A Singular Function Boundary Integral Method for Laplacian Problems with Boundary Singularities

Abstract

A singular function boundary integral method for Laplacian problems with boundary singularities is analyzed. In this method, the solution is approximated by the truncated asymptotic expansion for the solution near the singular point and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multiplier functions. The resulting discrete problem is posed and solved on the boundary of the domain, away from the point of singularity. The main result of this paper is the proof of convergence of the method; in particular, we show that the method approximates the generalized stress intensity factors, i.e., the coefficients in the asymptotic expansion, at an exponential rate. A numerical example illustrating the convergence of the method is also presented. Introduction.In many engineering problems (e.g., fracture mechanics applications) governed by elliptic partial differential equations (PDEs), boundary singularities arise when there is a sudden change in the boundary conditions and/or on the boundary itself. Singularities are known to affect adversely the accuracy and the convergence of standard numerical methods, such as finite element, boundary element, finite difference, and spectral methods. Grid refinement and high-order discretizations are common strategies aimed at improving the convergence rate and accuracy of the above-mentioned standard methods. If, however, the form of the singularity is taken into account and is properly incorporated into the numerical scheme, then a more effective method may be constructed (see, e.g., [10,15]). (For a recent survey of treatment of singularities in elliptic boundary value problems see [17] and the references therein.)In the case of the two-dimensional Laplace equation, for example, the form of the singularity is visible through the asymptotic expansion for the solution u near the singular point. When the boundaries sharing the singular point are not curved, u is given by [12]

The Singular Function Boundary Integral Method for the 2-D and 3-D Laplace Equation Problems in Mechanics, with a Boundary Singularity

“…The method has been tested on standard elliptic problems exhibiting exponential convergence and high accuracy with respect to the number of singular functions. This behavior of the method has been theoretically proved in [28]. The results obtained in the 2-D case have encouraged the extension of the method in 3-D problems with a straight-edge singularity [6,13].…”

“…As in previous implementations of the method [9][10][11][12][13][14], one may observe that the values of singular coefficients converge rapidly with N λ . In fact, in [28] a theoretical analysis of the method proved algebraic convergence in N λ . Also, very accurate estimates are obtained.…”

“…This behavior will be explained theoretically below. According to reference [28], if λ ∈ H k (∂Ω AB ), for some k≥1 and λ h is the approximation of the Lagrange multiplier function with h being always the mesh-width, then there exist positive constants C and γ∈(0,1), independent of N α and h such that…”

On solving the cracked‐beam problem by block method

“…where From an engineering standpoint, one of the interesting quantities is the stress intensity factor which is the coefficient 1 of the leading term in the asymptotic expression (1), (6) for the solution u of the boundary value problem (5). On the basis of (26) for the approximation of the coefficient 1 , we obtain…”

“…An exhaustive survey of treatment of singularities is provided in the recent papers by Li and Lu [4], Dosiyev [5], Xenophontos et al [6] and the references therein.…”