THEORY AND APPLICATION OF FRACTIONAL STEP CHARACTERISTIC FINITE DIFFERENCE METHOD IN NUMERICAL SIMULATION OF SECOND ORDER ENHANCED OIL PRODUCTION

A kind of second-order implicit fractional step characteristic finite difference method is presented in this paper for the numerically simulation coupled system of enhanced （chemical） oil production in porous media. Some techniques, such as the calculus of variations, energy analysis method, commutativity of the products of difference operators, decomposition of high-order difference operators and the theory of a priori estimates are introduced and an optimal order error estimates in l^2 norm is derived. This method has been applied successfully to the numerical simulation of enhanced oil production in actual oilfields, and the simulation results ate quite interesting and satisfactory.

L^p-L^q ESTIMATES FOR HIGHER ORDER PERTURBED HYPERBOLIC EQUATION

In this paper Lp Lq estimates for the solution u(x,t) to the following perturbed higher order hyperbolic equation are considered,(οtt-aΔ)(οtt-bΔ)u+V(x)u=0,\ x∈Rn,n≥6, οjtu(x,0)=0,\ ο3tu(x,0)=f(x),\ (j=0,1,2).We assume that the potential V(x) and the initial data f(x) are compactly supported, and V(x) is sufficiently small, then the solution u(x,t) of the above problem satisfies the same Lp Lq estimates as that of the unperturbed problem. Received November 25,1996. Revised April 14,1997.1991 MR Subject Classification:35L05,35B20,35B45.

Convergence of a Sinusoidal Series with an Infinite Integral

In this paper, we study the relationship between the convergence of the sinusoidal series and the infinity integrals (any real number α ∈[0,1], parameter p > 0). First of all, we study the convergence of the series (any real number α ∈[0,1], parameter p > 0), mainly using the estimation property of the order to obtain that the series diverges when 0 p ≤1-α, the series converges conditionally when 1-α p ≤1, and the series converges absolutely when p >1. In the next part, we study the convergence state of the infinite integral (any real number α ∈[0,1], parameter p > 0), and get that when 0 p ≤1-α, the infinite integral diverges;when 1-α p ≤1, the infinite integral conditionally converges;when p >1, the infinite integral absolutely converges. Comparison of the conclusions of the above theorem, it is not difficult to derive the theorem: the level of and the infinity integral with the convergence of the state (any real number α ∈[0,1], the parameter p >0), thus promoting the textbook of the two with the convergence of the state requires the function of the general term or the product of the function must be monotonically decreasing conditions.

A NEW NONCONFORMING MIXED FINITE ELEMENT SCHEME FOR THE STATIONARY NAVIER-STOKES EQUATIONS

In this article, a new stable nonconforming mixed finite element scheme is proposed for the stationary Navier-Stokes equations, in which a new low order Crouzeix- Raviart type nonconforming rectangular element is taken for approximating space for the velocity and the piecewise constant element for the pressure. The optimal order error estimates for the approximation of both the velocity and the pressure in L2-norm are established, as well as one in broken H1-norm for the velocity. Numerical experiments are given which are consistent with our theoretical analysis.

P_1-nonconforming triangular finite element method for elliptic and parabolic interface problems

The lowest order Pl-nonconforming triangular finite element method （FEM） for elliptic and parabolic interface problems is investigated. Under some reasonable regularity assumptions on the exact solutions, the optimal order error estimates are obtained in the broken energy norm. Finally, some numerical results are provided to verify the theoretical analysis.

CHARACTERISTIC FINITE DIFFERENCE ALTERNATING-DIRECTION METHOD ANDANALYSIS FOR NUMERICAL RESERVOIR SIMULATION

Petroleum science has made remarkable progress in organic geochemistry and in the research into the theories of petroleum origin, its transport and accumulation. In estimating the oil-gas resources of a basin, the knowledge of its evolutionary history and especially the numerical computation of fluid flow and the history of its changes under heat is vital. The mathematical model call be described as a coupled system of nonlinear partial differentical equations with initial-boundary value problems. This thesis, from actual conditions such as the effect of fluid compressibility and the characteristic of large-scal science-engineering computalion. puts forward a kind of characteristic finite difference alternating-direction scheme. Optimal order estimates in L-2 norm are derived for the error in the approximate solutions.

A LOW ORDER NONCONFORMING MIXED FINITE ELEMENT METHOD FOR NON-STATIONARY INCOMPRESSIBLE MAGNETOHYDRODYNAMICS SYSTEM

A low order nonconforming mixed finite element method(FEM)is established for the fully coupled non-stationary incompressible magnetohydrodynamics(MHD)problem in a bounded domain in 3D.The lowest order finite elements on tetrahedra or hexahedra are chosen to approximate the pressure,the velocity field and the magnetic field,in which the hydrodynamic unknowns are approximated by inf-sup stable finite element pairs and the magnetic field by H^(1)(Ω)-conforming finite elements,respectively.The existence and uniqueness of the approximate solutions are shown.Optimal order error estimates of L^(2)(H^(1))-norm for the velocity field,L^(2)(L^(2))-norm for the pressure and the broken L^(2)(H^(1))-norm for the magnetic field are derived.

A Second Order Nonconforming Rectangular Finite Element Method for Approximating Maxwell's Equations

Abstract The main objective of this paper is to present a new rectangular nonconforming finite element scheme with the second order convergence behavior for approximation of Maxwell＇s equations. Then the corresponding optimal error estimates are derived. The difficulty in construction of this finite element scheme is how to choose a compatible pair of degrees of freedom and shape function space so as to make the consistency error due to the nonconformity of the element being of order O（h^3）, properly one order higher than that of its interpolation error O（h^2） in the broken energy norm, where h is the subdivision parameter tending to zero.

A multiple maneuvering targets tracking algorithm based on a generalized pseudo-Bayesian estimator of first order

We describe the design of a multiple maneuvering targets tracking algorithm under the framework of Gaussian mixture probability hypothesis density(PHD) filter.First,a variation of the generalized pseudo-Bayesian estimator of first order(VGPB1) is designed to adapt to the Gaussian mixture PHD filter for jump Markov system models(JMS-PHD).The probability of each kinematic model,which is used in the JMS-PHD filter,is updated with VGPB1.The weighted sum of state,associated covariance,and weights for Gaussian components are then calculated.Pruning and merging techniques are also adopted in this algorithm to increase efficiency.Performance of the proposed algorithm is compared with that of the JMS-PHD filter.Monte-Carlo simulation results demonstrate that the optimal subpattern assignment(OSPA) distances of the proposed algorithm are lower than those of the JMS-PHD filter for maneuvering targets tracking.

THE OPTIMAL CONVERGENCE ORDER OF THE DISCONTINUOUS FINITE ELEMENT METHODS FOR FIRST ORDER HYPERBOLIC SYSTEMS

In this paper, a discontinuous finite element method for the positive and symmetric, first-order hyperbolic systems （steady and nonsteady state） is constructed and analyzed by using linear triangle elements, and the O（h^2）-order optimal error estimates are derived under the assumption of strongly regular triangulation and the Ha-regularity for the exact solutions. The convergence analysis is based on some superclose estimates of the interpolation approximation. Finally, we discuss the Maxwell equations in a two-dimensional domain, and numerical experiments are given to validate the theoretical results.

BOUNDARY DIFFERENCE-INTEGRAL EQUATION METHOD AND ITS ERROR ESTIMATES FOR SECOND ORDER HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION

Combining difference method and boundary integral equation method,we propose a new numerical method for solving initial-boundary value problem of second order hyperbolic partial differential equations defined on a bounded or unbounded domain in R~3 and obtain the error estimates of the approximate solution in energy norm and local maximum norm.

ANISOTROPIC CROUZEIX-RAVIART TYPE NONCONFORMING FINITE ELEMENT METHODS TO VARIATIONAL INEQUALITY PROBLEM WITH DISPLACEMENT OBSTACLE

In this paper, anisotropic Crouzeix-Raviart type nonconforming finite element meth- ods are considered for solving the second order variational inequality with displacement obstacle. The convergence analysis is presented and the optimal order error estimates are obtained under the hypothesis of the finite length of the free boundary. Numerical results are provided to illustrate the correctness of theoretical analysis.

A NOTE ON THE QUADRILATERAL MESH CONDITION RDP(N,ψ)

The main aim of this paper is to show that the quadrilateral mesh condition RDP（N, ψ） is only sufficient but not necessary for the optimal order error estimate of the Q isoparametric element in the Hi norm.

NODAL O(h^4)-SUPERCONVERGENCE IN 3D BY AVERAGING PIECEWISE LINEAR,BILINEAR,AND TRILINEAR FE APPROXIMATIONS

We construct and analyse a nodal O（h^4）-superconvergent FE scheme for approximating the Poisson equation with homogeneous boundary conditions in three-dimensional domains by means of piecewise trilinear functions. The scheme is based on averaging the equations that arise from FE approximations on uniform cubic, tetrahedral, and prismatic partitions. This approach presents a three-dimensional generalization of a two-dimensional averaging of linear and bilinear elements which also exhibits nodal O（h^4）-superconvergence （ultracon- vergence）. The obtained superconvergence result is illustrated by two numerical examples.

A Mixed-finite Volume Element Coupled with the Method of Characteristic Fractional Step Difference for Simulating Transient Behavior of Semiconductor Device of Heat Conductor And Its Numerical Analysis

The mathematical system is formulated by four partial differential equations combined with initial- boundary value conditions to describe transient behavior of three-dimensional semiconductor device with heat conduction. The first equation of an elliptic type is defined with respect to the electric potential, the successive two equations of convection dominated diffusion type are given to define the electron concentration and the hole concentration, and the fourth equation of heat conductor is for the temperature. The electric potential appears in the equations of electron concentration, hole concentration and the temperature in the formation of the intensity. A mass conservative numerical approximation of the electric potential is presented by using the mixed finite volume element, and the accuracy of computation of the electric intensity is improved one order. The method of characteristic fractional step difference is applied to discretize the other three equations, where the hyperbolic terms are approximated by a difference quotient in the characteristics and the diffusion terms are discretized by the method of fractional step difference. The computation of three-dimensional problem works efficiently by dividing it into three one-dimensional subproblems and every subproblem is solved by the method of speedup in parallel. Using a pair of different grids （coarse partition and refined partition）, piecewise threefold quadratic interpolation, variation theory, multiplicative commutation rule of differential operators, mathematical induction and priori estimates theory and special technique of differential equations, we derive an optimal second order estimate in L2-norm. This numerical method is valuable in the simulation of semiconductor device theoretically and actually, and gives a powerful tool to solve the international problem presented by J. Douglas, Jr.

Numerical analysis of history-dependent variational-hemivariational inequalities

In this paper,numerical analysis is carried out for a class of history-dependent variationalhemivariational inequalities by arising in contact problems.Three different numerical treatments for temporal discretization are proposed to approximate the continuous model.Fixed-point iteration algorithms are employed to implement the implicit scheme and the convergence is proved with a convergence rate independent of the time step-size and mesh grid-size.A special temporal discretization is introduced for the history-dependent operator,leading to numerical schemes for which the unique solvability and error bounds for the temporally discrete systems can be proved without any restriction on the time step-size.As for spatial approximation,the finite element method is applied and an optimal order error estimate for the linear element solutions is provided under appropriate regularity assumptions.Numerical examples are presented to illustrate the theoretical results.

IDENTIFICATION OF MULTIVARIATE ARMA MODELS

In this paper, we propose the estimates of orders and parameters in identifiable multivariate ARMA models. These estimates are direct and easy to be calculated, and can be proved to follow LIL (Law of iterated logarithm) and CLT (Central limit theorem) under some mild conditions.

CHARACTERISTICS-FINITE ELEMENT METHODS FOR SEAWATER INTRUSION NUMERICAL SIMULATION AND THEORETICAL ANALYSIS

Both numerical simulation and theoretical analysis of seawater intrusion in coastal regions are of great theoretical importance in environmental sciences. The mathematical model can be described as a problem of the initial boundary values for a system of 3-dimensional nonlinear parabolic partial differential equations, one being the pressure flow equation and the other is the concentration convection-dispersion equation of the salt contained. For a generic case of a 3-dimensional bounded region, a backward-difference time-stepping scheme is defined. It approximates the pressure by the standard Galerkin procedure and the concentration by a Galerkin method of charederistics, where calculus of variations, theory of prior estimates and techniques are made use of Optimal order estimates in H1 norm are derived for the errors in the approximate solution.

The Alternating-Direction Schemes and Numerical Analysis for the Three-dimensional Seawater Intrusion Simulation

Both numerical simulation and theoretical analysis of seawater intrusion in coastal regions are of great theoretical importance in environmental sciences. The mathematical model can be described as a coupled system of three dimensional nonlinear partial differential equations with initial-boundary value problems. In this paper, according to the actual conditions of molecular and three-dimensional characteristic of the problem, we construct the characteristic finite element alternating-direction schemes which can be divided into three continuous one-dimensional problems. By making use of tensor product algorithm, and priori estimation theory and techniques, the optimal order estimates in H1 norm are derived for the error in the approximate solution.