L^2-ERROR OF EXTRAPOLATION CASCADIC MULTIGRID (EXCMG)

Based on an asymptotic expansion of finite element, an extrapolation cascadic multigrid method （EXCMG） is proposed, in which the new extrapolation and quadratic interpolation are used to provide a better initial value on refined grid. In the case of multiple grids, both superconvergence error in H^1-norm and the optimal error in l2-norm are analyzed. The numerical experiment shows the advantage of EXCMG in comparison with CMG.

The Characteristics-mixed Finite Element Method for Enhanced Oil Recovery Simulation and Optimal Order L^2 Error Estimate

This article discusses the enhanced oil recovery numerical simulation of the chemical flooding(such as surfactants, alcohol, polymers) composed of two-dimensional multicomponent, ultiphase and incompressible mixed fluids. After the oil field is waterflooded, there is still a large amount of crude oil left in the oil deposit. By adding certain chemical substances to the fluid injected, its driving capacity can be greatly increased. The mathematical model of two-dimensional enhanced oil recovery simulation can be described

The Characteristic Finite Difference Methods for Enhanced Oil Recovery Simulation and L^2 Estimates

A 2-dimensional, multicomponent, multiphase, and incompressible compositional reservoir simulator has been developed and applied to chemical flooding (surfactants, alcohol and polymers) and convergence analysis. The characteristic finite difference methods for 2-dimensional enhanced oil recovery can be described as a coupled system of nonlinear partial differential equations. For a generic case of the cross interference and bounded region, we put forward a kind of characteristic finite difference schemes and make use of thick and thin grids to form a complete set, and of calculus of variations, the theory of prior estimates and techniques. Optimal order estimates in L^2 norm are derived for the error in the approximate solutions. Thus we have thoroughly solved the well-known theoretical problem proposed by a famous scientist, J. Douglas, Jr.

Superconvergence and L^(∞)-Error Estimates of the Lowest Order Mixed Methods for Distributed Optimal Control Problems Governed by Semilinear Elliptic Equations

In this paper, we investigate the superconvergence property and the L∞-errorestimates of mixed finite element methods for a semilinear elliptic control problem. Thestate and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions.We derive some superconvergence results for the control variable. Moreover, we derive L^(∞)-error estimates both for the control variable and the state variables. Finally, anumerical example is given to demonstrate the theoretical results.

A MIXED FINITE ELEMENT AND CHARACTERISTIC MIXED FINITE ELEMENT FOR INCOMPRESSIBLE MISCIBLE DARCY-FORCHHEIMER DISPLACEMENT AND NUMERICAL ANALYSIS

In this paper a mixed finite element-characteristic mixed finite element method is discussed to simulate an incompressible miscible Darcy-Forchheimer problem.The flow equation is solved by a mixed finite element and the approximation accuracy of Darch-Forchheimer velocity is improved one order.The concentration equation is solved by the method of mixed finite element,where the convection is discretized along the characteristic direction and the diffusion is discretized by the zero-order mixed finite element method.The characteristics can confirm strong stability at sharp fronts and avoids numerical dispersion and nonphysical oscillation.In actual computations the characteristics adopts a large time step without any loss of accuracy.The scalar unknowns and its adjoint vector function are obtained simultaneously and the law of mass conservation holds in every element by the zero-order mixed finite element discretization of diffusion flux.In order to derive the optimal 3/2-order error estimate in L^(2) norm,a post-processing technique is included in the approximation to the scalar unknowns.Numerical experiments are illustrated finally to validate theoretical analysis and efficiency.This method can be used to solve such an important problem.

FINITE DIFFERENCE FRACTIONAL STEP METHODS FOR THE TRANSIENT BEHAVIOR OF A SEMICONDUCTOR DEVICE

Characteristic finite difference fractional step schemes are put forward. The electric potential equation is described by a seven-point finite difference scheme, and the electron and hole concentration equations are treated by a kind of characteristic finite difference fractional step methods. The temperature equation is described by a fractional step method. Thick and thin grids are made use of to form a complete set. Piecewise threefold quadratic interpolation, symmetrical extension, calculus of variations, commutativity of operator product, decomposition of high order difference operators and prior estimates are also made use of. Optimal order estimates in l2 norm are derived to determine the error of the approximate solution. The well-known problem is thorongley and completely solred.

NONLINEAR STABILITY OF PLANAR SHOCK PROFILES FOR THE GENERALIZED KdV-BURGERS EQUATION IN SEVERAL DIMENSIONS

This paper is concerned with the nonlinear stability of planar shock profiles to the Cauchy problem of the generalized KdV-Burgers equation in two dimensions. Our analysis is based on the energy method developed by Goodman [5] for the nonlinear stability of scalar viscous shock profiles to scalar viscous conservation laws and some new decay estimates on the planar shock profiles of the generalized KdV-Burgers equation.

Characteristic finite difference method and application for moving boundary value problem of coupled system

The coupled system of multilayer dynamics of fluids in porous media is to describe the history of oil-gas transport and accumulation in basin evolution.It is of great value in rational evaluation of prospecting and exploiting oil-gas resources.The mathematical model can be described as a coupled system of nonlinear partial differential equations with moving boundary values.A kind of characteristic finite difference schemes is put forward,from which optimal order estimates in l~2 norm are derived for the error in the approximate solutions.The research is important both theoretically and practically for the model analysis in the field,the model numerical method and software development.

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.

SPACE-TIME DISCONTINUOUS GALERKIN METHOD FOR MAXWELL EQUATIONS IN DISPERSIVE MEDIA

In this paper, a unified model for time-dependent Maxwell equations in dispersive media is considered. The space-time DG method developed in [29] is applied to solve the un-derlying problem. Unconditional L2-stability and error estimate of order O？τr＋1＋hk＋1/2？ are obtained when polynomials of degree at most r and k are used for the temporal dis-cretization and spatial discretization respectively. 2-D and 3-D numerical examples are given to validate the theoretical results. Moreover, numerical results show an ultra-convergence of order 2r＋1 in temporal variable t.

THE CHARACTERISTIC FINITE ELEMENT ALTERNATING-DIRECTION METHOD AND ANALYSIS FOR THREE-DIMENSIONAL NUMERICAL RESERVOIR SIMULATION

Petroleum science has made remarkable progress in organic geochcmistry 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 can 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 three-dimensional characteristic of large-scale science-engineering computation, we put forward a kind of characteristic finite element alternating-direction schemes and obtain optimal order estimates in L^2 norm for the error in the approximate assumption.

THE CHARACTERISTIC FINITE DIFFERENCE METHOD FOR THREE-DIMENSIONAL MOVING BOUNDARY VALUE PROBLEM

The software for oil-gas transport and accumulation is to describe the history of oil-gas transport and accumulation in basin evolution. It is of great value in rational evaluation of prospecting and exploiting oil-gas resources. The mathematical model can be discribed as a coupled system of nonlinear partial differential equations with moving boundary value problem. For a generic case of the three-demensional bounded region, bi this thesis, the effects of gravitation、buoyancy and capillary pressure are considered, we put forward a kind of characteristic finite difference schemes and make use thick and thin grids to form a complete set, and of calculus of vaviations, the change of variable, the theory of prior estimates and techniques, Optimal order estimates in l^2 norm are derived for the error in approximate assumption, Thus we have completely solved the well-known theoretical problem proposed by J. Douglas, Jr.

On the Construction and Analysis of Finite Volume Element Schemes with Optimal L^(2) Convergence Rate

We provide a general construction method for a finite volume element(FVE)scheme with the optimal L^(2)convergence rate.The k-(k-1)-order orthogonal condition(generalized)is proved to be a sufficient and necessary condition for a k-order FVE scheme to have the optimal L^(2) convergence rate in 1D,in which the independent dual parameters constitute a(k-1)-dimension surface in the reasonable domain in k-dimension.In the analysis,the dual strategies in different primary elements are not necessarily to be the same,and they are allowed to be asymmetric in each primary element,which open up more possibilities of the FVE schemes to be applied to some complex problems,such as the convection-dominated problems.It worth mentioning that,the construction can be extended to the quadrilateral meshes in 2D.The stability and H^(1) estimate are proved for completeness.All the above results are demon-strated by numerical experiments.

Error Estimates for the Iterative Discontinuous Galerkin Method to the Nonlinear Poisson-Boltzmann Equation

This paper is devoted to the error estimate for the iterative discontinuous Galerkin(IDG)method introduced in[P.Yin,Y.Huang and H.Liu.Commun.Comput.Phys.16:491-515,2014]to the nonlinear Poisson-Boltzmann equation.The total error includes both the iteration error and the discretization error of the direct DG method to linear elliptic equations.For the DDG method,the energy error is obtained by a constructive approach through an explicit global projection satisfying interface conditions dictated by the choice of numerical fluxes.The L^(2) error of order O(h^(m+1))for polynomials of degree m is further recovered.The bounding constant is also shown to be independent of the iteration times.Numerical tests are given to validate the established convergence theory.

Statistical Estimation of the Shannon Entropy

The behavior of the Kozachenko–Leonenko estimates for the(differential) Shannon entropy is studied when the number of i.i.d. vector-valued observations tends to infinity. The asymptotic unbiasedness and L^2-consistency of the estimates are established. The conditions employed involve the analogues of the Hardy–Littlewood maximal function. It is shown that the results are valid in particular for the entropy estimation of any nondegenerate Gaussian vector.

EXPONENTIAL TIME DIFFERENCING-PADE FINITE ELEMENT METHOD FOR NONLINEAR CONVECTION-DIFFUSION-REACTION EQUATIONS WITH TIME CONSTANT DELAY

In this paper,ETD3-Padéand ETD4-PadéGalerkin finite element methods are proposed and analyzed for nonlinear delayed convection-diffusion-reaction equations with Dirichlet boundary conditions.An ETD-based RK is used for time integration of the corresponding equation.To overcome a well-known difficulty of numerical instability associated with the computation of the exponential operator,the Padéapproach is used for such an exponential operator approximation,which in turn leads to the corresponding ETD-Padéschemes.An unconditional L^(2) numerical stability is proved for the proposed numerical schemes,under a global Lipshitz continuity assumption.In addition,optimal rate error estimates are provided,which gives the convergence order of O(k^(3)+h^(r))(ETD3-Padé)or O(k^(4)+h^(r))(ETD4-Padé)in the L^(2)norm,respectively.Numerical experiments are presented to demonstrate the robustness of the proposed numerical schemes.

Convergence of Solutions of General Dispersive Equations Along Curve

In this paper, the authors give the local L^2 estimate of the maximal operator S_(φ,γ)~* of the operator family {S_(t,φ,γ)} defined initially by ■which is the solution(when n = 1) of the following dispersive equations(~*) along a curve γ:■where φ : R^+→R satisfies some suitable conditions and φ((-?)^(1/2)) is a pseudo-differential operator with symbol φ(|ξ|). As a consequence of the above result, the authors give the pointwise convergence of the solution(when n = 1) of the equation(~*) along curve γ.Moreover, a global L^2 estimate of the maximal operator S_(φ,γ)~* is also given in this paper.

FINITE ELEMENT METHOD AND ANALYSIS FOR CHEMICAL-FLOODING SIMULATION

This article discusses the enhanced oil recovery numerical simulation of the chemical-flooding (such as surfactallts, alcohol, polymers) composed of three-dimensional multicomponent, multiphase and incompressible mixed fluids. The mathematical model can be described as a coupled system of nonlinear partial differential equations with initialboundary value problerns. viom the actual conditions such as the effect of cross interference and the three-dimensional charederistic of large-scale science-engineering computation,this article puts forward a kind of characteristic finite element fractional step schemes and obtain the optimal order error estdriates in L2 norm. Thus we have thoroughly solved the well-known theoretical problem proppsed by a famous scientist, R. E. Ewing.

Implicit DG Method for Time Domain Maxwell’s Equations Involving Metamaterials

An implicit discontinuous Galerkin method is introduced to solve the timedomain Maxwell’s equations in metamaterials.The Maxwell’s equations in metamaterials are represented by integral-differential equations.Our scheme is based on discontinuous Galerkin method in spatial domain and Crank-Nicolson method in temporal domain.The fully discrete numerical scheme is proved to be unconditionally stable.When polynomial of degree at most p is used for spatial approximation,our scheme is verified to converge at a rate of O(τ^(2)+h^(p)+1/2).Numerical results in both 2D and 3D are provided to validate our theoretical prediction.

Global Existence and Blow-up for Semilinear Wave Equations with Variable Coefficients

The authors consider the critical exponent problem for the variable coefficients wave equation with a space dependent potential and source term. For sufficiently small data with compact support, if the power of nonlinearity is larger than the expected exponent, it is proved that there exists a global solution. Furthermore, the precise decay estimates for the energy, L^2 and L^(p+1) norms of solutions are also established. In addition, the blow-up of the solutions is proved for arbitrary initial data with compact support when the power of nonlinearity is less than some constant.