Accuracy Of Least-Squares Methods For - The Navier-Stokes .
NASATechnicalMemorandumICOMP-93.19-1 06 0.Accuracy of Least-Squares MethodsThe Navier-Stokes EquationsFor(x)0t',0i .N!Or.O"Z0t0PavelB. iniaand StateUniversity--"t&LUandl'-Z. (:3MaxD. Gu nzburgerInstitutefor ComputationalLewis io0andVirginiaB Iacksburg,PolytechnicVirginiaInstituteand StateU niversityU3 I/JOCII-'-II' "ZUJ N"." -J ZJune1993
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ACCURACYFOR THEOF rginia PolytechnicBlaeksburg,B. BochevInstituteVirginiaand State University24061-0531andMax D. Gunzburgerfor Computational MechanicsLewis Research CenterInstituteCleveland,and 44135Institute and State UniversityVA 24061-0531interestin least-squaresfiniteformulationsof the incompressibleNavier-Stokesequations.for the resultingdiscreteequationscan be devisedwhichsystemsa primaryof algebraicequations.oftenveryvariableyieldOn thepoorotherin Propulsionhand,approximations.it is well-documentedThus,hereand also commenton theoreticalerrorestimates,that computationalevidenceleast,in eThus,to thedesirablemethodsthatwe studya series of computationalexperiments,of standardmethodsfor derivingerroroneelementThe main cause forrequirethe propertiesfor velocity-vorticity-this interestis theof only symmetric,usingof thesethevorticitymethodsIt is found,despitethat these methodsyieldedfact thatpositiveasthroughthe failureare, at theby least-squaresmethods,approximations.1. ecentapproach.[3]-[5], [13]-[17],Here,pressureof thenumber)systemsdefinitenumbermethodsin thepolynomial*This workalso supportedNASAmethodfiniteof partialAgreementdifferentialideas;flow has[9], [10], or [11].see,have beene.g.,receivedAmong[7] for a recentdevelopedand applied;procedure.space mayfor Computationalinvolvingthesurveysee, e.g.,unknownfor example,a Newtonof theseto onlylinearadvantagevelocity,(withof thismethodOfficeof Scientificin PropulsionResearchunderat the ctinThe influencethroughtheto the Reynoldspositivedefinitelinearis thata singlepiecewisei.e.,one mayuse equalbe used for all test and trial functions,Mechanicsequationsscalar fields.is felt onlysymmetric,for thelinearization,of the solution.systemscontinuationencounteris examinedThesetheone has sevenimplementedA furtherprincipleequations.in a neighborhooddefinitenesscan expectvariationalequationswith,at leastif properlyoneNCC3-233.alongsystems,was suppo 'ted in part by the Air ForceCooperativee.g.,Navier-StokesIn three-dimensionslinearare used,by the Institutesee,on a sibleon the positivesolutionof incompressibleuse of least-squaresbasedvariables.size of the neighborhood.mathematicians;least-squaresof a least-squaresReynoldstheequations[22].systemas dependentpositiveandof the stationary,a first-ordersymmetric,Navier-Stokesengineershas beenelementsolutionthe applicationof theAlso, truly[20], anda finiteapproximateintofromdevelopmentsof one .WorkMDGfundedwasunder
order interpolationwith respect to a singlegrid for alldependent variablesand testfunctions. Afinaladvantage resultingfrom the use of a least-squaresprincipleisthat,unlikesome other methodsinvolvingthe s well,linearizationsincorrect,no artificialleavingissuesespeciallyare taken.openthe questionthestudywithto thein the orderof [3]-[5] andthe analyses[13]-[17];foundneed be devised.however,therearethe diseretization,in someof theseandpapersareof approximations.elementthe methodof the accuracyfor the-vortieityin which the least-squares,of the resconnecteda computationalboundaryis similarstepsIn §2, we definetheoreticalherenumericalmethod.describedIn §3, we discussin §2.of the algorithm.Then,Finally,somepracticalandin §4, we give the resultsin §5 we givesomeofconcludingremarks.2.THE2.1LEAST-SQUARES- oundeddimensionlesssetequationsMETHODequationsf C IR a denotegoverningthethesteadyflow domainandincompressiblelet rdenoteflow of a viscousitsboundary.fluid mayThebe writtenin the form(2.1)divu(2.2) 0inft,curl u - w 0(2.3)Pcurlwin it, w x u gradp fin it,and(2.4)wheredivwu, p, and ca .thattotalNotehead,Thetwo such(2.5)f a giventhe terminologyi.e., p fi boundaryllul2bodyforce."pressure", whereit seeemsalgorithmsystempressure,in it,and vorticityThefirst-orderthe velocity-vorticity-pressureIn view of (2.2),least-squaresthe velocity, 0thatto explicitly(2.1)-(2.4)conditions.shouldof partialsinceu the inversedifferentialof steady,the variableof theequationsincompressiblep is actuallythethe pressure.(2.4);However,withis to imposeU ----U 1it is crucialsee, e.g., [3] andbe supplementedThe firstsystemmisleading(2.4) is redundant.requirerespectively,form of the equationsis a little15 denotesfields,boundarythe velocityonF,to thestabilityof the[15].conditions.on the boundary,Herei.e.,we examine
whereU1ponentdenotesof thea givenvorticityfunctiondefinedis also lcom-i.e.,. n n. curIUton r,where n denotes the unit outer normal to f/.To see this,one merely needs to observe that n- curluinvolvesonly tangentialderivativesof the tangentialcomponents of u and these maybe deducedfrom (2.5).To these one must add a conditionto fixthe pressure;we choose to fixthe pressure ata singlepoint xo in ft,i.e.,(2.7)p(xo) Po,whereP0 is a givenThesecondcomponentnumber.combinationof boundaryof the velocity,conditionswe consideris thepressureandthenormali.e.,(2.8)p Ponrand(2.9)u.n U2whereP and U2 denoteall thattwousefulas ctionsconditionsa ns;2.2least-squares- Thegivendefinedalongfor theNavier-Stokesrigorousanalysiscan be given usingand(2.8)-(2.9)we will discussOne can use (2.1)-(2.4)S(u, ,p)(2.10)on F,r.standardissuesfinitetechniques;(2.8)-(2.9)we considerelementthemare nothereforapproximationsofthis is not the case for (2.5)-(2.7).to be relatedin moreconditionsequations;of least-squarescan be showntheseThe boundarydetailto secondorderellipticpartialbelow.principleto definethe least-squaresfunctional [ (Icurlu- I2 Idivu[2 Idiv lJf\ [ucurl w w x u ncipleThen(u,p,standardthenrequirestechniquesw) of ,7 necessarily/[(curluJNLthe minimizationfrom thecalculusof this- f[2)functionalof variationsmayd .over appropriatebe usedsatisftyw) curly (vcurlw divu gradpdivv w x u-f).(w x v)J&Q0,to deducefuncthat
jfn[divw div{:- (curlu - w)- C(2.12) (vcurlw gradp w u - f). (vcurl x u)] d 0,and(2.13)(vcurlIn (2.11)-(2.13)notgo intothedetailtestfunctionsheresince gradp w x u - f). grad q d 0.(v, q, ) are requiredwe are primarilyto belonginterestedto suitablein finitefunctionelementspaces;discretizationswe doof theseequations.Clearly2.3- Theany solutiontwo-dimensionalFor planaronly.Then,(u, ,p)of, say (2.1)-(2.7),satisfies(2.11)-(2.13).caseflows we havewe have thatthatu -- (ul,v , 0) T and ul,w -- (0, 0, w) T whereu2,w a /axland p are- aul/ax2.functionsof xl and x2In this case,the system(2.1)-(2.4)simplifiesto0ul0u(2.14)(2.15)h 2axl 0 ma,aulax2w Oop(2.1s)v- Oxxlinf/,- u2w f linnandop(2.17)whereina,f (fl,f2, O)r.Theboundaryconditions(2.5)-(2.7)the boundary conditions (2.8)-(2.9)remain unchanged.conditions (2.11)-(2.13)alsosimplifyin the obvious manner2.4- FiniteelementmethodsStartingwithweakdefinedh.thein a completelyFor example,for a givenpolynomialsof degreeIn thisthecaseS h {vlviformulationstandardpositiveless thanparameter(2.11)-(2.13),manner.or equalh maya finiter, S h couldto r with respectbe relatedto thefiniteelementconsist(2.5)size of the},elementspaceand(2.7)andgrid.methodcan beS h parametrizedof continuousto a subdivision S h, i 1,2,3},v0to justfortwo-dimensional problems.a conformingWe chooseintegerreduceThe functional (2.10) and the necessary(overof f intoWe thenf/) piecewisefinitedefinebyelements.thespaces
Zoh { eSh l .n Oonr},andQ {qESFor theboundaryconditionsWh E S h, and ph E S h such(2.5)-(2.7),thatI q(xo) 0}.thediscreteproblemis definedu h U h and w h n -- W h on F, ph(xo)(2.1s)seek u h E S h,as follows: Po,' divv (vcurlw[div w h div h -h gradp h w h u h - f).(w h Vh)] dY v h E V , 0(curl u h - wh).(2.19) (Vcurlwh gradph Wh U h f).(vcurlv h h Uh)] d.Q 0and(2.20)vcurlwhare satisfied.example,thatdefineby the sameThe discretemustbe suitablythatPRACTICAL3.1- Newton'smethodTheequationsbe solvedhereused,it shouldHessianfor the boundaryto accountfor theconsider(2.18)-(2.20)Newton'sbe chosenso thatand curl UI n, respectively.interpolantsof the latterdefined(2.8)-(2.9)respectare approxi-to a single grid.is also given by (2.18)-(2.20)on F and the testboundarypair.of u h and h,withfunctionsconditionsForspacesexceptin (2.18)-(2.20)v h n 0 andqh 0 on F.to U2 and P, respectively.THEORETICALmanner.UIV q h E Q othe componentspolynomialsconditions 0ISSUESarea nonlinearThereare manymethod.However,it preservessystemmethodsthatof algebraicone mightwe do remarksymmetrythatand positiveequationsthatuse for sucha purpose;if a quasi-Newtondefinitenessmustmethodisof the approximatematrices.Newton's(o),ANDin an iterativewe onlyqh andpieceuriseare approximations3.discretei.e.,u h n Uh and ph phredefinedU2h and phof the datapair to be boundaryvariables,degree continuousproblemnow we haveAgain,the formerall of the discretethat Wh X Uh --f)-gradqh&'2U h and W h are approximationswe couldNotematedHere gradphmethodfor the solutionof (2.18)-(2.20)and p(0) for u h, w h, and ph, respectively,is definedthe sequenceas follows.of NewtonGiveniteratesinitialguesses{ u (k), (k),u ( ),p(k) }k o
isgenerated recursivelyby solving,fork 1,2,.,the systemA[(curlu (k)- w( )) -curlv h divu (k)divv h (ucurlw( ) gradp (k) w (k)x u (k-i) w (k-i) u(k)) (w(k-i) v h)(3.1) (ucurlw(k-l) gradp (k-l)%w (k-l)x u ( -I)- f). (w(k)x vh)] d (ucurlw (k-l) gradp ( -i) 2w (k-i)x u(k-l)) (w( -i) x v h)dr/ V v h E V ,fn [(divw( )div- (curlu (k)- w( )) ( (ucurlw(k) gradp(k) w (#)x u ( -I) (k- ) x u(k)) ( curl ) (ucurlw(k) gradp (k) w (k)x u (k-l) w(k-i) uCk)).((.hx U (k-z))(3.2) (ucurlw(k-i) gradp(k-i) w(k-i) x u ( -i)--f). ( u(k))]d.Q /,[v(f (k-1)x uCk-1)) (r,curlw ( -l) curl h gradp (e-i) 2w ( -1)x u(*-l)) ( .h x u(k-1))]d.FtV Zoh ,andn( ,curlw(k) gradp(k) w (k) x u (k-i) w (k-i)x u(k)) gradqhdfl(3.3) j/a(f t ( -l)The systemiteratefromFirst,theit is easythe Hessianis exactly9, i.e.,ratheris symmetricvalueof themaySurelyonedifficultynumber.Thisnumber.is usedto determineit also hasMoreover,positiveTheseThissomethe k-thveryin a neighborhooddefinite;in a neighborhooddefinite.method,well ask whatfor solvingis knownobservationand positiveVq h E Qoh .observations,is independentalongare potentiallygoodfeatures.of a minimizer,but this Hessianof a solutionfeatureNewtonwiththematrixof (2.18)-(2.20),of the value ofguaranteedlocalvery advantageous.methodsonemethodthatof (2.10) is necessarilyThus,d.However,is symmetric.of Newton'spointiterative(3.1)-(3.3)of (3.1)-(3.3).Reynoldsconvergence gradqhformidable.this system(3.1)-(3.3)At thisequationsfor the functionalContinuationrelatedlooksmatrixof theincreases?1)-stto see thatmatrixquadratic3.2-(k -algebraicthe coefficientthe systemandof linearx u ( 'l))is thatis thatthenonlinearto be uations,decreasesfor otherdiscretizationsdefinitenessmethodfor Newton'sas themethod,in size withof theof the Hessiannumberor for uations.is guaranteedotheronlyAin a
neighborhoodReynoldsdoesn'tof a minimizer;number.convergeis, of course,methodsAs a result,and/orcontinuationthe linearthatfor an arbitrarywhileto determineand also is suchthe size of thisthat the coefficientunacceptable,for solvingIn orderagainsystemsor homotopymatrixmethods,we wantwitha solution.ReMany valueNow,in (3.1)-(3.3)among others. Re.ReynoldssupposeWe denoteof m, we obtainthedefineis withinnumber,(u ,wm,p, )thatNewton'smethodThe formerthe use of simplethe Newtoniterativeiterates.ball of Newton'sk 1 is positivedefinite,a simplemethod,one can usecontinuation(2.18)-(2.20)method;in the form 0.of increasingof (2.18)-(2.20)by Newton'swith increasingdefinite.i.e., the value of the Reynoldswe have a sequencesolutionprecludethe systemRe)havethe attractionwithexpressdecreasesis not positiveHere we describeF (uh, h,ph;the targetwouldthatthatsee, e.g., [18], [19], or [21]. Let us symbolicallyHere Re 1/ , denotesguess we mayoccuranceguesssurelyin (3.1)-(3.3)(3.1)-(3.3)an initialthe coefficientinitialmatrixthe latterneighborhoodReynoldsnumbersfor u,,, 1/Remethod,numberbyi.e., we solveat which{Re, )M I(u, ,a3, ,p, ).the sequenceForof linearsystems(3.4)Here,F' denotesthe Jacobian(u ),,(0) , .,(0) j to be used-.rnthatRe1is sufficientlyof F with respectto start,smallfor eachso thatbe the solutionof a discretelineartake for .ul" (o) ,t#l (o) ,Pl (o),] the solutionu 1.remainingi.e.,Thelatterinitialby solvingproblemguessesthe linearto (u h, a , ph).them,iterationStokesiteration(3.4)problem.For example,witha symmetric,positive{(u ), --m,(0) ,k. n. (0) IMJJrn 2withconvergesof (2.18)-(2.20)involvesalgebraictheWe need to specifyrespectto k in (3.4).are determinedguessesAssumeif l.(0) Wl .(0) F1. (0) ! is chosenkUlwe couldall cross productdefinitethe initialchoosetermslinearalgebraicby "continuingRe;deletedalongto 1 andandwithsystem.Thethe andcoefficientFRe denotesmatrixof the linearfor the last iterationThe combinedsufficientlysmall,denotesthe Frechetof (3.4)theconvergedderivativesystem(3.5)themayat the (m - 1)-stNewton-continuationuse of (3.5) shoulditeratesdeterminedof F with respectbe chosen(3.4)to be the sameat theRe.as the(m dfromto the parameteris nowinitial7completelyguessesthatdefined.are withinIf (Remthe attractionRein-l)ballisof
Newton'smethodandsuchthat the coefficientmatrices(Tileysince Newton'sare alwayssymmetric,of course.)In fact,convergent,i.e., its attractionball is nontrivial,the Hessianmatrixdefiniteis positiveone can guaranteepositivethatdefinitecan be madeby an iterativeonly for eitherone can restartWe notemethodby choosingNewton-continuationmethod,positiveis too large,eitherk 1 are positiveis guaranteedto be locallyof a minimizer(Re,n-Re l)methoddefmite.shouldfor whichsufficientlysmall,only deal with symmetric,matrices.The methodare solvedwithand since the neighboorhoodis also nontrivial,the combinedin (3.4)oftentheFor example,e.g., theconjugatematrices.Then,Newtonthe m-thstagethatmethod,if we have choseniterationthe even simplersupposegradientor theby choosingalongsystemsworks,an incrementconjugatea smaller"continuationthe linearthatgradientvalue(Reiterationof Rea constant"in (3.4)or workswell,-Re, l)will fail.in (3.4)Inand (3.5).method(3.6)for generatingdisadvantage(3.5)initialguesseshas beenfoundto work well in viscousof (3.6)it thatit breaksdownin the vicinitycan be ammendedWe alsodifficultynoteso thatthatof not havingtherepositiveis to add to the diagonalthe residualmatrixdefiniteentriesof the previousF' on theit can handleHessianiterate.singularpossiblematrices.of the coefficientNewtonleft-handsuchare modificationsmatrixflow calculations;of bifurcationpoints;points,see [18], [19], andto Newton'smethodFor example,one simplein (3.1)-(3.3)In the notationsee [12]. Theor turningthata multipleof (3.4),while[21].circumventthesuch modificationof the magnitudewe wouldreplacetheofJacobianside with(3.7)where]]. ]] denotessurethatof thematrixproblem,properties3.3-thethe Euclideanrepresentedthetermof Newton'sEnhancingmass(2.1)div u h0 exactly.elementspace,if we replacetheIn fact,oneTheHowever,zero so thatlargewe can makeas we approachwe recoverthe solutionthe localconvergenceof the solution.methodpotentialin conservingdiscussedshowcontainsthatby a[divulto 1/v .at besta weight2, whereThus,advantagehereall polynomialsC by introducingin (2.10)to be proportionalis anotherinterestedcan easilyto any element,]divul 2 termThis7 sufficietnlyconservationthe size of the constantC can be showndiv u h small./ approachesone is especiallyrestrictedthe constantdefinite.in the neighborhoodas well as possible.We can reduceBy choosingby (3.7) is positivemultiplyingmethodIn many instancesequationlength.i.e., satisfyingis not exactlydiv u h -. Ch"of degreewheneverless thaninto the functionala largeleast-squaresthe continuitymass conserving,a 0 is a constant,by choosingof themass,valuemethod.theor equal(2.10).then/ e.,thefiniteto r.Indeed,constantof a, one can makeHowever,one must
keepin mindthat thelargerthevalueofa, theto the sfor the nonlinearproblem,concerningfor thefollowingStokesone satisfiesthe momentumequationrelativeequation.- TheoreticalErrorworseFirst,at leastleast-squaresusingthe theorydementof [2] (seeNavier-Stokesawayaccuracyfinitealso [6] andequationsfrom singularpoints.the sameNow for the Stokesdiv u 0(3.9)curl u - w 0pcurlware derived[9]), one can showare essentially(3.8)(3.10)approximationsthroughthatas thosethethe errorfor the linearproblemin f , gradpin f , fin r estimatingbe ellipticthe errorandthatacrossof solutionsof e complementingmerelysquareof (2.1)-(2.7)On the otherall the requisiteare usedsmoothhand,withrespecterrorto use of the theoryestimates.andand, undermildrestrictionsthatare merelyderivativesconditions(2.5)-(2.7)the derivatives(2.5)doesnotof u, w, and p befor least-squaresshow thatof [1] andFor example,of fl, thenthe systemtherefore(3.8)-(3.11)one mayif continuousandfiniteone can ultimatelyUh[1 IW -whll [p-philthatpolynomialsconclude Ch"on the domain,Ilu - uhllo ll" -- hllo lip-- philo nof theFor the one requiresestimationtheapproximationselementbe used.to a triangulation(3.12)(3.13)satisfyof the problem.errorelementbe able to boundof [1] whenone can econditionsin orderof thebe carriedThus,of least-squaresone mustconditionsintegrable.discretizationsmay obtainboundaries,in termsin 2the boundaryof [1]; see, e.g., [4], [5], and [23]. Moreover,continuous 0,thatdoesin thatsatisfycase oneof degree rfor sufficiently
whereh denotesL2(f )-normof theNothing(2.1)'(2.7)usinga measurewe havederivatives,saidstudy- veof (2.8)-(2.9)withand thefiniteAll we ationsso far is thatofwe cannot,this is the motivationfor theellipticproblemsthe problemAp A denotestheLaplacianoperator,gin ,andp PonF.letThenu gradpwe haveThusdivu(3.16)-(3.17)first-order(3.17)is a first-ordersystemis not ellipticare not stable;(3.18)thenit can be shownthenp is alsoto (3.14).However,and thatin generalleast-squaresfiniteif one considerssameterms,as (2.8).for appropriatelyare identicalWe can thendefinedaccuracyessentiallyof dimension.theis elliptic.of (3.14)-(3.15).sameas thatThus,hereoptimalsystemsSTUDYfor thewe restrictifpis a solutionmethodsthisfor (3.16)-inf/,of
certain differences as well, especially in the order in which the least-squares, the diseretization, and the linearizations steps are taken. Furthermore, the analyses found in some of these papers are incorrect, leaving open the question of the accuracy of approximations. In §2, we define the least-squares finite element method.
Linear Least Squares ! Linear least squares attempts to find a least squares solution for an overdetermined linear system (i.e. a linear system described by an m x n matrix A with more equations than parameters). ! Least squares minimizes the squared Eucliden norm of the residual ! For data fitting on m data points using a linear
5.8.1 The Compatible Least-Squares Finite Element Method with a Reaction Term 177 5.8.2 The Compatible Least-Squares Finite Element Method Without a Reaction Term 181 5.9 Practicality Issues 182 5.9.1 Practical Rewards of Compatibility 184 5.9.2 Compatible Least-Squares Finite Element Methods on Non-Affine Grids 190
least-squares finite element models of nonlinear problems – (1) Linearize PDE prior to construction and minimization of least-squares functional Element matrices will always be symmetric Simplest possible form of the element matrices – (2) Linearize finite element equations following construction and minimization of least-squares. functional
An adaptive mixed least-squares finite element method for . Least-squares Raviart–Thomas Finite element Adaptive mesh refinement Corner singularities 4:1 contraction abstract We present a new least-squares finite element method for the steady Oldroyd type viscoelastic fluids.
3.2 Least-squares regression, Interpreting a regression line, Prediction, Technology: Least-Squares Regression Lines on the Calculator Interpret the slope and y intercept of a least-squares regression line in context. Use the least-squares regression line to predict y f
Bodies Moving About the Sun in Conic Sections", and in it he used the method of least squares to calculate the shapes of orbits. Legendre published about least squares in 1805, 4 years before. However, Gauss claimed to have known about least squares in 1795. .
For best fitting theory curve (red curve) P(y1,.yN;a) becomes maximum! Use logarithm of product, get a sum and maximize sum: ln 2 ( ; ) 2 1 ln ( ,., ; ) 1 1 2 1 i N N i i i N y f x a P y y a OR minimize χ2with: Principle of least squares!!! Curve fitting - Least squares Principle of least squares!!! (Χ2 minimization)
ANALISIS PENERAPAN AKUNTANSI ORGANISASI NIRLABA ENTITAS GEREJA BERDASARKAN PERNYATAAN STANDAR AKUNTANSI KEUANGAN NO. 45 (STUDI KASUS GEREJA MASEHI INJILI DI MINAHASA BAITEL KOLONGAN) KEMENTERIAN RISET TEKNOLOGI DAN PENDIDIKAN TINGGI POLITEKNIK NEGERI MANADO – JURUSAN AKUNTANSI PROGRAM STUDI SARJANA TERAPAN AKUNTANSI KEUANGAN TAHUN 2015 Oleh: Livita P. Leiwakabessy NIM: 11042103 TUGAS AKHIR .
