The Singular Function Boundary Integral Method (SFBIM) for solving two-dimensional elliptic problems with boundary singularities is revisited. In this method the solution is approximated by the leading terms of the asymptotic expansion of the local solution, which are also used to weight the governing partial differential equation. The singular coefficients, i.e., the coefficients of the local asymptotic expansion, are thus primary unknowns. By means of the divergence theorem, the discretized equations are reduced to boundary integrals and integration is needed only far from the singularity. The Dirichlet boundary conditions are then weakly enforced by means of Lagrange multipliers, the discrete values of which are additional unknowns. In the case of two-dimensional Laplacian problems, the SFBIM converges exponentially with respect to the numbers of singular functions and Lagrange multipliers. In the present work the method is applied to Laplacian test problems over circular sectors, the analytical solution of which is known. The convergence of the method is studied for various values of the order p of the polynomial approximation of the Lagrange multipliers (i.e., constant, linear, quadratic, and cubic), and the exact approximation errors are calculated. These are compared to the theoretical results provided in the literature and their agreement is demonstrated. In the last few decades there has been an extensive study of planar elliptic boundary value problems with boundary singularities. The methods that have been proposed for the solution of such problems range from special mesh-refinement schemes to sophisticated techniques that incorporate, directly or indirectly, the form of the local asymptotic expansion, which is known in many occasions. These methods aim to improve the accuracy and resolve the convergence difficulties that are known to appear in the neighborhood of singular points. The local solution, centered at the singular point, in polar coordinates (r,h) is of the general form
['Evgenia Christodoulou', 'Christos Xenophontos', 'Georgios C. Georgiou']
The Singular Function Boundary Integral Method for singular Laplacian problems over circular sections