The solution of Laplacian problems over L‐shaped domains with a singular function boundary integral method

Abstract

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.

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

“…Table 1 presents the convergence in the values of the four leading coefficients, with respect to the number of Lagrange multipliers N λ and for N a =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 .…”

“…The implementation of the SFBIM to both a 2-D and a 3-D Laplacian model problems, yielded highly accurate results for the singular coefficients and the EFIFs and exhibited fast convergence, as in other two-dimensional applications [9][10][11][12] of the method. For the planar problem, the numerical results are favorably compared with the analytic solution.…”

“…Now, in order to obtain the optimum combination of N a and N λ , several series of runs have been made. Previous applications of the method in 2-D problems [9][10][11][12][13][14], indicated that N λ should be large enough. This means that we must have an adequate number of elements on the boundary which will also help in having a better accuracy in numerical integration.…”

“…In the SFBIM the solution of the problem is approximated by the leading terms of the local solution expansion given by (9). Thus, in (9) we employ the first N a terms and we substitute the EFIFs with their polynomial expression given by (10).…”

“…Following the notation used in previous applications of the SFBIM (e.g. [9][10][11][12][13][14]), the above expansion is written as follows:…”