Analysis of the singular function boundary integral method for a biharmonic problem with one boundary singularity

Abstract

In this article, we analyze the singular function boundary integral method (SFBIM) for a two-dimensional biharmonic problem with one boundary singularity, as a model for the Newtonian stick-slip flow problem. In the SFBIM, the leading terms of the local asymptotic solution expansion near the singular point are used to approximate the solution, and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multiplier functions. By means of Green's theorem, the resulting discretized equations are posed and solved on the boundary of the domain, away from the point where the singularity arises. We analyze the convergence of the method and prove that the coefficients in the local asymptotic expansion, also referred to as stress intensity factors, are approximated at an exponential rate as the number of the employed expansion terms is increased. Our theoretical results are illustrated through a numerical experiment.

Untitled

“…It has been developed by G. Georgiou and co-workers and has been used in many studies to tackle planar harmonic and biharmonic equation problems and three-dimensional Laplace equation problems in the fields of theoretical and fracture Mechanics, fluid flow, etc. (Christodoulou et al 2012a, b, Elliotis et al 2010, Elliotis et al 2002, Elliotis et al 2005a, b, Elliotis et al 2006, Elliotis et al 2007, Elliotis et al 2014, Elliotis 2016, Elliotis 2019, Georgiou et al 1996. It exhibits exponential convergence, a feature which was observed in all previous applications of the method.…”