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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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