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The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. (English) Zbl 0811.65083
The authors develop finite difference methods for elliptic equations of the form \[ \nabla\cdot (\beta(x)\nabla u(x))+ k(x) u(x)= f(x) \] in a region \(\Omega\in \mathbb{R}^ n\), \(n= 1,2\), where \(\Omega\) is a segment for \(n=1\) and a rectangle for \(n=2\). It is supposed that a uniform rectangular grid is used. Also it is supposed that the functions \(\beta(x)\), \(k(x)\), \(f(x)\) are piecewise smooth with discontinuous points situated on an irregular surface \(\Gamma\) of codimension \(n-1\) contained in \(\Omega\), and along which \(f(x)\) may have a delta function singularity.
It is shown that modifying the standard centered difference schemes when \(\Gamma\) is not aligned with the grid it is possible to maintain second- order accuracy in grid points along \(\Gamma\). For \(n=1\) the modification of the stencil in such points still consists of three points, and for \(n=2\) the stencil consists of six points. At the end three numerical examples are supplied, when the known solution of the differential problem is piecewise infinitely differentiable.
Reviewer: V.Makarov (Kiev)

65N06 Finite difference methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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