The Convergence of the Finite Mass Method for Flow

TheConvergeneoftheFiniteMassMethodforFlowsinGivenForeandVeloityFieldsHarryYserentantMathematishesInstitut,UniversitatTubingen,72076Tubingen,GermanyAbstrat.ThenitemassmethodisanewLagrangianmethodtosolveproblemsinontinuummehanis,primarilytosimulateompressibleows.Itisdiretlyfoundedonadisretizationofmass,notofspaeaswithlassialdisretizationshemes.Massissubdividedintosmallmasspaketsofniteextensioneahofwhihisequippedwithnitelymanyinternaldegreesoffreedom.Thesemasspaketsmoveundertheinueneofinternalandexternalforesandthelawsofthermodynamisandanhangetheirshapetofollowthemotionoftheuid.Inthepresentpaper,seondorderonvergeneofthemethodisshownfortheaseofowsinagivenveloityeldorsolelydrivenbyexternalfores.1IntrodutionThenitemassmethodwasintroduedin[3℄andisbasedononeptsthathavebeendevelopedin[5℄,[6℄,[7℄.ItisagridlessLagrangianmethodtosolveproblemsinontinuummehanisandis,inontrasttoniteelementandnitevolumemethods,diretlybasedonadisretizationofmass,notofspae.Massissubdividedintosmallmasspaketseahofwhihisequippedwithnitelymanyinternaldegreesoffreedom.Thesemasspaketsanmoveindependentlyofeahother,anrotate,expandandontrat,andanevenhangetheirshapetofollowthedeformationofthematerial.Theapproxi-mationsthenitemassmethodproduesaredierentiablefuntionsandnotdisretemeasuressuhthatinsomewayitomesmuhlosertolassialdisretizationsshemesthantotheusualLagrangianpartilemethods.In[3℄,thenitemassmethodhasbeenusedtosimulatefreegasows,astheyareusuallydesribedbytheompressibleEulerandNavier{Stokesequations.Beauseofourlakofknowledgeontheexistene,uniqueness,andregularityofsolutionsoftheseequations,itisveryhardtoproveon-vergeneforthissituation.Soitisalreadysurprisingthat,analogouslytotheonsiderationsin[5℄and[7℄,theexisteneoflimitssatisfyingthebasiphysialpriniplesunderlyinggasdynamisanbeshown.Thereforeinthisnotewerestritourattentiontotwomuhsimplerases,themotionofmassundertheinueneofanexternalforeeldandthetransportofmassinagivenveloityeld.Fortheseases,amuhmoredetailedonvergeneanalysisispossible,andinfat,oneyieldsseondorderonvergeneasoneouldexpetfromtheonstrutionofthemethod.Thusourresultsalsodemonstratethattheansatzfuntionsusedinthenitemassmethodareprinipallyapableofproduingseondorderapproximations.420HarryYserentantThemethodofproofisbasiallysimple.Asthemasspaketsdonotinter-atunderthegivenirumstanes,themotionofeahpaketanbeanalyzedseparatelyusingtehniquesfromthetheoryofordinarydierentialequationsandloallinearizationsoftheeldsunderonsideration.Theresultingloalmaximumnormerrorestimatesarethenassembledtoglobalestimatesfortheveloityeldandtheowitselfusingthepartitionofunitygivenbythemassfrationsofthesinglepakets.Thesemassfrationsalreadyplayadominantroleintheonstrutionofthemethodandareusedtodenetheapproximateveloity,forexample.Inthisrespetourtehniqueofproofthereforeresemblesthetehniquesappliedinthepartitionofunityniteel-ementmethodofBabuskaandMelenk[1℄orinthereentpaperofGriebelandShweitzer[4℄,forexample.Theargumentationforthemassdensityisquitedierentfromtheargu-mentationfortheveloityeldortheowandisbasedonthefatthatthedisretemassdensityandthedisreteveloityarelinkedtoeahotherbyonstrutionandexatlyfullltheontinuityequationdesribingtheon-servationofmass.Thereforetheapproximationofthemassdensityisau-tomatiallyasgoodasthatoftheveloityinthesenseofabakwarderroranalysis.Weshowseondorderonvergeneofertainloalspatialmeanval-ues,morepreisely,seondorderonvergeneinthedualspaeofthespaeoftheontinuouslydierentiablefuntionswithompatsupport.2ThebasiapproahInthissetion,weshortlyreallhowthemotionofmassorarelatedphysialquantityismodeledinthenitemassmethod.Asindiatedabove,thebasiideaistosubdividethetotalmassintosmallmasspakets.Theinternalmassdistributioninsidesuhamasspaketisdesribedbyashapefuntion:Rd!R,dthespaedimension,withompatsupportthatattainsonlyvalues0.WesupposethatZ(y)dy=1;Z(y)ydy=0;(2.1)andthatZ(y)ykyldy=JÆkl;(2.2)withtheyktheomponentsofy.Inthepresentontext,needsonlytobeintegrableandould,uptonormalization,betheharateristifuntionofaube,forexample,buttoobtainsmoothapproximationsandforappliationsinuiddynamisasin[3℄,shouldbeatleastontinuouslydierentiable.Thesinglemasspaketsanmoveinspaeandanundergoarbitrarylineardeformations.Thepointsyofthemasspaketimovealongthetra-jetoriest!qi(t)+Hi(t)y;detHi(t)0:(2.3)FiniteMassMethod421Thevetorqi(t)determinesthepositionofthepaketandthematrixHi(t)itssize,shapeandorientationinspae.Conversely,y=Hi(t)1(xqi(t))(2.4)arethebodyoordinatesofthepointatpositionxinspaeattimet.Themassdensityofthepaketirelatedtospaeoordinatesistherefore,uptoitsonstanttotalmass,i(x;t)=Hi(t)1(xqi(t))detHi(t);(2.5)thedeterminantensuresthatZi(x;t)dx=1(2.6)independentofqiandHi,thatisthetotalmassontainedinthispaketremainsxed.Thepointsyofthepaketihavetheveloityt!q0i(t)+H0i(t)y:(2.7)Insertingtheexpression(2.4)fory,onegetstheveloityeldvi(x;t)=q0i(t)+H0i(t)Hi(t)1(xqi(t))(2.8)ofthepaketirelatedtothespaeoordinates.Letmi0denotethemassofthepaketi.Thetotalmassdensity(x;t)=NXi=1mii(x;t)(2.9)thenresultsfromthesuperpositionofthemassdensitiesofthesi