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Solutions of boundary value problems in three‐dimensional domains with edges may exhibit singularities which are known to influence both the accuracy of the finite element solutions and the rate of convergence in the error estimates. This paper considers boundary value problems for the Poisson equation on typical domains Ω ⊂ ℝ3 with edge singularities and presents, on the one hand, explicit computational formulas for the flux intensity functions. On the other hand, it proposes and analyzes a nonconforming finite element method on regular meshes for the efficient treatment of the singularities. The novelty of the present method is the use of the explicit formulas for the flux intensity functions in defining a postprocessing procedure in the finite element approximation of the solution. A priori error estimates in H1(Ω) show that the present algorithm exhibits the same rate of convergence as it is known for problems with regular solutions. [ABSTRACT FROM AUTHOR]

Let M ‾ be a compact smoothly stratified pseudomanifold with boundary, satisfying the Witt assumption. In this paper we introduce the de Rham signature and the Hodge signature of M ‾ , and prove their equality. Next, building also on recent work of Albin and Gell-Redman, we extend the Atiyah-Patodi-Singer index theory established in our previous work under the hypothesis that M ‾ has stratification depth 1 to the general case, establishing in particular a signature formula on Witt spaces with boundary. In a parallel way we also pass to the case of a Galois covering M ‾ Γ of M ‾ with Galois group Γ. Employing von Neumann algebras we introduce the de Rham Γ-signature and the Hodge Γ-signature and prove their equality, thus extending to Witt spaces a result proved by Lück and Schick in the smooth case. Finally, extending work of Vaillant in the smooth case, we establish a formula for the Hodge Γ-signature. As a consequence we deduce the fundamental result that equates the Cheeger-Gromov rho-invariant of the boundary ∂ M ‾ Γ with the difference of the signatures of and M ‾ and M ‾ Γ : sign dR (M ‾ , ∂ M ‾) − sign dR Γ ( M ‾ Γ , ∂ M ‾ Γ) = ρ Γ (∂ M ‾ Γ). We end the paper with two geometric applications of our results. [ABSTRACT FROM AUTHOR]

High order techniques are known to be effective for the electromagnetic analysis of smooth structures. In the following, high order representations developed for the current density at edges and junctions are incorporated into the locally-corrected Nyström method. Conducting structures used for purposes of illustration include a strip, a structure with three fins and a junction, and a hexagonal cylinder. Results suggest that the accuracy of the numerical results obtained with the new approach is comparable to that obtained for problems with smooth surfaces. [ABSTRACT FROM AUTHOR]

n L-estimate of the finite element error is proved for a Dirichlet and a Neumann boundary value problem on a three-dimensional, prismatic and non-convex domain that is discretized by an anisotropic tetrahedral mesh. To this end, an approximation error estimate for an interpolation operator that is preserving the Dirichlet boundary conditions is given. The challenge for the Neumann problem is the proof of a local interpolation error estimate for functions from a weighted Sobolev space. [ABSTRACT FROM AUTHOR]

Abstract: The elastic solution in a vicinity of a re-entrant wedge can be described by a Williams like expansion in terms of powers of the distance to a point on the edge. This expansion has a particular structure due to the invariance of the problem by translation parallel to the edge. We show here that some terms, so-called primary solutions, derive directly from solutions to the 2-D corner problem posed in the orthogonal cross section of the domain. The others, baptized shadow functions, derive of the primary solutions by integration along the axis parallel to the edge. This 3-D Williams expansion is shown to be equivalent to the edge expansion proposed by Costabel et al. [M. Costabel, M. Dauge, Z. Yosibash, A quasidual function method for extracting edge stress intensity functions, SIAM J. Math. Anal. 35 (5) (2004) 1177–1202]. To cite this article: T. Apel et al., C. R. Mecanique 336 (2008). [Copyright &y& Elsevier]

Abstract: The present work deals with an evaluation of stress intensity factors (SIFs) along straight crack fronts and edges in three-dimensional isotropic elastic solids. A new numerical approach is developed for extraction, from a solution obtained by the boundary element method (BEM), of those SIFs, which are relevant for a failure assessment of mechanical components. In particular, the generalized SIFs associated to eigensolutions characterized by unbounded stresses at a neighbourhood of the crack front or a reentrant edge and also that associated to T-stress at the crack front can be extracted. The method introduced is based on a conservation integral, called H-integral, which leads to a new domain-independent integral represented by a scalar product of the SIF times some element shape function defined along the crack front or edge. For sufficiently small element lengths these weighted averages of SIFs give reasonable pointwise estimation of the SIFs. A proof of the domain integral independency, based on the bi-orthogonality of the classical two-dimensional eigensolutions associated to a corner problem, is presented. Numerical solutions of two three-dimensional problems, a crack problem and a reentrant edge problem, are presented, the accuracy and convergence of the new approach for SIF extraction being analysed. [Copyright &y& Elsevier]

This paper is concerned with a priori error estimates and convergence analysis of the Fourier-finite-element solutions of the Neumann problem for the Lamé equations in axisymmetric domains [InlineMediaObject not available: see fulltext.] with reentrant edges. The Fourier-FEM combines the approximating Fourier method with respect to the rotational angle using trigonometric polynomials of degree N ( N→∞), with the finite element method on the plane meridian domain of [InlineMediaObject not available: see fulltext.] with mesh size h ( h→0) for approximating the Fourier coefficients. The asymptotic behavior of the solution near reentrant edges is described by singularity functions in non-tensor product form and treated numerically by means of finite element method on locally graded meshes. For [InlineMediaObject not available: see fulltext.] the rate of convergence of the combined approximations in [InlineMediaObject not available: see fulltext.] is proved to be of the order [InlineMediaObject not available: see fulltext.] [ABSTRACT FROM AUTHOR]

In this paper we present the basic mathematical tools for treating boundary value problems for the Maxwell equations in three-dimensional axisymmetric domains with reentrant edges using the Fourier-finite-element method. We consider both the classical and the regularized time-harmonic Maxwell equations subject to perfect conductor boundary conditions. The partial Fourier decomposition reduces the three-dimensional boundary value problem into an infinite sequence of two-dimensional boundary value problems in the plane meridian domain of the axsiymmetric domain. Here, suitable weighted Sobolev spaces that characterize the solutions of the reduced problems are given, and their trace properties on the rotational axis are proved. In these spaces, it is proved that the reduced problems are well posed, and the asymptotic behavior of the solutions near reentrant corners of the meridian domain is explicitly described by suitable singularity functions. Finally, a finite number of the two-dimensional problems is considered and treated using H1-conforming finite elements. An approximation of the solution of the three-dimensional problem is obtained by Fourier synthesis. For domains with reentrant edges, the singular field method is employed to compensate the singular behavior of the solutions of the reduced problems. Emphases are given to convergence analysis of the combined approximations in H1 under different regularity assumptions on the solution. [ABSTRACT FROM AUTHOR]

The asymptotics of solutions to scalar second order elliptic boundary value problems in three-dimensional polyhedral domains in the vicinity of an edge is provided in an explicit form. It involves a family of eigen-functions with their shadows, and the associated edge flux intensity functions (EFIFs), which are functions along the edges. Utilizing the explicit structure of the solution in the vicinity of the edge we present a new method for the extraction of the EFIFs called quasidual function method. It can be interpreted as an extension of the dual function contour integral method in 2-D domains, and involves the computation of a surface integral J[R] along a cylindrical surface of radius R away from the edge as presented in a general framework in (Costabel et al., 2004). The surface integral J[R] utilizes special constructed extraction polynomials together with the dual eigen-functions for extracting EFIFs. This accurate and efficient method provides a polynomial approximation of the EFIF along the edge whose order is adaptively increased so to approximate the exact EFIF. It is implemented as a post-solution operation in conjunction with the p-version finite element method. Numerical realization of some of the anticipated properties of the J[R] are provided, and it is used for extracting EFIFs associated with different scalar elliptic equations in 3-D domains, including domains having edge and vertex singularities. The numerical examples demonstrate the efficiency, robustness and high accuracy of the proposed quasi-dual function method, hence its potential extension to elasticity problems. [ABSTRACT FROM AUTHOR]

This paper analyses the regularity of the weak solution of the linear elasticity system in three-dimensional axisymmetric domains with reentrant edges under prescribed traction on the boundary by means of Fourier series. Using partial Fourier analysis with respect to one space direction (rotational angle), the three-dimensional boundary value problem (BVP) is decomposed into a sequence of decoupled two-dimensional boundary value problems on the meridian of the axisymmetric domain. The splitting of the 2D solutions near corners of the meridian domain into regular and singular parts provides coefficients from which the 3D edge singularity functions (generalized stress intensity factors) are derived. Two types of singularity functions are presented, namely, a tensorial type, which needs more smoothness assumption on the right hand side and a non-tensorial type, which does not demand any further smoothness assumptions. [ABSTRACT FROM AUTHOR]

In this paper we derive new two-dimensional (2-D) quadrature formulas for the discretization of boundary integral equations in the presence of conducting or dielectric edges. The proposed formulas allow us to exactly integrate polynomials of degree less than or equal to five, multiplied by an algebraic singular factor that diverges along one side of the triangular integration domain. This is the kind of singularity that occurs when physical edges are present in both conducting and dielectric bodies. Numerical tests are performed on the presented formulas, in order to validate the achieved improvement in accuracy, and examples are given of their application to the determination of radar cross-section of 3-D metallic objects. [ABSTRACT FROM AUTHOR]

We present a methodfor the computation of the coefficients of singularities along the edges of a polyhedron for second-order elliptic boundary value problems. The class of problems considered includes problems of stress concentration along edges or crack fronts in general linear three-dimensional elasticity. Our method uses an incomplete construction of three-dimensional dual singular functions, based on explicitly known dual singular functions of two-dimensional problems tensorizedby test functions along the edge andcombinedwith complementary terms improving their orthogonality properties with respect to the edge singularities. Our method is aimed at the numerical computation of the stress intensity functions. It is suitable for a postprocessing procedure in the finite element approximation of the solution of the boundary value problem. [ABSTRACT FROM AUTHOR]

In a series of papers, Harriman et al. have presented a reliable means of simulating steady state currents with adaptive finite element. They have demonstrated that multi-step, even non-linear, mechanisms, alongside convection, can be incorporated. However, there is considerable complication in the mathematical approach taken, and it would seem to be limited as it stands to certain types of electrode reaction models – notably those without heterogeneous kinetics or transient effects. In this paper we discuss alternative approaches to error estimation and adaptivity, and present a simpler formulation, capable of simulating systems with heterogeneous kinetics; transient simulations also appear more attainable. We introduce, apparently for the first time in electrochemistry, the use of gradient recovery methods to both error estimation and accurate current calculations. The result is an algorithm with considerably more potential for generalisation, closer to the ideal of an entirely flexible automatic simulation program, capable of dealing with any mechanism or electrode geometry. In tests we find our method to perform more efficiently than that cited above, producing accurate results with simpler meshes in less time. [Copyright &y& Elsevier]

This review paper presents a unified formulation of the Kutta condition for steady and unsteady flows, implemented by removing all unbounded velocity singularities (of power law and logarithmic type) at the trailing edge, and including nonlinear wakes and thick swept back wings. A suitable boundary integral approach is adopted and the uniqueness issue is discussed for several wing configurations of interest in aerodynamics. Sommario. Si presenta una formulazione unificata della condizione di Kutta per flussi stazionari e non stazionari, ottenuta imponendo la limitatezza della velocità al bordo d'uscita, e valida nel caso nonlineare anche per ali a freccia. Si utilizza un opportuno approccio integrale al contorno e si discute il problema dell'unicità per svariate configurazioni alari di interesse nelle applicazioni. [ABSTRACT FROM AUTHOR]

Microstrip lines are analyzed by considering edge-singular behavior using closed-form Green's functions in a spatial domain. A Maxwell function which incorporates the appropriate edge conditions of the line is introduced for the derivations of a transverse correlation function. From calculations of excess lengths of an open-end discontinuity, the results of the proposed method using the edge conditions are in better agreement with the quasistatic results than those of transverse uniform current variations for conductor strips with relatively wider width [ABSTRACT FROM PUBLISHER]