No abstract

SUMMARYA singular function boundary integral method (SFBIM) is proposed for solving biharmonic problems with boundary singularities. The method is applied to the Newtonian stick-slip ow problem. The streamfunction is approximated by the leading terms of the local asymptotic solution expansion which are also used to weight the governing biharmonic equation in the Galerkin sense. By means of the divergence theorem the discretized equations are reduced to boundary integrals. The Dirichlet boundary conditions are weakly enforced by means of Lagrange multipliers, the values of which are calculated together with the singular coe cients. The method converges very fast with the number of singular functions and the number of Lagrange multipliers, and accurate estimates of the leading singular coe cients are obtained. Comparisons with the analytical solution and results obtained with other numerical methods are also made.

SUMMARYThe singular function boundary integral method is applied for the solution of a Laplace equation problem over an L-shaped domain. The solution is approximated by the leading terms of the local asymptotic solution expansion, while the Dirichlet boundary conditions are weakly enforced by means of Lagrange multipliers. Estimates of great accuracy are obtained for the leading singular coe cients, as well as for the Lagrange multipliers. Comparisons are made with recent numerical results in the literature.

No 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]