EIGENVALUE EXTRAPOLATION FORBIQUADRATIC FINITE ELEMENT APPROXIMATION IN RECTANGULAR DOMAIN

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

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…     

THE FINITE ELEMENT PROBABILITY COMPUTATIONAL METHOD FOR PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS

1.IntroductionThefiaiteelementprobabilitycomputationalmethodisakindofnumericalcalculationtecboqueofauniquestyle.Themethodshavebeendevelopedduringthefiftiestosixesforuseindeferencemethodll'2'3)andforsolvingsystemofalgebraicequations.Inengineeringandphysicsproblems,oftenwehavetosolvetheconcelltratedloadproblem.Inthiscase,weneedonlytoknowthevaluesofthefunctionatafewnodalpoilltsofthedomain.Itwillwastealotoftimetocomputetheunnecessaryvaluesofthefunctionbythefiniteelemelltmethods(ordeferencemethods…     

FINITE ELEMENT ERROR EXPANSION FOR NONUNIFORM QUADRILATERAL MESHES

An error expansion for isoparametric bilinear finite element approximation isestablished with some kind of nonuniform quadrilateral mesh constructed in the followingway:We first decompose the considered polygonal domain into several fixed convex macro-quadrilaterals and then link up some equi-proportionate points of the opposite edges in eachmacro-quadrilateral and form a quadrilateral mesh which may not be uniform and so isuseful in the adaptive refinement.     

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, […     

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…     

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

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

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.     

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.     

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.     

THE SPLITTING EXTRAPOLATION FOR FINITEELEMENT METHOD

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

FINITE ELEMENT EXPANSION FOR VARIABLE COEFFICIENT ELLIPTIC PROBLEMS

The Ritz projections of the elliptic problem(?)(a_(ij)(?)_ju) c_0u=f in Ω,u=0 on (?)Ω,and the quasilinear eiliptic problem(?)(a_i(x,Du)) a_0(x,Du)=0 in Ω,u=0 on (?)Ωwith linear finite elements admit an asymptotic error expansion,respectively,for certain classes of“uniform” meshes.This provides the theoretical justification for the use of Richardson extrapolationfor increasing the accuracy of the scheme from O(h~2) to O(h~4 |ln h|).     

HIGHLY NONCONFORMING FINITE ELEMENT APPROXIMATIONS FOR A FOURTH ORDERVARIATIONAL INEQUALITY WITH CURVATURE OBSTACLE

The purpose of this paper is to obtain the optimal error estimates of O(h) for the highly nonconforming elements to a fourth order variational inequality with curvature obstacle in a convex domain with simply supported boundary by using the novel function splitting method and the orthogonal properties of the nonconforming finite element spaces.Morley‘s element approximation is our special case.     

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.     

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.     

ACCURACY ANALYSIS FOR QUASI-CAREY ELEMENT*

In this paper, a new triangular element （Quasi-Carey element） is constructed by the idea of Specht element. It is shown that this Quasi-Carey element possesses a very special property, i.e., the consistency error is of order O（h^2）, one order higher than its interpolation error when the exact solution belongs to H^3（Ω）. However, the interpolation error and consistency error of Carey element are of order O（h）. It seems that the above special property has never been seen for other triangular elements for the second order problems.     

THE SUPERCONVERGENCE ANALYSIS OF AN ANISOTROPIC FINITE ELEMENT

This paper deals with the high accuracy analysis of bilinear finite element on the class of anisotropic rectangular meshes. The inverse inequalities on anisotropic meshes are established. The superclose and the superconvergence are obtained for the second order elliptic problem. A numerical test is given, which coincides with our theoretical analysis.     

CHARACTERISTIC ALTERNATING-DIRECTION FINITE ELEMENT METHODS FOR NONRECTANGULAR REGIONS FOR COUPLED SYSTEM OF DYNAMICS OF FLUIDS IN POROUS MEDIA AND ITS ANALYSIS

For coupled system of multilayer dynamics of fluids in porous media, the characteristic alternating-direction finite element methods for nourectangular regions applicable to parallel arithmetic are put forward and two-dimensional and three-dimensional schemes are used to form a complete set. Some techniques, such as calculus of variations, isoparametric transformation, patch approximation, operator-splitting, characteristic method, negative norm estimate, energy method, the theory of prior estimates and techniques are used. For the nourectangular regions case, optimal order estimates in L^2 norm are derived for the error in the approximation solution. Thus the well-known theoretical problem has been thoroughly and completely solved. These methods have been successfully used in multilayer oil resources migration-accumulation numerical simulation.     

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.