nonstationary results are thought of within the prolonged types of Landau, Lifshitz and Gilbert for (electrically undertaking) ferromagnets.

the target of this monograph is a numerical research of those types. half I discusses convergence habit of other finite aspect schemes for fixing the desk bound challenge. half II bargains with numerical analyses of other penalization / projection innovations in nonstationary micromagnetism; it closes with a bankruptcy on nematic liquid crystals to teach applicability of those new how you can additional applications.

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1. A stable non-conforming discretization 45 From the monotonicity of the subgradients we infer from Ajmj E a'l/J(mj) that (Ajmj, mj+l - mj)1R2 :S 0 for j = 1,2 and m3 := mI. This shows o :S (A2m2 - AIml,6)1R2 almost everywhere in w. 7) 0, and are non-negative and hence vanish: e = 0, (A2m2 - AIml,6)1R2 (Dcp**(m2) - Dcp**(ml), 6)1R2 = 0 almost everywhere. 6) imply that 6 is divergence-free in the sense of distributions. This means div 6 = 0 almost everywhere in w but also (6, n)1R2 = 0 on the boundary aw, with n the unit normal vector (in a weak sense according to the formula of integration by parts).

2 in the uniaxial case (cf. 1). 5) while {Uh' mh, Ah} is a solution to (P N) or (PE,N). 22) 50 Chapter 2. Convexified Micromagnetism where {Uh' mh, Ah} denotes an arbitrary solution to (P N) or (Pc;,N). 22) is uniformly positive (for a mesh-size tending to zero) if, for instance, the set {x E W : Im(x)1 < 1 - 6"} has no interior point for all 0 < 6" < l. 22). Let /lj := max{lmhIK2j_p ImhlK2J and suppose without loss of generality that /lj = ImhlK2j for all j = 1, ... ,J. For mh E £,O(T) and MhIK2j_2+k := (1- /lj)+ (_1)k (1, l)T /V2 for j = 1, ...

2) implies IV'u l 12 dx = (V'uI, pI) . 24) where n = (nl,n2f is the unit exterior normal to f E {8A,8B,8C,8D} at x E f. 24) is zero, according to our construction. 2nd step: We consider the same domain w C ]R2, as in the first step, but the given (quasiuniform) triangulation is now supposed to be arbitrary. 21) is not that immediate. In the following, we use the notation of step 1. 2. 26) where pilK denotes an extension of any of the involved pIlKnD that are locally in question, for D E {A, B, C, D}.

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