A posteriori error estimates for optimal control problems governed by parabolic equations

Summary. In this paper, we derive a posteriori error estimates for the finite element approximation of quadratic optimal control problem governed by linear parabolic equation. We obtain a posteriori error estimates for both the state and the control approximation. Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive finite element approximation schemes for the control problem. Received July 7, 2000 / Revised version received January 22, 2001 / Published online January 30, 2002 RID="*" ID="*" Supported by EPSRC research grant GR/R31980     

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

1.IntroductionPartialdifferentialequationssubjecttounilateralboundaryconditionsareusuallycalledSignoriniproblemsintheliterature.TheseproblemshavebeenstudiedbymanyauthodssincetheappearenceofthehistoricalpaperbyA.Signoriniin1933[25].Signoriniproblemsaroseinmanyareasofapplicationse.g.,theelasticitywithunilateralconditions[lo],thefluidmechnicsproblemsinmediawithsemipermeableboundaries[8,12],theelectropaintprocess[1]etc.Fortheexistence,uniquenessandregularityresultsforSignorinitypeproblemswerefer…     

Characterization of the Existence of an Enriched Linear Finite Element Approximation Using Biorthogonal Systems

The present paper is intended to give a characterization of the existence of an enriched linear finite element approximation based on biorthogonal systems. It is shown that the enriched element exists if and only if a certain multivariate generalized trapezoidal type cubature formula has a nonzero approximation error. Furthermore, for such an enriched element we derive simple explicit formulas for the basis functions, and show that the approximation error can be written as the sum of the error of the (non-enriched) element plus a perturbation that depends on the enrichment function. Finally, we estimate the approximation error in L² norm. We also give an alternative approach to estimate the approximation error, which relies on an appropriate use of the Poincaré inequality.     

THE APPROXIMATIONS OF THE EXACT BOUNDARY CONDITION AT AN ARTIFICIAL BOUNDARY FOR LINEARIZED INCOMPRESSIBLE VISCOUS FLOWS

1.IntroductionManyproblemsarisinginfluidmechanicsaregiveninanunboundeddomain,suchasfluidflowaroundobstacles.Whencomputingthenumericalsolutionsoftheseproblems,oneoftenintroducesartificialboundariesandsetsupaxtificialboundaryconditionsonthem.Thentheoriginal…     

On the convergence of finite element solutions to the interface problem for the Stokes system

The Stokes system with a discontinuous coefficient (Stokes interface problem) and its finite element approximations are considered. We firstly show a general error estimate. To derive explicit convergence rates, we introduce some appropriate assumptions on the regularity of exact solutions and on a geometric condition for the triangulation. We mainly deal with the MINI element approximation and then consider P1-iso-P2/P1 element approximation. Results are expected to give an instructive remark in numerical analysis for two-phase flow problems.     

A combined finite element and Marker and cell method for solving Navier-Stokes equations

Summary A new method is developed to approximate Navier-Stokes equations for incompressible fluids on polygonal domains. This method is based on a simple finite element approximation adapted to the Marker and Cell technique. Its main originality lies in the fact that the components of the velocity are approximated on distinct interlaced networks. The error analysis shows that this method is of order one.     

Lavrentiev regularization + Ritz approximation = uniform finite element error estimates for differential equations with rough coefficients

We consider a parametric family of boundary value problems for a diffusion equation with a diffusion coefficient equal to a small constant in a subdomain. Such problems are not uniformly well-posed when the constant gets small. However, in a series of papers, Bakhvalov and Knyazev have suggested a natural splitting of the problem into two well-posed problems. Using this idea, we prove a uniform finite element error estimate for our model problem in the standard parameter-independent Sobolev norm. We also study uniform regularity of the transmission problem, needed for approximation. A traditional finite element method with only one additional assumption, namely, that the boundary of the subdomain with the small coefficient does not cut any finite element, is considered.

One interpretation of our main theorem is in terms of regularization. Our FEM problem can be viewed as resulting from a Lavrentiev regularization and a Ritz-Galerkin approximation of a symmetric ill-posed problem. Our error estimate can then be used to find an optimal regularization parameter together with the optimal dimension of the approximation subspace.

     

Numerical Verification Methods for Solutions of the Free Boundary Problem

We propose two methods to enclose the solution of an ordinary free boundary problem. The problem is reformulated as a nonlinear boundary value problem on a fixed interval including an unknown parameter. By appropriately setting a functional space that depends on the finite element approximation, the solution is represented as a fixed point of a compact map. Then, by using the finite element projection with constructive error estimates, a Newton-type verification procedure is derived. In addition, numerical examples confirming the effectiveness of current methods are given.     

A posteriori error estimates of finite element methods for nonlinear quadratic boundary optimal control problem

This paper is aimed at studying finite element discretization for a class of quadratic boundary optimal control problems governed by nonlinear elliptic equations. We derive a posteriori error estimates for the coupled state and control approximation. Such estimates can be used to construct a reliable adaptive finite element approximation for the boundary optimal control problem. Finally, we present a numerical example to confirm our theoretical results.     

A new methodology for anisotropic mesh refinement based upon error gradients

We introduce a new strategy for controlling the use of anisotropic mesh refinement based upon the gradients of an a posteriori approximation of the error in a computed finite element solution. The efficiency of this strategy is demonstrated using a simple anisotropic mesh adaption algorithm and the quality of a number of potential a posteriori error estimates is considered.     

A formulation of the eddy current problem in the presence of electric ports

The time-harmonic eddy current problem with either voltage or current intensity excitation is considered. We propose and analyze a new finite element approximation of the problem, based on a weak formulation where the main unknowns are the electric field in the conductor, a scalar magnetic potential in the insulator and, for the voltage excitation problem, the current intensity. The finite element approximation uses edge elements for the electric field and nodal elements for the scalar magnetic potential, and an optimal error estimate is proved. Some numerical results illustrating the performance of the method are also presented.     

A characteristic-mixed finite element method for time-dependent convection-diffusion optimal control problem

In this paper, we examine the method of characteristic-mixed finite element for the approximation of convex optimal control problem governed by time-dependent convection-diffusion equations with control constraints. For the discretization of the state equation, the characteristic finite element is used for the approximation of the material derivative term (i.e., the time derivative term plus the convection term), and the lowest-order Raviart-Thomas mixed element is applied for the approximation of the diffusion term. We derive some a priori error estimates for both the state and control approximations.     

HOW TO PROVE THE DISCRETE RELIABILITY FOR NONCONFORMING FINITE ELEMENT METHODS

Optimal convergence rates of adaptive finite element methods are well understood in terms of the axioms of adaptivity.One key ingredient is the discrete reliability of a residualbased a posteriori error estimator,which controls the error of two discrete finite element solutions based on two nested triangulations.In the error analysis of nonconforming finite element methods,like the Crouzeix-Raviart or Morley finite element schemes,the difference of the piecewise derivatives of discontinuous approximations to the distributional gradients of global Sobolev functions plays a dominant role and is the object of this paper.The nonconforming interpolation operator,which comes natural with the definition of the aforementioned nonconforming finite element in the sense of Ciarlet,allows for stability and approximation properties that enable direct proofs of the reliability for the residual that monitors the equilibrium condition.The novel approach of this paper is the suggestion of a right-inverse of this interpolation operator in conforming piecewise polynomials to design a nonconforming approximation of a given coarse-grid approximation on a refined triangulation.The results of this paper allow for simple proofs of the discrete reliability in any space dimension and multiply connected domains on general shape-regular triangulations beyond newest-vertex bisection of simplices.Particular attention is on optimal constants in some standard discrete estimates listed in the appendices.     

Finite element approximationof a nonlinear elliptic equation arising from bimaterial problemsin elastic-plastic mechanics

Summary. In [13], a nonlinear elliptic equation arising from elastic-plastic mechanics is studied. A well-posed weak formulation is established for the equation and some regularity results are further obtained for the solution of the boundary problem. In this work, the finite element approximation of this boundary problem is examined in the framework of [13]. Some error bounds for this approximation are initially established in an energy type quasi-norm, which naturally arises in degenerate problems of this type and proves very useful in deriving sharper error bounds for the finite element approximation of such problems. For sufficiently regular solutions optimal error bounds are then obtained for some fully degenerate cases in energy type norms. Received June 12, 1998 / Revised version received June 21, 1999 / Published online June 8, 2000     

Analysis on a Finite Volume Element Method for Stokes Problems

Finite volume element method for the Stokes problem is considered. We use a conforming piecewise linear function on a fine grid for velocity and piecewise constant element on a coarse grid for pressure. For general triangulation we prove the equivalence of the finite volume element method and a saddle-point problem, the inf-sup condition and the uniqueness of the approximation solution. We also give the optimal order H^1 norm error estimate. For two widely used dual meshes we give the L^2 norm error estimates, which is optimal in one case and quasi-optimal in another ease. Finally we give a numerical example.     

退化椭圆问题的最小二乘混合元逼近

１．引言我们将考虑具有退化系数的椭圆问题其中 Ω为 ＩＲ２中的一个凸多边形区域,定义为这里的ｇ＞０为分片线性连续函数．从物理背景来看,问题（１．１）来源于轴对称共振结构中的电磁场研究．Ｍａｒｉｎｉ,Ｐｉｅｔｒａ（１９９５）［４］研究了问题（１．１）的混合有限元逼近,并得到了最优误差估计． 此文,我们采用一种新的混合元,即最小二乘混合元方法［５］;对退化问题（１．１）进行逼近,利用插值投影证明了近似解具有最优阶精度的收敛性．比较起经典混合元方法来,最小二乘混合元方法有两个优越性：有限元空间不必满足Ｌ…     

An optimal order convergence for a weak formulation of the compressible Stokes system with inflow boundary condition

Summary. We formulate the compressible Stokes system given in (1.1) into a (new) weak formulation (2.1). A finite element method for this is presented. Existence and uniqueness of the finite element method is shown. An optimal error estimate for the numerical approximation is obtained. Numerical examples are given, showing its efficiency and rates of convergence of the approximate solutions that results from the discrete problem (3.1). Received October 20, 1996 / Revised version received January 21, 1999 / Published online: April 20, 2000     

Mixed finite element approximation of a degenerate elliptic problem

Summary. We present a mixed finite element approximation of an elliptic problem with degenerate coefficients, arising in the study of the electromagnetic field in a resonant structure with cylindrical symmetry. Optimal error bounds are derived. Received May 4, 1994 / Revised version received September 27, 1994     

A POSTERIORI ERROR ESTIMATE OF OPTIMAL CONTROL PROBLEM OF PDE WITH INTEGRAL CONSTRAINT FOR STATE

In this paper, we study adaptive finite element discretization schemes for an optimal control problem governed by elliptic PDE with an integral constraint for the state. We derive the equivalent a posteriori error estimator for the finite element approximation, which particularly suits adaptive multi-meshes to capture different singularities of the control and the state. Numerical examples are presented to demonstrate the efficiency of a posteriori error estimator and to confirm the theoretical results.     

一类广义插值函数与广义有限元方法的后验估计

１．问题的提出具有快速振荡系数的微分方程大量出现在复合材料、多孔介质渗流等实际问题中．因为这类方程系数的激烈振荡性（受小尺度ε控制）,通常的有限元方法需花费巨大的计算工作量才能获得有意义的数值解（文［７］）,这对多维问题是无法承受的．８０年代发展起来的广义有限元法（文［１］［３］［５］）,为这类问题的解决提供了一条有效的新途径,它可在剖分步长ｈ＞＞　ε的情况下得到令人满意的数值结果．在广义有限元的理论分析中,因方程解的正则性估计通常与小尺度ε　有关,所以通常的有限元分析方法有一定的困难,目前尚未…