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

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.

[1]  Z. C. Li,et al.  Special boundary approximation methods for laplace equation problems with boundary singularities - Applications to the motz problem , 2006, Comput. Math. Appl..

[2]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[3]  Juhani Pitkäranta,et al.  Boundary subspaces for the finite element method with Lagrange multipliers , 1979 .

[4]  Susanne C. Brenner,et al.  Multigrid methods for the computation of singular solutions and stress intensity factors I: Corner singularities , 1999, Math. Comput..

[5]  Andreas Poullikkas,et al.  Comparison of two methods for the computation of singular solutions in elliptic problems , 1997 .

[6]  Serge Nicaise,et al.  General Interface Problems-II , 1994 .

[7]  Miltiades Elliotis,et al.  The singular function boundary integral method for a two-dimensional fracture problem , 2006 .

[8]  Zi-Cai Li,et al.  Penalty combinations of the Ritz-Galerkin and finite difference methods for singularity problems , 1997 .

[9]  I. Babuska.  The finite element method with Lagrangian multipliers , 1973 .

[10]  Barna A. Szabó,et al.  NUMERICAL ANALYSIS OF SINGULARITIES IN TWO DIMENSIONS. PART 2: COMPUTATION OF GENERALIZED FLUX/STRESS INTENSITY FACTORS , 1996 .

[11]  Lorraine G. Olson,et al.  A SINGULAR FUNCTION BOUNDARY INTEGRAL METHOD FOR THE LAPLACE EQUATION , 1996 .

[12]  Miltiades Elliotis,et al.  The solution of Laplacian problems over L-shaped domains with a singular function boundary integral method , 2002 .

[13]  Serge Nicaise,et al.  TRANSMISSION PROBLEMS FOR THE LAPLACE AND ELASTICITY OPERATORS: REGULARITY AND BOUNDARY INTEGRAL FORMULATION , 1999 .

[14]  Miltiades Elliotis,et al.  Solution of the planar Newtonian stick–slip problem with the singular function boundary integral method , 2005 .

[15]  S. K. Maiti,et al.  A review of SIF evaluation and modelling of singularities in BEM , 2000 .

[16]  T. Lu,et al.  Singularities and treatments of elliptic boundary value problems , 2000 .

[17]  P. Grisvard.  Singularities in Boundary Value Problems , 1992 .

[18]  A. Yakhot,et al.  Computing flux intensity factors by a boundary method for elliptic equations with singularities , 1998 .

[19]  Hajime Igarashi,et al.  A boundary element method for potential fields with corner singularities , 1996 .

[20]  Rudolf Mathon,et al.  Boundary methods for solving elliptic problems with singularities and interfaces , 1987 .

[21]  Miltiades Elliotis,et al.  Solving Laplacian problems with boundary singularities: a comparison of a singular function boundary integral method with the p/hp version of the finite element method , 2005, Appl. Math. Comput..

[22]  I Babuska,et al.  The p and h-p Versions of the Finite Element Method; State of the Art. , 1986 .

[23]  Barna A. Szabó,et al.  Numerical analysis of singularities in two-dimensions part 1: Computation of eigenpairs , 1995 .

[24]  Zi-Cai Li,et al.  Combined Methods for Elliptic Equations with Singularities, Interfaces and Infinities , 1998 .

[25]  I. Babuska,et al.  Finite Element Analysis , 2021 .