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.     

Singularity functions at axisymmetric edges and their representation by Fourier series

The regularity of solutions of the Dirichlet problem for the Poisson equation in three-dimensional axisymmetric domains with reentrant edges is studied by means of Fourier series. The decomposition of the 3D problem into variational equations in 2D, a priori estimates of their solutions, a theorem of Riesz–Fischer type and two singularity functions (of tensor and non-tensor product type) are given.     

Graded mesh refinement and error estimates of higher order for DGFE solutions of elliptic boundary value problems in polygons

Error estimates for DGFE solutions are well investigated if one assumes that the exact solution is sufficiently regular. In this article, we consider a Dirichlet and a mixed boundary value problem for a linear elliptic equation in a polygon. It is well known that the first derivatives of the solutions develop singularities near reentrant corner points or points where the boundary conditions change. On the basis of the regularity results formulated in Sobolev–Slobodetskii spaces and weighted spaces of Kondratiev type, we prove error estimates of higher order for DGFE solutions using a suitable graded mesh refinement near boundary singular points. The main tools are as follows: regularity investigation for the exact solution relying on general results for elliptic boundary value problems, error analysis for the interpolation in Sobolev–Slobodetskii spaces, and error estimates for DGFE solutions on special graded refined meshes combined with estimates in weighted Sobolev spaces. Our main result is that there exist a local grading of the mesh and a piecewise interpolation by polynoms of higher degree such that we will get the same order O (h^α) of approximation as in the smooth case. © 2011 Wiley Periodicals, Inc. Numer Mehods Partial Differential Eq, 2012     

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.     

Null Controllability of Three-dimensional Heat Equation in Singular Domains

We study in this paper a class of parabolic equations in singular domains. These equations are defined in a singular cylindrical domain whose cross-section contains one reentrant corner or one straight emerging crack. We assume that the diffusion coefficients are non-smooth in the normal direction. We will show some spectral inequality thanks to Carleman type estimates and the construction of a suitable weight function satisfying some properties. As in Benabdallah et al. (C. R. Acad. Sci. Paris 344(6):357–362, 2007), we deduce the null-controllability of these equations with the help of the Lebeau-Robbiano method.     

On a fast direct elliptic solver by a modified Fourier method

We describe high order numerical algorithms for the solution of second order elliptic equations in rectangular domains. These algorithms are based on the Fourier method in combination with a subtraction procedure. The singularities at the corner points, arising due to non-smoothness of the boundaries, are treated explicitly using properly constructed singular corner functions. The present algorithm is a generalization of the Fast Poisson Solver developed in our previous paper. This revised version was published online in August 2006 with corrections to the Cover Date.     

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.     

Singular Matched Eigenfunction Expansions for Stokes Flow around a Corner

A singular matched eigenfunction expansion method is describedfor solving Stokes flow around a corner. The flow region isdecomposed into a number of simpler rectangular subregions;this enables the stream function to be represented by meansof an expansion of Papkovich-Fadle eigenfunctions in each ofthese subregions. The coefficients in these expansions are obtainedby matching them across common interfaces in a weak sense. Theresulting solution is used in a post-processing technique todetermine the coefficients in the known locally convergent expansionof the stream function at reentrant and salient corners. A smallnumber of terms in this expansion is necessary to produce accurateapproximations.     

Necessary optimality conditions for a singular surface in the form of synthesis

For a linear control problem using the traditional open-loop approach, a new representation for the singular control and generalized, invariant conditions for optimality are found. The phase portrait of a nonlinear control problem is considered in the neighborhood of singular trajectories. The singular paths form a hypersurface, approached by regular paths from both sides. The Bellman function for this problem is a classical (smooth) solution to a first-order PDE with nonsmooth Hamiltonian over two smooth (regular) branches, related to the halfneighborhoods of the surface. These solutions are at least twice differentiable and have first discontinuous derivatives of odd order. The invariant form for these necessary conditions is found in terms of Jacobi (Poisson) brackets, consisting of several equalities and inequalities. The latter relations guarantee the validity of the Kelley condition as well as the geometrical constraints for the singular control variables. Thus, the Kelley condition appears to be just a certain property of a smooth solution to a first-order PDE with nonsmooth Hamiltonian. All the relations, including the Hamiltonian equations of singular motion, do not use singular controls; they are based on regular Hamiltonians depending only upon the state vector and the gradient of the Bellman function (adjoint vector).This work was suported by Grant No. 93-013-16285 of the Russian Fund for Fundamental Research.     

Decompositions in Edge and Corner Singularities for the Solution of the Dirichlet Problem of the Laplacian in a Polyhedron

The solution of the three-dimensional Dirichlet problem for the Laplacian in a polyhedral domain has Special singular forms at corners and edges. The main result of this paper is a “tensor-product” decomposition of those singular forms along the edges. Such a decomposition with both edge singularities, additional corner singularities and a smoother remainder refines known regularity results for the solution where either the edge singularities are of non-tensor product form or the remainder term belongs to an anisotropic Sobolev space for data given in an isotropic Sobolev space.     

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.     

A regularity result of solution to the compressible Stokes equations on a convex polygon

A compressible Stokes problem is analyzed in a convex polygon D. The goal of this paper is to sort out a singularity of the pressure function at the corner and to establish the corresponding regularity result of the resulted remainder part of the solution. For this a solution formula is derived and the singular function of the Stokes problem is considered. It is seen that the lowest order of the regularity of the system is the same as that of the (incompressible) Stokes one.     

On the boundary value problem of the biharmonic operator on domains with angular corners

The paper is concerned with boundary singularities of weak solutions of boundary value problems governed by the biharmonic operator. The presence of angular corner points or points at which the type of boundary condition changes in general causes local singularities in the solution. For that case the general theory of V. A. Kondrat'ev provides a priori estimates in weighted Sobolev norms and asymptotic singular representations for the solution which essentially depend on the zeros of certain transcendental functions. The distribution of these zeros will be analysed in detail for the biharmonic operator under several boundary conditions. This leads to sharp a priori estimates in weighted Sobolev norms where the weight function is characterized by the inner angle of the boundary corner. Such estimates for “negative” Sobolev norms are used to analyse also weakly nonlinear perturbations of the biharmonic operator as, for instance, the von Kármán model in plate bending theory and the stream function formulation of the steady state Navier-Stokes problem. It turns out that here the structure of the corner singularities is essentially the same as in the corresponding linear problem.     

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.     

两非均匀半平面粘结的非均匀平面的裂纹问题*

本文对两个非均匀半平面粘结的非均匀平面的裂纹问题作了分析,文中假定两种材料的泊松比v相同,但杨氏模量随坐标x按不同形式的指数函数变化.本文使用非均匀平面问题的单裂纹解及富氏变换方法, 使问题归为一个柯西型奇异积分方程,最后对应力强度因子的计算给出了若干数值例子.     

Analytical Behaviour of Solutions of Boundary Integral Equations for a Nonsmooth Region

The Dirichlet problem for Helmholtz's equation in a domain exterior to some bounded smooth boundary in two dimensionsmay be solved by means of a combined potential of the singleand double layers. In this paper, the problem arising from allowingcorner points on the boundary is investigated. The resultingnoncompact operator is effectively split into singular and compactparts. By using the Mellin transforms, the equation can be convertedinto some Cauchy-type singular integral equations. Consequently,the singular form of the solution is found in terms of r^ßat a corner with 0>ß>1. As a first step towarddeveloping new numerical methods for the problem, one typicalexample is presented to demonstrate the slow convergence ofexisting methods without any modifications. Then the mesh-gradingtechnique designed for singular equations is successfully implementedto restore the order of convergence.     

A hybrid method for numerical solution of Poisson’s equation in a domain with a dielectric corner

An electrostatic problem of determining a potential in a domain containing an incoming dielectric corner, which reduces to solving Poisson’s equation in this domain, is considered. A specific feature of the solution of this problem is that it is bounded in a neighborhood of the dielectric corner but its gradient increases without limit. An efficient hybrid algorithm for the numerical solution of the problem, based on the finite element method and taking into account the known asymptotic representation of the solution in the neighborhood of the dielectric corner, is proposed.     

Impacts with Global Dissipation Index

A geometric interpretation of Moreau's frictionless multi‐contact impact law is presented and extended to the case of reentrant corners, describing for example the corner‐corner impact problem of two rectangular blocks. The methods used are based on the geometry of cones. The main construction of the impact law for the convex case is done by the unique orthogonal decomposition of the pre‐impact velocity with respect to two orthogonal closed convex cones in the tangent space of the differential manifold, one of them approximating the non‐smooth boundary of the admissible domain in some neighborhood of the point of impact.     

Parabolic finite volume element equations in nonconvex polygonal domains

We study spatially semidiscrete and fully discrete finite volume element approximations of the heat equation with homogeneous Dirichlet boundary conditions in a plane polygonal domain with one reentrant corner. We show that, as a result of the singularity in the solution near the reentrant corner, the convergence rate is reduced from optimal second order, similarly to what was shown for the finite element method in the earlier work 2 . Optimal order convergence may be restored by mesh refinement near the corners of the domain. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

On M–multisplittings of singular M–matrices with application to Markov chains

Given a singular M–matrix of a linear system, convergent conditions under which iterative schemes based on M–multisplittings are studied. Two of those conditions, the index of the iteration matrix and its spectral radius are investigated and related to those of the M-matrix. Furthermore, a parallel multisplitting iteration scheme for solving singular linear systems is suggested which can be applied to practical problems such as Poisson and elasticity problems under certain boundary conditions, the Neumann problem, and in Markov chains. A discussion of that multisplitting scheme, based on Gauss–Seidel type splittings is given for computing the stationary distribution vector of Markov chains. In this case a computational viable algorithm can be constructed, since only the nonsingularity of one weighting matrix of the multisplitting is needed. © 1998 John Wiley & Sons, Ltd.