THE DOMAIN DECOMPOSITION TECHNIQUES FOR THE FINITE ELEMENT PROBABILITY COMPUTATIONAL METHODS

1. IntroductionWe have discussed probability computational methods of finite element methods in [1,2,3,4].Probability computational methods have many advalltages. The advantages are more obviousin three dimensions for finite element problems. In probability model, the time that the particlemoves from one place to the boundary of the domain and is absorbed by the boundary (whichis called the time of running up to the boundary for short) is only dependent on the averagedistance between the start…     

GLOBAL SUPERCONVERGENCE FOR A MIXED FINITE ELEMENT METHOD FOR THE WAVE EQUATION

1.IntroductionWeconsiderthesecondorderhyperbolicequationwithinitialconditionswhereaCRZisaboundeddomainwithboundariesparalleltotheaxes.Forconvenience,weassumehomogeneousDirichletboundaryconditions'Weassumethatthefunctionsailandfareuniformlyboundedandmeasurable,andthespatialoperatorisuniformlyellipticinthesensethatthereexistpositiveconstantsorandPsuchthatInamixedfiniteelementformulationbothdisplacementsandstressesareapproximatedsimultaneously.ThispropertyisimportantincomputingDarcyvelocitieso…     

A NEWTON MULTIGRID METHOD FOR QUASILINEAR PARABOLIC EQUATIONS

A combination of the classical Newton Method and the multigrid method, i.e., a Newton multigrid method is given for solving quasilinear parabolic equations discretized by finite elements. The convergence of the algorithm is obtained for only one step Newton iteration per level. The asymptotically computational cost for quasilinear parabolic problems is O（NNk） similar to multigrid method for linear parabolic problems.     

EXTRAPOLATION FOR FINITE ELEMENT METHODS OF THE FIRST KIND FREDHOLM INTEGRAL EQUATIONS

In this paper,a numerical analysis of finite element methods for Fredholm equationsof the first kind,which is based upon regularization,is made.It shows that extrapolationalgorithms can accelerate the convergence of approximate solutions.     

A HIGH ACCURACY ANALYSIS OF THE MIXED FINITE ELEMENT METHOD FOR BIHARMONIC EQUATIONS

In this paper,the linear mixed finite element method is considered forthe biharmonic Dirichlet boundary value problem.Almost optimal error estimates,superconvergence estimates and the derivative error expansion are obtained.     

A NEW STABILIZED FINITE ELEMENT METHOD FOR SOLVING THE STOKES AND NAVIER-STOKES EQUATIONS PART II. THE NONLINEAR CASE

1 IntroductionIt is necessary to make some conuneots on some of the re1ated methodologies lately emp1oyedto stabilize the Ga1erkin finite e1ement method for the incompressible NavierStokes equations,including the Stokes equation and the advectiondiffusion problem. Owing to the fact that thehigh Reynold number amounts to a small diffusive term in the advection--diffusion problem,here we will repeatedly mention the diffosion, not the Reynold number or the viscosity in the.. Jincompressible Na…     

A NEW STABILIZED FINITE ELEMENT METHOD FOR SOLVING THE STOKES AND NAVIER-STOKES EQUATIONS PART I. THE LINEAR CASE

1 IntroductionDuring the past decade, many stabilized finite element methods were designed for solvingproblems arising from fluid and solid mechanics. For example, the Navier--Stokes problem hasbeen playing an active role in the development of stabilized methods, e.g. see [1--6], etc andreferences therein. It is well known that the standard GaJerkin method for the Navier--Stokesproblem is subject to two main obstacles. One comes from the high Reynolds nUmber or highadvection, or the limiti…     

THE SPLITTING EXTRAPOLATION FOR FINITEELEMENT METHOD

1.IntroductionDespitethesignificantprogressmadeindigitalcomputersandcomputationalmethodsoaferthelasttwodecades,computationalworkaswellascomputerstoragerequiremelltintheappliedfieldissogreatthatsuper-computersarenotabletoaft'ord.111recentyears)thehotpoilltsincomputationalmathematicsarehowtodevelopanealcomPlltationalmethodwhichwouldfulfilthefollowingrequirelllents;1)ItshouldbeaparallelcomputationalIned]'>(lwhichcan11eimplementedonrnultiprocessorcomputersreqlliringminimalcommunicationanloll!{pro(…     

SUPERCONVERGENCE OF THE MORTAR FINITE ELEMENT APPROXIMATION TO A PARABOLIC PROBLEM

1.IntroductionDomaindecompositionapproachesaremostimportanttechniqllesforsolvingpartialdifferentialequations.Suchreformulationsareusuallymotivatedbytheneedtocreatesolverswhichareeasilyparallelised,see[1]and[2].MostapplicationsusediscretizationsgridswhicharedeAnedgloballyoverthewholedomainandthensplitilltosubdomains.Inmechanics,.thisresultsinanoverallconformingapprotimatiollofthevelocityfield.However,itmightbemoreconvethelltandefficienttouseapproximationswhicharedefinedindependelltlyoneachsubd…     

AN EXTRAPOLATION METHODFOR FINITE ELEMENT APPROXIMATION OF INTEGRODIFFERENTIAL EQUATIONS WITH PARAMETERS

In this paper,we discuss the accelerating convergence method for finite elementapproximation of integro-differential equations with parameters.As applications,we give theerror estimates of finite element for the first kind of Fredholm integral equation,particularlyfor the Volterra integral equation with kernel condition k(x,x)=0.     

FINITE ELEMENT METHOD AND ANALYSIS FOR CHEMICAL-FLOODING SIMULATION

1.IntroductionThisarticlediscussestheenhanced0ilrecoverynumericalsimulationofthechemicalflood-ing(suchassurfactats,alcohol,polymers)c0mposed0f3-dimensional,multicomponent,muLtiphaseandincompressiblemixedfluids.Aftertheoilfieldiswaterfi0oded,thereisstillalargeamountofcrudeoilleftintheoildeposit.Thisamountofoil,boundedbycapillarypressure,cannotflow;or)asaresultofthedisadvantageousrateofflowbetweenthedisplacinganddisplacedphases,thevolumethattheinjectionfiowaffectsistoosmalltodrivethecrudeoil.By…     

HIGH-ACCURACY FINITE ELEMENT METHOD FOR OPTIMAL CONTROL PROBLEM

1 IntroductionOptimal design or control is playing an increasingly importal role in engineering designwork. Efficient numerical methods are among the keys to successful application of optimalcontrol in practical work. With its wide range of application in scielltific and engineeringnumerical simulation, finite element approximation of optimal control problems plays a veryimportant role in numerical method of these problems. There have been eXtensive studies inthis respect, see, for example, […     

THE EXTRAPOLATION METHOD FOR A CLASS OF NON-SMOOTH FREDHOLM INTEGRAL EQUATIONS

For a class of non-smooth Fredholm integral equations we prove thatRichardson extrapolation method can be used to increase accuracy for finite ele-ment methods.On the basis of local refinement mesh,a superconvergence result isshown.Our theory also covers a class of non-smooth Wiener-Hopf equations andan application includes the calculation in certain linear elastic fracture problems.     

FINITE ELEMENT ERROR EXPANSIONS FOR THE EIGENVALUE APPROXIMATION TO THE MULTIGROUP DIFFUSION EQUATIONS

Asymptotic expansions for the finite element approximation to the eigenvalues of the mult-igroup diffusion equations on Ω(?)R~n(n=2,3)of reactor theory are given,firstly for a piecewiseuniform triangulation,then for nonuniform quadrilateral meshes,and finally for nonuniformhexahedral meshes.The effect of certain classes of numerical integrations is studied.As applicationsof the expansions,several extrapolation formulas and a posteriori error estimates are obtained.     

EIGENVALUE EXTRAPOLATION FORBIQUADRATIC FINITE ELEMENT APPROXIMATION IN RECTANGULAR DOMAIN

1.IntroductionItiswellknownthattheextrapolationmethodscanimprovetheaccuracyoffiniteelementsolutionandarewellstudiedinmanypapers.Wereferto[1--4]forafundamentalworkonasymptoticerrorexpansionsandextrapolationsandto[5]forasurveyonextrapolationtechniques.Forth…     

THE MODIFIED METHOD OF CHARACTERISTICS WITH FINITE ELEMENT OPERATOR—SPLITTING PROCEDURES FOR COMPRESSIBLE MULTICOMPONENT DISPLACEMENT PROBLEM

For the three-dimensional compressible multicomponent displacement problem we put forward the modified method of characteristics with finite element operator-splitting procedures and make use of operator-splitting,characteristic method,calculus of variations,energy method,negative norm estimate,two kinds of test functions and the theory of prior estimates and techniques.Optimal order estimates in L^2 norm are derived for the error in the approximate solution.These methods have been successfully used in oil-gas resources estimation,enhanced oil recovery simulation and seawater intrusion numerical simulation.     

NONLINEAR GALERKIN METHOD FOR STEADYNONLINEAR DIFFERENTIAL EQUATIONS

1.IntroductionNonlinearGalerkinmethodsarenumericalschemesforthedissipativeevolutionpartialdifferelltialequationswherethespatialdiscretizationreliesonanonlinearmanifoldinsteadofalinearspaceasintheclassicalGalerkinmethod.Moreprecisely,oneconsidersafinitedim…     

IMPLICIT—EXPLICIT MULTISTEP FINITE ELEMENT METHODS FOR THE SEMICONDUCTOR DEVICE PROBLEM

The transient behavior of a semiconductor device consists of a Poisson equation for the electric potential and of two nonlinear parabolic equations for the electron density and hole density.The electric potential equation is discretized by the finite element method.The electron and hole density equations are treated by implicit-explicit multistep finite element methods.The optimal L^2-norm error estimates are derived.     

MULTILEVEL CORRECTION FOR COLLOCATION SOLUTIONS OF VOLTERRA NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS

1. IntroductionWe consider a Volterra integro-differential equationwith initial condition u(0) = not where f, P, k are continuous on their respective domainsJ, JxR and axs (n:~ {(t,s): 05s5t5T}) such that (1.l) possesses a unique solutionu E C'(J).It has been shown in [1] that the collocation approximation for (1.1) by discolltinuous piece--wise polynomial spline collocation at the Gauss points restore optimal local superconvergenceat the knots but does not yield global superconvergence on…     

INTERPOLATED FINITE ELEMENT METHODS FOR SECOND ORDER HYPERBOLIC EQUATIONS AND THEIR GLOBAL SUPERCONVERGENCE

We develop the interpolated finite element method to solve second-order hy-perbolic equations. The standard linear finite element solution is used to generate a newsolution by quadratic interpolation over adjacent elements. We prove that this interpo-lated finite element solution has superconvergence. This method can easily be applied togenerating more accurate gradient either locally or globally, depending on the applications.This method is also completely vectorizable and parallelizable to take the advantages ofmodern computer structures. Several numerical examples are presented to confirm ourtheoretical analysis.