The Dual Singular Function Method for 2D Boundary Integral Equations

The accuracy of standard boundary element methods for elliptic boundary value problems deteriorates if the boundary of the domain contains corners or if the boundary conditions change along the boundary. Here we first investigate the convergence behaviour of standard spline Galerkin approximation on quasi-uniform meshes for boundary integral equations on polygonal domains. It turns out, that the order of convergence depends on some constant describing the singular behaviour of solutions near corner points of the boundary. In order to recover the full order of convergence for the Galerkin approximation we propose the dual singular function method which is often used for improving the accuracy of finite element methods. The theoretical convergence results are confirmed and illustrated by a numerical example.     

Error Analysis of Local Refinements of Polygonal Domains

This paper gives a thorough analysis of the local refinement method on plane polygonal domains with special attention to the treatment of reentrant corner. Convergence rates of the finite element method under various norms are derived via a systematic treatment of the interpolation theory in weighted Sobolev spaces. It is proved that by refining the mesh suitably, the finite element approximations for problems with singularities achieve the same convergence rates as those for smooth solutions.     

Optimal control of the Stokes equations: conforming and non-conforming finite element methods under reduced regularity

This paper deals with a control-constrained linear-quadratic optimal control problem governed by the Stokes equations. It is concerned with situations where the gradient of the velocity field is not bounded. The control is discretized by piecewise constant functions. The state and the adjoint state are discretized by finite element schemes that are not necessarily conforming. The approximate control is constructed as projection of the discrete adjoint velocity in the set of admissible controls. It is proved that under certain assumptions on the discretization of state and adjoint state this approximation is of order 2 in L ²(Ω). As first example a prismatic domain with a reentrant edge is considered where the impact of the edge singularity is counteracted by anisotropic mesh grading and where the state and the adjoint state are approximated in the lower order Crouzeix-Raviart finite element space. The second example concerns a nonconvex, plane domain, where the corner singularity is treated by isotropic mesh grading and state and adjoint state can be approximated by a couple of standard element pairs.     

Corner asymptotics of the magnetic potential in the eddy‐current model

In this paper, we describe the magnetic potential in the vicinity of a corner of a conducting body embedded in a dielectric medium in a bidimensional setting. We make explicit the corner asymptotic expansion for this potential as the distance to the corner goes to zero. This expansion involves singular functions and singular coefficients. We introduce a method for the calculation of the singular functions near the corner, and we provide two methods to compute the singular coefficients: the method of moments and the method of quasi‐dual singular functions. Estimates for the convergence of both approximate methods are proven. We eventually illustrate the theoretical results with finite element computations. The specific nonstandard feature of this problem lies in the structure of its singular functions: They have the form of series whose first terms are harmonic polynomials, and further terms are genuine nonsmooth functions generated by the piecewise constant zeroth order term of the operator. Copyright © 2013 John Wiley & Sons, Ltd.     

Finite element treatment of boundary singularities by augmentation with non-exact singular functions

A two-dimensional Poisson problem which contains both an interface and a reentrant corner is considered. For this problem the singular form of the solution at the reentrant corner is not known explicitly, with the result that a (nonexact) approximation to the singular form has to be calculated. The finite element method is applied to the Poisson problem, with the test and trial function spaces augmented with the nonexact singular functions. An error analysis for the nonexact augmentation is presented.     

用新的逼近空间求解奇性问题

用新的逼近空间求解奇性问题梁国平，孔林，何江衡（中国科学院数学研究所）ＡＮＥＷＦＩＮＩＴＥＥＬＥＭＥＮＴＳＰＡＣＥＦＯＲＳＩＮＧＵＬＡＲＩＴＹＰＲＯＢＬＥＭＳ￥ＬｉａｎｇＧｕｏ－ｐｉｎｇ；ＫｏｎｇＬｉｎ；ＨｅＪｉａｎｇ－ｈｅｎｇ（Ｉｎｓｔｉｔｕｔｅｏ...     

Extrapolation techniques for reducing the pollution effect or reentrant corners in the finite element method

Summary Standard finite element schemes applied to boundary value problems on domains with reentrant corners suffer from a global loss of accuracy caused by the influence of the corner singularities. For a model situation, it is shown that this pollution effect can be completely described in terms of an asymptotic error expansion with respect to fractional powers of the mesh size parameter. This provides the basis of an extrapolation procedure which recovers the full order of accuracy for the solution as well as for the corresponding stress intensity factors.     

The Convergence of Multigrid Methods for Solving Finite Element Equations in the Presence of Singularities

1.IntroductionMultigridMethodsprovideoptimalordersolversforlinearsysternsoffiniteele-mentequationsarisingfromellipticboundaryvalueproblems.Theconvergenceofmultigridmethodswasprovedbymanya.tho.s[2-6,9-12l.AlltheseproofS,requirestrongregularitiesandquasi-uniformityofgridsl',,']-Forexample,assumingH1+oregularityandquasi-uniformtriangulations,Bank&Dupollt[3]showedaconvergencerateofo(mY),foragrowingnumbermofsmoothingstepsperlevel.Intheoptimalcasecr=1,theproblemhastobeH'-regUlar.Whentheregionhas…     

On the solution of Maxwell's equations in polygonal domains

This paper is concerned with the structure of the singular and regular parts of the solution of time‐harmonic Maxwell's equations in polygonal plane domains and their effective numerical treatment. The asymptotic behaviour of the solution near corner points of the domain is studied by means of discrete Fourier transformation and it is proved that the solution of the boundary value problem does not belong locally to H² when the boundary of the domain has non‐acute angles. A splitting of the solution into a regular part belonging to the space H², and an explicitly described singular part is presented. For the numerical treatment of the boundary value problem, we propose a finite element discretization which combines local mesh grading and the singular field methods and derive a priori error estimates that show optimal convergence as known for the classical finite element method for problems with regular solutions. Copyright © 2006 John Wiley & Sons, Ltd.     

Finite element error estimates for Neumann boundary control problems on graded meshes

A specific elliptic linear-quadratic optimal control problem with Neumann boundary control is investigated. The control has to fulfil inequality constraints. The domain is assumed to be polygonal with reentrant corners. The asymptotic behaviour of two approaches to compute the optimal control is discussed. In the first the piecewise constant approximations of the optimal control are improved by a postprocessing step. In the second the control is not discretized; instead the first order optimality condition is used to determine an approximation of the optimal control. Although the quality of both approximations is in general affected by corner singularities a convergence order of 3/2 can be proven provided that the mesh is sufficiently graded.     

EXTRAPOLATION OF BILINEAR FINITE ELEMENT SOLUTION WITH LOCAL REFINEMENT MESH

For the less smooth solution caused by the reentrant domain it is shown that one step of extrapolation increases the order of bilinear finite element solution from 2 to 3 when the rectangular mesh satisfies certain local refinement condition.     

Carleman inequalities for the heat equation in singular domains

We consider the Cauchy problem associated to the heat equation firstly in a plane domain with a reentrant corner, then in a cracked domain. By constructing a weight function, we show a result of null controllability using Carleman estimates.     

Superelements for the finite element solution of two-dimensional elliptic problems with boundary singularities

A novel singular superelement (SSE) formulation has been developed to overcome the loss of accuracy encountered when applying the standard finite element schemes to two-dimensional elliptic problems possessing a singularity on the boundary arising from an abrupt change of boundary conditions or a reentrant corner. The SSE consists of an inner region over which the known analytic form of the solution in the vicinity of the singular point is utilized, and a transition region in which blending functions are used to provide a smooth transition to the usual linear or quadratic isoparametric elements used over the remainder of the domain. Solution of the finite element equations yield directly the coefficients of the asymptotic series, known as the flux/stress intensity factors in linear heat transfer or elasticity theories, respectively. Numerical examples using the SSE for the Laplace equation and for computing the stress intensity factors in the linear theory of elasticity are given, demonstrating that accurate results can be attained for a moderate computational effort.     

Finite element methods with numerical quadrature for elliptic problems with smooth interfaces

The purpose of this paper is to study the effect of the numerical quadrature on the finite element approximation to the exact solution of elliptic equations with discontinuous coefficients. Due to low global regularity of the solution, it seems difficult to achieve optimal order of convergence with classical finite element methods [Z. Chen, J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math. 79 (1998) 175-202]. We derive error estimates in finite element method with quadrature for elliptic interface problems in a two-dimensional convex polygonal domain. Optimal order error estimates in L² and H¹ norms are shown to hold even if the regularity of the solution is low on the whole domain. Finally, numerical experiment for two dimensional test problem is presented in support of our theoretical findings.     

Non-Conforming Domain Decomposition with Hybrid Method

We present a non-conforming domain decomposition technique for solving elliptic problems with the finite element method. Functions in the finite element space associated with this method may be discontinuous on the boundary of subdomains. The sizes of the finite meshes, the kinds of elements and the kinds of interpolation functions may be different in different subdomains. So, this method is more convenient and more efficient than the conforming domain decomposition method. We prove that the solution obtained by this method has the same convergence rate as by the conforming method, and both the condition number and the order of the capacitance matrix are much lower than those in the conforming case.     

The Fourier Singular Complement Method for the Poisson problem. Part II: axisymmetric domains

This paper is the second part of a threefold article, aimed at solving numerically the Poisson problem in three-dimensional prismatic or axisymmetric domains. In the first part of this series, the Fourier Singular Complement Method was introduced and analysed, in prismatic domains. In this second part, the FSCM is studied in axisymmetric domains with conical vertices, whereas, in the third part, implementation issues, numerical tests and comparisons with other methods are carried out. The method is based on a Fourier expansion in the direction parallel to the reentrant edges of the domain, and on an improved variant of the Singular Complement Method in the 2D section perpendicular to those edges. Neither refinements near the reentrant edges or vertices of the domain, nor cut-off functions are required in the computations to achieve an optimal convergence order in terms of the mesh size and the number of Fourier modes used.     

Boundary behavior near reentrant corners for solutions of certain elliptic equations

The behavior near a reentrant corner of quasilinear non-uniformly elliptic partial differential equations which satisfy certain conditions is examined. For an elliptic equation satisfying a general maximum principle and having «convex barriers», we show that if a solution has radial limits at the corner, these limits have the same qualitative behavior as those for nonparametric minimal surfaces. For certain equations of mean curvature type, radial limits at the reentrant corner are shown to exist.     

鱼骨形网格上二阶方程混合元的超收敛

In this paper, superconvergence of the lowest order Raviart-Thomas mixed finite element approximation for second order Neumann boundary value problem on fishbone shape meshes is analyzed. The main term of the error between the exact solution and the finite element interpolating function is determined by Bramble-Hilbert lemma on the individual finite element. A part of the main term of the error on two adjacent finite elements can be cancelled along the special direction, and thus the higher order error estimate is obtained on the whole domain by summation. Compared with the general finite element error estimate,the convergence rate can be increased from order one to order two in L2-norm by postprocessing superconvergence technique.     

Interior estimates for a low order finite element method for the Reissner–Mindlin plate model

Interior error estimates are obtained for a low order finite element introduced by Arnold and Falk for the Reissner–Mindlin plates. It is proved that the approximation error of the finite element solution in the interior domain is bounded above by two parts: one measures the local approximability of the exact solution by the finite element space and the other the global approximability of the finite element method. As an application, we show that for the soft simply supported plate, the Arnold–Falk element still achieves an almost optimal convergence rate in the energy norm away from the boundary layer, even though optimal order convergence cannot hold globally due to the boundary layer. Numerical results are given which support our conclusion. This revised version was published online in June 2006 with corrections to the Cover Date.     

-curved finite elements and applications to plate and shell problems

Many thin-plate and thin-shell problems are set on plane reference domains with a curved boundary. Their approximation by conforming finite-elements methods requires ¹-curved finite elements entirely compatible with the associated ¹-rectilinear finite elements. In this contribution we introduce a ¹-curved finite element compatible with the P₅-Argyris element, we study its approximation properties, and then, we use such an element to approximate the solution of thin-plate or thin-shell problems set on a plane-curved boundary domain. We prove the convergence and we get a priori asymptotic error estimates which show the very high degree of accuracy of the method. Moreover we obtain criteria to observe when choosing the numerical integration schemes in order to preserve the order of the error estimates obtained for exact integration.