on stabilized finite element methods in relaxed micromagnetism

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1、ONSTABILIZEDFINITEELEMENTMETHODSINRELAXEDMICROMAGNETISMSTEFANA.FUNKENANDANDREASPROHLAbstract.Themagnetizationstateofaferromagneticbodyisgivenasthesolutionofanon-convexvariationalproblem.Arelaxationofthismodelbyconvexifyingtheenergydensityresolvesessentialm

2、acroscopicinformationthatappliedphysicistsandengineersareafter.Thenumericalanalysisoftherelaxedmodelfacesanonlinearlyconstrainedconvexbutdegeneratedenergyfunctionalinmixedformulationformagneticpotentialudandmagnetizationm.In[CPr],theconformingP1?(P0)-eleme

3、ntind=2;3spatialdimensionsisshowntoleadtoanill-poseddiscreteprobleminrelaxedmicromagnetismandsuboptimalconvergencebehavior,eveniftheoriginalproblemiswell-posed.Thisobservationmotivatedtointroduceanon-conformingmethodtotheproblemthatimpliesawell-poseddiscre

4、teproblemwithsolutionsconvergingatoptimalrate.Inthiswork,weintroduceanddiscusstwostabilizedconformingmethodsthatemployinter-elementjumpsofthemagnetizationinnormaldirectionandtotalmagnetizationjumps,respectively.Bothstabilizedschemesconvergeatoptimalrates,w

5、hereasonlythelatteroneleadstoadiscreteproblemwithuniquesolution.1.IntroductiondThemathematicaldescriptionofmagnetizationstatesm:!!Rinrigidferromagneticdbodiescoveringadomain!R,d=2;3reliesontheclassicalmodelofWeiss,LandauandLifshitz[B],wherethemagnetizatio

6、nminimizesthemagnetostaticenergyZZZZ122(1.1)E(m)=jrmjdx+(m)dx?f
mdx+jrujdx2!!!subjecttothefollowingconsequenceofMaxwell'sequations,?1(1.2)?
u+div(m)=0inH():!Here,isthecharacteristicfunctionoftheset!,i.e.,(x)=1ifx2!and(x)=0if!!!not.Further,wechooseabou

7、ndedsetc!forpracticalsimplicationsinthesubsequentdanalysisinsteadofthewholeRtoconsidernon-localenergycontributions,seealso[CPr]onthismatter.Foradierentapproachusingthecouplingofpenalizedniteelementmethodsandboundaryelementmethodswereferto[CF].Themagneti

8、cenergyisassembledfromdierentcontributionsattributingtodierentpropertiesofminimizers:therstcontribution(surfaceenergy")isscaledby>0andfavorsnon-oscillatorystructures.Thenon-convexanisotropicenergydensity0