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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)

MSC:
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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