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Analysis of the singular function boundary integral method for a biharmonic problem with one boundary singularity
Author(s) -
Christodoulou E.,
Elliotis M.,
Georgiou G.,
Xenophontos C.
Publication year - 2012
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.20654
Subject(s) - mathematics , biharmonic equation , mathematical analysis , singularity , asymptotic expansion , singular boundary method , singular integral , singular solution , boundary value problem , boundary (topology) , mixed boundary condition , free boundary problem , dirichlet problem , integral equation , boundary element method , finite element method , physics , thermodynamics
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. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011