Finite difference solution to the Poisson equation at an intersection of interfaces

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.