The singular function boundary integral method for 3D Laplacian problems with a boundary straight edge singularity
Date
2012Source
Applied Mathematics and ComputationVolume
219Issue
3Pages
10731081Google Scholar check
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Threedimensional Laplace problems with a boundary straightedge singularity caused by two intersecting flat planes are considered. The solution in the neighbourhood of the straight edge can be expressed as an asymptotic expansion involving the eigenpairs of the analogous twodimensional problem in polar coordinates, which have as coefficients the socalled edge flux intensity functions (EFIFs). The EFIFs are functions of the axial coordinate, the higher derivatives of which appear in an inner infinite series in the expansion. The objective of this work is to extend the singular function boundary integral method (SFBIM), developed for twodimensional elliptic problems with point boundary singularities [G.C. Georgiou, L. Olson, G. Smyrlis, A singular function boundary integral method for the Laplace equation, Commun. Numer. Methods Eng. 12 (1996) 127134] for solving the above problem and directly extracting the EFIFs. Approximating the latter by either piecewise constant or linear polynomials eliminates the inner infinite series in the local expansion and allows the straightforward extension of the SFBIM. As in the case of twodimensional problems, the solution is approximated by the leading terms of the local asymptotic solution expansion. These terms are also used to weight the governing harmonic equation in the Galerkin sense. The resulting discretized equations are reduced to boundary integrals by means of the divergence theorem. The Dirichlet boundary conditions are then weakly enforced by means of Lagrange multipliers. The values of the latter are calculated together with the coefficients of the EFIFs. The SFBIM is applied to a test problem exhibiting fast convergence of order k + 1 (k being the order of the approximation of the EFIFs) in the L 2norm and leading to accurate estimates for the EFIFs. © 2012 Elsevier Inc. All rights reserved.
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