Finite difference solution to the Poisson equation at an intersection of interfaces
Date
2004
Authors
Jarvis, D.
Noye, B.
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Journal Title
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Journal article
Citation
Australia and New Zealand Industrial and Applied Mathematics (ANZIAM) Journal, 2004; 45(E):C632-C645
Statement of Responsibility
D. A. Jarvis and B. J. Noye
Conference Name
Abstract
We consider the solution u to Poisson’s equation L(pu) = f on a polygonal domain Ω ∈ 2 R ², which itself is composed of polygonal subdomains Ωi, where L is the Laplacian operator and the coefficient p is piecewise constant, with value pi in region Ωi. At a point S of intersection of the interfaces between Ωi and adjacent regions the solution may have singular components. These, if present, may be severe and will degrade the convergence of the basic methods of numerical approximation to the solution u in the locality of S. Elaborate methods are required to accurately estimate the singular components, or stress intensity factors, or to improve the accuracy of the numerical solution near S. When the interfaces are straight lines on a Cartesian grid, with homogeneous interface conditions, we show that a remarkable pattern of symmetries of the singular components leads to a simple finite difference solution at the point of intersection S, and to an estimate of the stress intensity factors enabling extraction of the singular components and improved accuracy at points close to S.
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Paper presented at the 11th Biennial Computational Techniques and Applications Conference (CTAC2003) held at ICIAM, Sydney 2003
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© Austral. Mathematical Soc. 2004