On the Finite Element Approximation of an Elliptic Variational Inequality Arising from an Implicit Time Discretization of the Stefan Problem

A semi-discretization in time of the weak formulation of thetwo phase Stefan moving boundary problem results in an ellipticboundary value problem with a non-linear jump discontinuitywhich can be set as an elliptic variational inequality. Thepurpose of this paper is to consider a finite element approximationto the inequality. Assuming that the solution is in H² and thatthe length of the free boundary is finite an error estimateis proved. The resulting algebraic problem is one of solvinga system of nonlinear equations associated with a diagonal multivaluedmonotone mapping. An S.O.R. method is given and shown to beglobally convergent.     

Finite Element Approximation of a Contact Vector Eigenvalue Problem

We consider a nonstandard elliptic eigenvalue problem of second order on a two-component domain consisting of two intervals with a contact point. The interaction between the two domains is expressed through a coupling condition of nonlocal type, more specifically, in integral form. The problem under consideration is first stated in its variational form and next interpreted as a second-order differential eigenvalue problem. The aim is to set up a finite element method for this problem. The error analysis involved is shown to be affected by the nonlocal condition, which requires a suitable modification of the vector Lagrange interpolant on the overall finite element mesh. Nevertheless, we arrive at optimal error estimates. In the last section, an illustrative numerical example is given, which confirms the theoretical results.     

一类抛物型变分不等式的有限元近似收敛估计

本文讨论了一类抛物型变分不等式的近似收敛问题．对有限元离散中引起较大误差的质量矩阵,采用了近似形式的集总质量矩阵来代替,时间项采用向后差分,得到了一个隐式的计算格式,证明了计算格式的收敛性及其收敛速度估计．文末给出了数值算例．     

On a Variational Inequality Formulation of an Electrochemical Machining Moving Boundary Problem and its Approximation by the Finite Element Method

An electrochemical machining moving boundary problem is formulated,after a change of variable, as an elliptic variational inequality.The unknown anode surface may now be found by solving just oneelliptic free boundary problem. The variational inequality isapproximated by the finite element method and numerical resultsare presented.     

四阶障碍问题的稳定化混合有限元方法

本文讨论了四阶障碍问题的稳定化混合有限元方法.首先,引入网格依赖范数,通过加罚方法得到了与四阶障碍问题的等价的混合变分形式.随后给出了基于C~0协调有限元空间(W_h,V_h)的混合有限元逼近,例如P_k-P_k三角形有限元.在网格依赖范数下,(W_h,V_h)满足离散的inf-sup条件.最后,我们在不同的假设下,得到了一些误差估计.     

On Stability and Convergence of a Finite Difference Approximation to a Parabolic Variational Inequality Arising From American Option Valuation

Abstract

In this article we study the numerical solution of a degenerate variational inequality of parabolic type arising from the valuation of American options by a finite difference method. Stability and convergence of the finite difference method are established. Numerical results are also presented to show the effectiveness and usefulness of the discretization scheme.     

Stokes问题在各向异性网格下的Bernadi-Raugel有限元逼近

在各向异性网格下,得到了Stokes问题著名的Bernadi-Raugel混合有限元格式的超逼近性质,而且通过构造插值后处理算子得到了关于速度的超收敛结果.     

抛物问题的mortar有限元逼近的瀑布型多重网格法

用瀑布型多重网格法解决椭圆、抛物问题，已有不少研究工作[1-2]，本文对抛物问题的mortar有限元的全离散格式提出瀑布型多重网格法，证明了该方法是最优的，即具有最优精确度和复杂度．     

Finite Element Approximation of Hemivariational Inequalities

The paper deals with a finite element approximation of elliptic and parabolic variational inequalities. Elliptic hemivariational inequalities are equivalently expressed as a system consisting of one equation and one inclusion for a couple of unknowns, namely a primal variable u and an element belonging to a multivalued mapping at u. Both components of the solution are approximated independently each other by a finite element method. Parabolic inequalities are transformed into a system of elliptic ones by using an appropriate time discretization. A numerical experiment is realized by using non-smooth optimization methods.     

Finite Element Approximation of Eigenvalue Problem for a Coupled Vibration Between Acoustic Field and Plate

1.IntroductionInthispaper,westudyanumericalmethodtocalculateeigen-frequenciesofacoupledvibrationbetweenacousticfieldandplate.Atypicalapplicationofthisresearchistoreduceanoiseinsideacarcausedbyanengineorothersourcesofthesound.OurstudywasmotivatedbytheworkofHagiwaraetal.15].Thebackgroundoftheresearchandsomeapplicationscanbeseenin[5].Werestrictourresearchtotheproblemswhereexactsolutionscanbegiveninaspecialcase.Themainfeatureofourresearchisthemathematicallyrigorousapproachtotheproblem.Weformulat…     

Finite Element Approximation of Space Fractional Optimal Control Problem with Integral State Constraint

In this paper finite element approximation of space fractional optimal control problem with integral state constraint is investigated. First order optimal condition and regularity of the control problem are discussed. A priori error estimates for control, state, adjoint state and lagrange multiplier are derived. The nonlocal property of the fractional derivative results in a dense coefficient matrix of the discrete state and adjoint state equation. To reduce the computational cost a fast projection gradient algorithm is developed based on the Toeplitz structure of the coefficient matrix. Numerical experiments are carried out to illustrate the theoretical findings.     

Extrapolation of Finite Element Approximation in a Rectangular Domain

Recently, the Richardson extrapolation for the elliptic Ritz projection with linear triangular elements on a general convex polygonal domain was discussed by Lin and Lu. We go back in this note to the simplest case, i.e. the bilinear rectangular elements on a rectangular domain which is a parallel case of the one-triangle model in the early work of Lin and Liu. We find that the finite element argument for the Richardson extrapolation with an accuracy of $O(h^4)$ needs only the regularity of $H^{4,\infty}$ for the solution $u$ but the finite difference argument for extrapolation with $O(h^{s+\alpha})$ accuracy needs $u\in C^{5+\alpha}(0＜\alpha＜1)$. Moreover, a formula is suggested to guarantee the extrapolation of $O(h^4)$ accuracy at fine gridpoints as well as at coarse gridpoints.     

Finite Element Approximation of a Rigid Punch Indenting a Membrane

Optimal order H¹ and L error bounds are obtained for a continuouspiecewise linear finite element approximation of an obstacleproblem, where the obstacle's height as well as the contactzone, _c, are a priori unknown. The problem models the indentationof a membrane by a rigid punch. For R², given ,g R⁺ and an obstacle defined over E we consider the minimization of |v|²_1,+gµover (v, µ) H¹₀() x R subject to v+µ on E. In additionwe show under certain nondegeneracy conditions that dist (_c,^h_c)Ch ln 1/h, where ^h_c is the finite element approximation to_c. Finally we show that the resulting algebraic problem canbe solved using a projected SOR algorithm.     

三维有限元本征值的外推

通过二维和三维积分恒等式,探讨泊松方程本征值问题三角线元和四面体线元Richardson外推的可行性.理论分析表明,如果剖分为均匀一致和拟一致,外推均可将解的精度提高二阶.     

Finite Element Approximation of Vector-Valued Hemivariational Problems

In this paper we develop a finite element approximationfor vector-valuedhemivariational inequalities.This class of hemivariational problems wasintroducedin [12],[13]. We study two differentproblems: unconstrained oneand constrained one witha nonempty, closed, convex constraint set K.We shall show firstly that the discrete problemsare solvable by usingconsequences of Kakutanifixed point theorem and secondly that the solutionsof the discrete problemsare close on subsequences to the continuous ones.     

非协调有限元与本征值下界

与协调有限元相比,非协调有限元常常给出本征值下界.这种现象已经由很多数值例子观察到.但是理论上的证明最近才做的更多.以Morley非协调元作为更简单的范例,来说明这种理论为何成功.     

Numerical Approximation of Solutions of a Nonlinear Inverse Problem Arising in Olfaction Experimentation

Identification of detailed features of neuronal systems is an important challenge in the biosciences today. Cilia are long thin structures that extend from the olfactory receptor neurons into the nasal mucus. Transduction of an odor into an electrical signal occurs in the membranes of the cilia. The cyclic-nucleotide-gated (CNG) channels which reside in the ciliary membrane and are activated by adenosine 3',5'-cyclic monophosphate (cAMP) allow a depolarizing influx of Ca(2+) and Na(+) and are thought to initiate the electrical signal.In this paper, a mathematical model consisting of two nonlinear differential equations and a constrained Fredholm integral equation of the first kind is developed to model experiments involving the diffusion of cAMP into cilia and the resulting electrical activity. The unknowns in the problem are the concentration of cAMP, the membrane potential and, the quantity of most interest in this work, the distribution of CNG channels along the length of a cilium. A simple numerical method is derived that can be used to obtain estimates of the spatial distribution of CNG ion channels along the length of a cilium. Certain computations indicate that this mathematical problem is ill-conditioned.     

A Posteriori Error Estimators and Adaptivity for Finite Element Approximation of the Non-Homogeneous Dirichlet Problem

Techniques are developed for a posteriori error analysis of the non-homogeneous Dirichlet problem for the Laplacian giving computable error bounds for the error measured in the energy norm. The techniques are based on the equilibrated residual method that has proved to be reliable and accurate for the treatment of problems with homogeneous Dirichlet data. It is shown how the equilibrated residual method must be modified to include the practically important case of non-homogeneous Dirichlet data. Explicit and implicit a posteriori error estimators are derived and shown to be efficient and reliable. Numerical examples are provided illustrating the theory.     

Approximation properties of the Generalized Finite Element Method

In this paper, we have obtained an approximation result in the Generalized Finite Element Method (GFEM) that reflects the global approximation property of the Partition of Unity (PU) as well as the approximability of the local approximation spaces. We have considered a GFEM, where the underlying PU functions reproduce polynomials of degree l. With the space of polynomials of degree k serving as the local approximation spaces of the GFEM, we have shown, in particular, that the energy norm of the GFEM approximation error of a smooth function is O(h ^l + k ). This result cannot be obtained from the classical approximation result of GFEM, which does not reflect the global approximation property of the PU.