HIGH ACCURACY FINITE VOLUME ELEMENT METHOD FOR TWO-POINT BOUNDARY VALUE PROBLEM OF SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

In this paper, a high accuracy finite volume element method is presented for two-point boundary value problem of second order ordinary differential equation, which differs from the high order generalized difference methods. It is proved that the method has optimal order error estimate O(h3) in H1 norm. Finally, two examples show that the method is effective.

Boundary Value Problems of p-Laplace Equations with Finite Time Delay

By establishing equivalent fixed point theorem, the boundary value problems of p Laplace equations with finite time delay are studied. It’s the first time that the functional differential equation is discussed with p Laplacian. The topological degree and fixed point theorem on cone are used to prove the existence of solution and positive solution. The conditions are all easy to check.

A New Analytical Approach for Solving Nonlinear Boundary Value Problems in Finite Domains

Based on the homotopy analysis method (HAM), we propose an analytical approach for solving the following type of nonlinear boundary value problems in finite domain. In framework of HAM a convenient way to adjust and control the convergence region and rate of convergence of the obtained series solutions, by defining the so-called control parameter h , is provided. This paper aims to propose an efficient way of finding the proper values of h.Such values of parameter can be determined at the any order of approximations of HAM series solutions by solving of a nonlinear polynomial equation. Some examples of nonlinear initial value problems in finite domain are used to illustrate the validity of the proposed approach. Numerical results confirm that obtained series solutions agree very well with the exact solutions.

Exponential Convergence of Finite-dimensional Approximations to Linear Bond-based Peridynamic Boundary Value Problems

In this paper, we demonstrate that the finite-dimensional approximations to the solutions of a linear bond-based peridynamic boundary value problem converge to the exact solution exponentially with the analyticity assumption of the forcing term, therefore greatly improve the convergence rate derived in literature.

Generalized variational principles for boundary value problem of electromagnetic field in electrodynamics

An expression of the generalized principle of virtual work for the boundary value problem of the linear and anisotropic electromagnetic field is given. Using Chien＇s method, a pair of generalized variational principles （GVPs） are established, which directly leads to all four Maxwell＇s equations, two intensity-potential equations, two constitutive equations, and eight boundary conditions. A family of constrained variational principles is derived sequentially. As additional verifications, two degenerated forms are obtained, equivalent to two known variational principles. Two modified GVPs are given to provide the hybrid finite element models for the present problem.

Mathematical analysis of EEP method for one-dimensional finite element postprocessing

For a class of two-point boundary value problems, by virtue of onedimensional projection interpolation, it is proved that the nodal recovery derivative obtained by Yuan＇s element energy projection （EEP） method has the accuracy O（h^min{2k,k＋4}） The theoretical analysis coincides the reported numerical results.

Finite Element Analysis of the Ramberg-Osgood Bar

In this work, we present a priori error estimates of finite element approximations of the solution for the equilibrium equation of an axially loaded Ramberg-Osgood bar. The existence and uniqueness of the solution to the associated nonlinear two point boundary value problem is established and used as a foundation for the finite element analysis.

Adaptive Finite Element Method for Steady Convection-Diffusion Equation

This paper examines the numerical solution of the convection-diffusion equation in 2-D. The solution of this equation possesses singularities in the form of boundary or interior layers due to non-smooth boundary conditions. To overcome such singularities arising from these critical regions, the adaptive finite element method is employed. This scheme is based on the streamline diffusion method combined with Neumann-type posteriori estimator. The effectiveness of this approach is illustrated by different examples with several numerical experiments.

Characteristic Finite Difference Methods for Moving Boundary Value Problem of Numerical Simulation of Oil Deposit

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 described as a coupled system of nonlinear partial differential equations with moving boundary value problem. This paper puts forward a kind of characteristic finite difference schemes, and derives from them optimal order estimates in l^2 norm for the error in the approximate solutions. The research is important both theoretically and practically for the model analysis in the field, for model numerical method and for software development.

A RIEMANN-HILBERT APPROACH TO THE INITIAL-BOUNDARY PROBLEM FOR DERIVATIVE NONLINEAR SCHRDINGER EQUATION

We use the Fokas method to analyze the derivative nonlinear Schrodinger （DNLS） equation iqt （x, t） = -qxx （x, t）＋（rq^2）x on the interval [0, L]. Assuming that the solution q（x, t） exists, we show that it can be represented in terms of the solution of a matrix Riemann- Hilbert problem formulated in the plane of the complex spectral parameter ξ. This problem has explicit （x, t） dependence, and it has jumps across {ξ∈C｜Imξ^4 = 0}. The relevant jump matrices are explicitely given in terms of the spectral functions {a（ξ）, b（ξ）}, {A（ξ）, B（ξ）}, and {A（ξ）, B（ξ）}, which in turn are defined in terms of the initial data q0（x） = q（x, 0）, the bound- ary data g0（t）= q（0, t）, g1（t） = qx（0, t）, and another boundary values f0（t） = q（L, t）, f1（t） = qx（L, t）. The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.

UPWIND FINITE VOLUME SCHEMES FOR SEMICONDUCTOR DEVICE

The mathematical model of semiconductor devices is described by the initial boundary value problem of a system of three nonlinear partial differential equations.One equation in elliptic form is for the electrostatic potential;two equations of convection-dominated diffusion type are for the electron and hole concentrations.Finite volume element procedure are put forward for the electrostatic potential,while upwind volume element schemes for the two concentration equations.Error estimates in L2norm for our numerical schemes are derived.

VARIABLE MESH FINITE DIFFERENCE METHOD FOR SELF-ADJOINT SINGULARLY PERTURBED TWO-POINT BOUNDARY VALUE PROBLEMS

A numerical method based on finite difference method with variable mesh is given for self-adjoint singularly perturbed two-point boundary value problems. To obtain parameter- uniform convergence, a variable mesh is constructed, which is dense in the boundary layer region and coarse in the outer region. The uniform convergence analysis of the method is discussed. The original problem is reduced to its normal form and the reduced problem is solved by finite difference method taking variable mesh. To support the efficiency of the method, several numerical examples have been considered.

NEW SECOND ORDER NONCONFORMING TRIANGULAR ELEMENT FOR PLANAR ELASTICITY PROBLEMS

In the use of finite element methods to the planar elasticity problems,one diffculty is to overcome locking when elasticity constant λ→∞.In the case of traction boundary condition,another diffculty is to make the discrete Korn＇s second inequality valid.In this paper,a triangular element is presented.We prove that this element is locking-free,the discrete Korn＇s second inequality holds and the convergence order is two.

A New Iterative Method for Multi-Moving Boundary Problems Based Boundary Integral Method

The present paper deals with very important practical problems of wide range of applications. The main target of the present paper is to track all moving boundaries that appear throughout the whole process when dealing with multi-moving boundary problems continuously with time up to the end of the process with high accuracy and minimum number of iterations. A new numerical iterative scheme based the boundary integral equation method is developed to track the moving boundaries as well as compute all unknowns in the problem. Three practical applications, one for vaporization and two for ablation were solved and their results were compared with finite element, heat balance integral and the source and sink results and a good agreement were obtained.

A Finite Volume Backward Euler Difference Method for Nonlinear Parabolic Integral-differential Equation

The Finite volume backward Euler difference method is established to discuss two-dimensional parabolic integro-differential equations.These results are new for finite volume element methods for parabolic integro-differential equations.

Petrov-Galerkin Spectral Element Method for Mixed Inhomogeneous Boundary Value Problems on Polygons

The authors investigate Petrov-Galerkin spectral element method. Some results on Legendre irrational quasi-orthogonal approximations are established, which play important roles in Petrov-Galerkin spectral element method for mixed inhomogeneous boundary value problems of partial differential equations defined on polygons. As examples of applications, spectral element methods for two model problems, with the spectral accuracy in certain Jacobi weighted Sobolev spaces, are proposed. The techniques developed in this paper are also applicable to other higher order methods.

LOCALLY STABILIZED FINITE ELEMENT METHOD FOR STOKES PROBLEM WITH NONLINEAR SLIP BOUNDARY CONDITIONS

Based on the low-order conforming finite element subspace （Vh, Mh） such as the P1-P0 triangle element or the Q1-P0 quadrilateral element, the locally stabilized finite element method for the Stokes problem with nonlinear slip boundary conditions is investigated in this paper. For this class of nonlinear slip boundary conditions including the subdifferential property, the weak variational formulation associated with the Stokes problem is an variational inequality. Since （Vh, Mh） does not satisfy the discrete inf-sup conditions, a macroelement condition is introduced for constructing the locally stabilized formulation such that the stability of （Vh, Mh） is established. Under these conditions, we obtain the H1 and L2 error estimates for the numerical solutions.

MULTIGRID ALGORITHM FOR THE COUPLING SYSTEM OF NATURAL BOUNDARY ELEMENT METHOD AND FINITE ELEMENT METHOD FOR UNBOUNDED DOMAIN PROBLEMS

In this paper, some V-cycle multigrid algorithms are presented for the coupling system arising from the discretization of the Dirichlet exterior problem by coupling the natural boundary element method and finite element method. The convergence of these multigrid algorithms is obtained even with only one smoothing on all levels. The rate of convergence is found uniformly bounded independent of the number of levels and the mesh sizes of all levels, which indicates that these multigrid algorithms are optimal. Some numerical results are also reported.

ON THE COUPLING OF BOUNDARY INTEGRAL AND FINITE ELEMENT METHODS FOR SIGNORINI PROBLEMS

In this paper, a exterior Signorini problem is reduced to a variational inequality on a bounded inner region with the help of a coupling of boundary integral and finite element methods. We established a equivalence between the original exterior Signorini problem and the variational inequality on the bounded inner region coupled with two integral equations on an auxiliary boundary. We also introduce a finite element approximation of the variational inequality and a boundary element approximation of the integral equations. Furthermore, the optimal error estimates are given.

THE COUPLING METHOD OF FINITE ELEMENTS AND BOUNDARY ELEMENTS FOR RADIATION PROBLEM

The paper presents the variational formulation and well posedness of the coupling method offinite elements and boundary elements for radiation problem. The convergence and optimal errorestimate for the approximate solution and numerical experiment are provided.