A finite element convergence analysis for 3D Stokes equations in case of variational crimes

We investigate a finite element discretization of the Stokes equations with nonstandard boundary conditions, defined in a bounded three-dimensional domain with a curved, piecewise smooth boundary. For tetrahedral triangulations of this domain we prove, under general assumptions on the discrete problem and without any additional regularity assumptions on the weak solution, that the discrete solutions converge to the weak solution. Examples of appropriate finite element spaces are given.     

解一类半线性外边值问题的自然边界元与有限元耦合法

In this paper, we combine a finite element approach with the natural boundary element method to stduy the weak solvability and Galerkin approximations of a class of semilinear exterior boundary value problems. Our analysis is mainly based on the variational formulation with constraints. We discuss the error estimate of the finite element solution and obtain the asymptotic rate of convergence O（h^n) Finally, we also give two numerical examples.     

Finite element approximation of the elasticity spectral problem on curved domains

We analyze the finite element approximation of the spectral problem for the linear elasticity equation with mixed boundary conditions on a curved non-convex domain. In the framework of the abstract spectral approximation theory, we obtain optimal order error estimates for the approximation of eigenvalues and eigenvectors. Two kinds of problems are considered: the discrete domain does not coincide with the real one and mixed boundary conditions are imposed. Some numerical results are presented.     

边界层问题的小波—有限元解

本文将小波分析与有限元法结合起来，建立了一种小波－有限元计算格式，并用该算法计算了一个典型的边界层问题，探讨了寻找边界层位置的过程以及计算边界层区的内部解及外部解的步骤。计算结果表明，用该法寻找的边界层位置以及所求得的内部解与真实结果完全符合。     

Boundary elements solution of stokes flow between curved surfaces with nonlinear slip boundary condition

This work presents a boundary integral equation formulation for Stokes nonlinear slip flows based on the normal and tangential projection of the Green's integral representational formulae for the velocity field. By imposing the surface tangential velocity discontinuity (slip velocity) in terms of the nonlinear slip flow boundary condition, a system of nonlinear boundary integral equations for the unknown normal and tangential components of the surface traction is obtained. The Boundary Element Method is used to solve the resulting system of integral equations using a direct Picard iteration scheme to deal with the resulting nonlinear terms. The formulation is used to study flows between curved rotating geometries: i.e., concentric and eccentric Couette flows and single rotor mixers, under nonlinear slip boundary conditions. The numerical results obtained for the concentric Couette flow is validated again a semianalytical solution of the problem, showing excellent agreements. The other two cases, eccentric Couette and single rotor mixers, are considered to study the effect of different nonlinear slip conditions in these flow configurations. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

ON THE ERROR ESTIMATE OF LINEAR FINITE ELEMENT APPROXIMATION TO THE ELASTIC CONTACT PROBLEM WITH CURVED CONTACT BOUNDARY

1. PreliminaryThe error estimate of linear finite element approximation to the elastic contact problem with curved contact boundary was considered in [2], in which the authors obtainedthe error bound of O(hl) with a much complex proof, requirement of three times continuously differentiable for contact boundary and extra regularities of triangulation (c.f. [2, Theorem 3.3, p.149]). In this paper, we obtained the error bound of O(hi) withonly requirement of two times continuously differentiable…     

一种有限元-边界元耦合分域算法

提出了一种有限元-边界元耦合分域算法．该算法将所分析问题的区域分解成有限元和边界元子域,在满足两子域界面上位移和面力协调连续的条件下,通过迭代求解得到问题的解．在迭代求解过程中,引入动态松弛系数,使收敛得以加速．该方法在两子域界面上有限单元结点和边界单元结点的位置相互独立,无需协调一致,对诸如裂纹扩展过程的模拟具有独特的优势．用所提出的耦合算法分析算例,得到的结果与有限元法、边界元法和另一种耦合算法的数值计算结果一致,验证了这种算法的正确性和可行性．     

Approximation of boundary values in the boundary element method

Various techniques may be applied to the approximation of the unknown boundary functions involved in the boundary element method (BEM). Several techniques have been examined numerically to find the most efficient. Techniques considered were: Lagrangian polynomials of the zeroth, first and second orders; spline functions; and the novel weighted minimization technique used successfully in the finite difference method (FDM) for arbitrarily irregular meshes. All these approaches have been used in the BEM for the numerical analysis of plates with various boundary conditions.Both coarse and fine grids on the boundary have been assumed. Maximal errors of the deflections of each plate and the bending moments have been found and the effective computer CPU times determined.Analysis of the results showed that, for the same computer time, the greatest accuracy was obtained by the weighted FDM approach. In the case of the Lagrange approximation, higher order polynomials have proved more efficient. The spline technique yielded more accurate results, but with a higher CPU time.Two discretization approaches have been investigated: the least-squares technique and the collocation method. Despite the fact that the simultaneous algebraic equations obtained were not symmetric, the collocation approach has been confirmed as clearly superior to the least-squares technique, because of the amount of computation time used.     

Weak Galerkin finite element method for valuation of American options

We introduce a weak Galerkin finite element method for the valuation of American options governed by the Black-Scholes equation. In order to implement, we need to solve the optimal exercise boundary and then introduce an artificial boundary to make the computational domain bounded. For the optimal exercise boundary, which satisfies a nonlinear Volterra integral equation, it is resolved by a higher-order collocation method based on graded meshes. With the computed optimal exercise boundary, the front-fixing technique is employed to transform the free boundary problem to a one- dimensional parabolic problem in a half infinite area. For the other spatial domain boundary, a perfectly matched layer is used to truncate the unbounded domain and carry out the computation. Finally, the resulting initial-boundary value problems are solved by weak Galerkin finite element method, and numerical examples are provided to illustrate the efficiency of the method.     

Boundary element method for calculation of effective elastic moduli in 3D linear elasticity

A boundary integral formulation of the linear elasticity problem for a multi‐component composite is given. The fast BEM solver based on the adaptive cross approximation is then obtained by the data‐sparse representation of the resulting Galerkin matrices. The solver is used to obtain effective elastic moduli of fibre and particle reinforced composites in three dimensions by means of the strain energy equivalence principle. Copyright © 2009 John Wiley & Sons, Ltd.     

弹性力学问题解唯一的边界积分方程

从积分方程式出发,应用基本解的特性分析,说明在力边值问题中,位移边界积分方程和面力边界积分方程的位移解不唯一.提出了位移解唯一的条件,建立了唯一解的位移边界积分方程和面力边界积分方程.实例计算结果表明唯一解的边界积分方程是有效的.     

一种改进边界元法在弹性扭转问题中的应用

本文用一种改进边界元法分析与计算了椭圆截面等直杆的扭转问题.并与边界元法的解进行比较,其结果极为符合.然而,改进边界元法较边界元法所需要的数据量少得多,计算时间也将大大减少了.因此,本文方法对求解Poisson方程问题是一种经济而行之有效的数值计算方法.     

On the problem of superconvergence of finite element method algorithms

The coincidence of an approximate solution to the boundary value problem for an ordinary differential equation with the exact solution at mesh nodes is proved for a certain class of the generalized finite element methods.     

三维Poisson方程外问题的高阶局部人工边界条件

１引言假设Ｒ３是一分片光滑的闭曲面．是以为边界的无界区域,=Ｒ3是以为边界的有界区域,并且存在球Ｂ0＝ｘｘＲ0我们考虑下面Ｐｏｉｓｓｏｎ方程的外问题：这里f(x）,ｇ（x）是,上的已知函数,f(x)的支集是紧的,即存在一个球面=x·x=R1,使得x＝xｘＲ１,有ｆx＝０．令＝,则ｆ（ｘ）的支集包含在中,令＝xx＝,表示u在上的外法向微商．用流量为零的条件代替无限远处条件（３）,则我们得到一个新的外问题：我们将分别讨论问题（１）－（３）和（４）－（７）的数值解．由于求解区域的无界性,给数值计算带来了本质性的困难．克服此…     

A system of boundary integral equations for polygonal plates with free edges

We consider the problem of a polygonal plate with free edges. It is a boundary value problem for the biharmonic operator on a polygon with Neumann boundary conditions. Its resolution is studied via boundary integral equations. A variational formulation of the boundary problem obtained by a double-layer potential is given. Finally, we implement the method and give numerical results. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Stabilization of the trial method for the Bernoulli problem in case of prescribed Dirichlet data

We apply the trial method for the solution of Bernoulli's free boundary problem when the Dirichlet boundary condition is imposed for the solution of the underlying Laplace equation, and the free boundary is updated according to the Neumann boundary condition. The Dirichlet boundary value problem for the Laplacian is solved by an exponentially convergent boundary element method. The update rule for the free boundary is derived from the linearization of the Neumann data around the actual free boundary. With the help of shape sensitivity analysis and Banach's fixed‐point theorem, we shed light on the convergence of the respective trial method. Especially, we derive a stabilized version of this trial method. Numerical examples validate the theoretical findings.Copyright © 2014 John Wiley & Sons, Ltd.     

美式期权自由边界的计算方法

美式期权的自由边界问题在金融工程文献中已经引起了广泛的关注,然而,它的数值计算方法一直是一个难点.基于差分技巧,给出了满足具有有限到期日的美式期权自由边界的两种计算方法,即,根据股票期权价格和相应的偏导数来确定自由边界条件.数值结果表明了上述两种方法下自由边界是一致性的.此外研究结果对自由边界的计算提供很好的科学依据.     

Error estimates for the finite element solution of elliptic problems with nonlinear newton boundary conditions

The paper is concerned with the study of an elliptic boundary value problem with a nonlinear Newton boundary condition. The existence and uniqueness of the solution of the continuous pioblem is a consequence of the monotone operator theory. The main attention is paid to the investigation of the finite element approximation using numeriral integration for the evaluation of boundary integrals. The error estimates for the solution of the discrete finite element problem are derived     

一反应扩散模型解的有限元方法和误差分析

１引言在地下水运移过程中,污染物（溶质）随地下水在含水层中运移,并常常发生各种化学反应．文献［１－３］等提出并论述了三种化学物质（如Ｍ１,Ｍ２和Ｍ３）之间发生的一类化学反应．文献［４,５］等建立和描述了这类反应的数学模型（Ｐ）．文献［６,７」的作者首次对模型（Ｐ）进行了理论上的定性分析,主要是利用上,下解方法,算子半群理论和Ｓｏｂｏｌｅｖ空间的般人定理等论证了模型（Ｐ）的整体古典解的存在唯一性和渐近性质,文［６,７］也讨论了整体解的极限性态和收敛性估计．此外,文［８,９］等也就模型（Ｐ）的一类特…     

A BOUNDARY ELEMENT PROCEDURE FOR THE SIGNORINI PROBLEM IN THREE-DIMENSIONAL ELASTICITY

In this paper, a new numerical method for the Signorini problem in three-dimensional elasticity is presented. The problem is reduced to a boundary variational inequality based on a new representation of the derivative of the doublelayer potential. Furthermore, a boundary element procedure is described for the numerical approximation of its solution and an abstract error estimate is given.