A NEW CLASS OF BILEVEL GENERALIZED MIXED EQUILIBRIUM PROBLEMS IN BANACH SPACES

A new class of bilevel generalized mixed equilibrium problems involving setvalued mappings is introduced and studied in a real Banach space. By using the auxiliary principle technique, new iterative algorithms for solving the generalized mixed equilibrium problems and bilevel generalized mixed equilibrium problems involving set-valued mappings are suggested and analyzed. Existence of solutions and strong convergence of the iterative sequences generated by the algorithms are proved under quite mild conditions. The behavior of the solution set of the generalized mixed equilibrium problems and bilevel generalized mixed equilibrium problems is also discussed. These results are new and generalize some recent results in this field.     

Application of auxiliary space preconditioning in field-scale reservoir simulation

We study a class of preconditioners to solve large-scale linear systems arising from fully implicit reservoir simulation.These methods are discussed in the framework of the auxiliary space preconditioning method for generality.Unlike in the case of classical algebraic preconditioning methods,we take several analytical and physical considerations into account.In addition,we choose appropriate auxiliary problems to design the robust solvers herein.More importantly,our methods are user-friendly and general enough to be easily ported to existing petroleum reservoir simulators.We test the efciency and robustness of the proposed method by applying them to a couple of benchmark problems and real-world reservoir problems.The numerical results show that our methods are both efcient and robust for large reservoir models.     

AN EMBEDDED BOUNDARY METHOD FOR ELLIPTIC AND PARABOLIC PROBLEMS WITH INTERFACES AND APPLICATION TO MULTI-MATERIAL SYSTEMS WITH PHASE TRANSITIONS

The embedded boundary method for solving elliptic and parabolic problems in geometrically complex domains using Cartesian meshes by Johansen and Colella （1998, J. Comput. Phys. 147, 60） has been extended for elliptic and parabolic problems with interior boundaries or interfaces of discontinuities of material properties or solutions. Second order accuracy is achieved in space and time for both stationary and moving interface problems. The method is conservative for elliptic and parabolic problems with fixed interfaces. Based on this method, a front tracking algorithm for the Stefan problem has been developed. The accuracy of the method is measured through comparison with exact solution to a two-dimensional Stefan problem. The algorithm has been used for the study of melting and solidification problems.     

An orthogonal basis for non-uniform algebraic-trigonometric spline space

Non-uniform algebraic-trigonometric B-splines shares most of the properties as those of the usual polynomial B-splines. But they are not orthogonal. We construct an orthogonal basis for the n-order(n ≥ 3) algebraic-trigonometric spline space in order to resolve the theoretical problem that there is not an explicit orthogonal basis in the space by now. Motivated by the Legendre polynomials, we present a novel approach to define a set of auxiliary functions,which have simple and explicit expressions. Then the proposed orthogonal splines are given as the derivatives of these auxiliary functions.     

On the Quaternion Ball and the Quaternion Projective Space

Since the quaternion ball was used to study the AdS/CFT problems tor spinor fields, it is worthwhile to study further the geometry （in sense of Klein） and analysis on it and on its extended space （in the sense of Behnke-Thullen）, the quaternion projective space.     

Banach空间中的一类二阶积分-微分方程边值问题的解

This paper investigates the maximal and minimal solutions of initial value problems for second order nonlinear integro-differential equations of Volterra type on a finite interval in a Banach space by establishing a comparison result and using the monotone iterative technique.     

Universal inequalities for lower order eigenvalues of self-adjoint operators and the poly-Laplacian

In this paper, we first establish an abstract inequality for lower order eigenvalues of a self-adjoint operator on a Hilbert space which generalizes and extends the recent results of Cheng et al. （Calc. Var. Partial Differential Equations, 38, 409-416 （2010））. Then, making use of it, we obtain some universal inequalities for lower order eigenvalues of the biharmonic operator on manifolds admitting some speciM functions. Moreover, we derive a universal inequality for lower order eigenvalues of the poly-Laplacian with any order on the Euclidean space.     

Fractional order boundary value problem with integral boundary conditions involving Pettis integral

In this article, we investigate the existence of Pseudo solutions for some fractional order boundary value problem with integral boundary conditions in the Banach space of continuous function equipped with its weak topology. The class of such problems constitute a very interesting and important class of problems. They include two, three, multi-point and nonlocal boundary-value problems as special cases. In our investigation, the right hand side of the above problem is assumed to be Pettis integrable function. To encompass the full scope of this article, we give an example illustrating the main result.     

SURFACE FINITE ELEMENTS FOR

In this article we define a surface finite element method （SFEM） for the numerical solution of parabolic partial differential equations on hypersurfaces F in R^n＋1. The key idea is based on the approximation of F by a polyhedral surface Гh consisting of a union of simplices （triangles for n = 2, intervals for n = 1） with vertices on Г. A finite element space of functions is then defined by taking the continuous functions on Гh which are linear affine on each simplex of the polygonal surface. We use surface gradients to define weak forms of elliptic operators and naturally generate weak formulations of elliptic and parabolic equations on Г. Our finite element method is applied to weak forms of the equations. The computation of the mass and element stiffness matrices are simple and straightforward. We give an example of error bounds in the case of semi-discretization in space for a fourth order linear problem. Numerical experiments are described for several linear and nonlinear partial differential equations. In particular the power of the method is demorrstrated by employing it to solve highly nonlinear second and fourth order problems such as surface Allen-Cahn and Cahn-Hilliard equations and surface level set equations for geodesic mean curvature flow.     

对流占优扩散问题的经济型流线扩散有限元法

In this paper, the economical finite difference-streamline diffusion (EFDSD) schemes based on the linear F.E. space for time-dependent linear and non-linear convection-dominated diffusion problems are constructed. The stability and error estimation with quasi-optimal order approximation are established in the norm stronger than L^2 - norm for the schemes considered. It is indicated by the results obtained that,for linear F.E. space, the EFDSD schemes have the same specific properties of stability and convergence as the traditional FDSD schemes for the problems discussed.     

Postprocessing and higher order convergence for the mixed finite element approximations of the eigenvalue problem

In this paper, we propose a method to improve the convergence rate of the lowest order Raviart-Thomas mixed finite element approximations for the second order elliptic eigenvalue problem. Here, we prove a supercloseness result for the eigenfunction approximations and use a type of finite element postprocessing operator to construct an auxiliary source problem. Then solving the auxiliary additional source problem on an augmented mixed finite element space constructed by refining the mesh or by using the same mesh but increasing the order of corresponding mixed finite element space, we can increase the convergence order of the eigenpair approximation. This postprocessing method costs less computation than solving the eigenvalue problem on the finer mesh directly. Some numerical results are used to confirm the theoretical analysis.     

Efficient and stable perfectly matched layer for CEM

An efficient unsplit perfectly matched layer for numerical simulation of electromagnetic waves in unbounded domains is derived via a complex change of variables. In order to surround a Cartesian grid with the PML, the time-dependent PML requires only one (scalar) auxiliary variable in two space dimensions and six (scalar) auxiliary variables in three space dimensions. It is therefore cheap and straightforward to implement. We use Fourier and energy methods to prove the stability of the PML. We extend the stability result to a semi-discrete PML approximated by central finite differences of arbitrary order of accuracy and to a fully discrete problem for the ‘Leap-Frog’ schemes. This makes precise the usefulness of the derived PML model for longtime simulations. Numerical experiments are presented, illustrating the accuracy and stability of the PML.     

An auxiliary space preconditioner for linear elasticity based on the generalized finite element method

We construct and analyze a preconditioner of the linear elasticity system discretized by conforming linear finite elements in the framework of the auxiliary space method. The auxiliary space preconditioner is based on two auxiliary spaces corresponding to discretizations of the scalar Poisson equation by linear finite elements and the generalized finite element method. Copyright © 2011 John Wiley & Sons, Ltd.     

On the maximal space analyticity radius for the 3D Navier–Stokes equations and energy cascades

In this paper, we study the maximal space analyticity radius associated with a regular solution of the Navier–Stokes equations and its connection to turbulence. In order to do this, we introduce a new auxiliary ODE for the evolution of the analyticity radius involving the Gevrey class norms. We further show that jumps in the maximal space analyticity radius are an intermittent event and are connected to inverse energy cascade. Our approach also leads to a new type of global regularity test for the Navier–Stokes equation.     

EFFICIENT LINEAR SCHEMES WITH UNCONDITIONAL ENERGY STABILITY FOR THE PHASE FIELD MODEL OF SOLID-STATE DEWETTING PROBLEMS

In this paper, we study linearly first and second order in time, uniquely solvable and unconditionally energy stable numerical schemes to approximate the phase field model of solid-state dewetting problems based on the novel "scalar auxiliary variable" (SAV) approach, a new developed efficient and accurate method for a large class of gradient flows. The schemes are based on the first order Euler method and the second order backward differential formulas (BDF2) for time discretization, and finite element methods for space discretization. The proposed schemes are proved to be unconditionally stable and the discrete equations are uniquely solvable for all time steps. Various numerical experiments are presented to validate the stability and accuracy of the proposed schemes.     

Antiperiodic solutions of fourth‐order impulsive differential equation

In this paper, the existence of antiperiodic solutions for fourth‐order impulsive differential equation is obtained by variational approaches and results on the auxiliary system. It is interesting that there is no growth restraint on nonlinear terms and impulsive terms. Besides, any minimizing sequence is bounded in a closed convex set of a space composed of Lipschitzian functions with the appearance of antiperiodic boundary value conditions.     

Preconditioners for higher order edge finite element discretizations of Maxwell's equations

In this paper,we are concerned with the fast solvers for higher order edge finite element discretizations of Maxwell's equations.We present the preconditioners for the first family and second family of higher order N′ed′elec element equations,respectively.By combining the stable decompositions of two kinds of edge finite element spaces with the abstract theory of auxiliary space preconditioning,we prove that the corresponding condition numbers of our preconditioners are uniformly bounded on quasi-uniform grids.We also present some numerical experiments to demonstrate the theoretical results.     

Auxiliary space preconditioning in H 0(curl; Ω)

We adapt the principle of auxiliary space preconditioning as presented in [J. Xu, The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids, Computing, 56 (1996), pp. 215–235.] to H (curl; ω)-elliptic variational problems discretized by means of edge elements. The focus is on theoretical analysis within the abstract framework of subspace correction. Employing a Helmholtz-type splitting of edge element vector fields we can establish asymptotic h-uniform optimality of the preconditioner defined by our auxiliary space method. This author was fully supported by Hong Kong RGC grant (Project No. 403403) This author acknowledges the support from a Direct Grant of CUHK during his visit at The Chinese University of Hong Kong.     

对称算子自共轭扩张的酉参数化描述(英文)

利用构造性的方法,给出了边值空间理论中几个结果新的证明,其中,边值空间理论是有关对称算子自共轭扩张的一种方法.同时,得到了几个新的结果.如发现了一般的边界三元组所具有的结构.进一步地,利用这个结果证明了辅助Hilbert空间H上的酉变换与亏空间K-和K+之间的等距同构映射间存在一个双解析的映射.发现并证明了一般边界条件:B(ψ):=MΓ1ψ+NΓ2ψ=0(其中M,N是阶数为亏指数的方阵)是自共轭的充要条件以及相应的酉变换和边界映射.